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Classification of 2F-modules, II

Robert M. Guralnick and Gunter Malle^{∗}

To John Thompson, on the occasion of his 70th birthday

Abstract. We classify the irreducible 2F-modules for the quasi-simple groups of Lie type with the modules in the natural characteristic (leaving three open cases). In concert with the results of our first paper, this gives the complete classification of such modules that arise in the context of the Aschbacher-Smith classification of quasithin groups. As a direct application we also obtain the list of irreducible F-modules of finite quasi-simple groups.

2000 Mathematics Subject Classification: 20G05, 20D06

1. Introduction and statement of results

In this paper we study the following situation: Let G be a finite group such that F^{∗}(G) is quasi-simple. For an absolutely irreducible faithful KG-modules V over a field K of characteristic ℓ and a subgroup A ≤ G of G we let

f(A) := f^{V} (A) := |A|^{2} · |C_{V} (A)|.

We want to classify those modules V for which G has an elementary abelian ℓsubgroup A satisfying

|V | ≤ f(A) .  (1.1)

Modules satisfying this and similar conditions have been studied for quite a while – going back to Thompson, Glauberman, Mason and others (see the remarks and references in [6]). Usually, there are conditions imposed in addition to (1.1) involving an assumption of quadratic or cubic action of the subgroup A on V.

These modules play a crucial role in a new approach to the revision of the classification of finite simple groups. In particular, for ℓ = 2, the results of this paper and [6] are used by Aschbacher and Smith in the proof of the classification of quasithin simple groups [2]. The Aschbacher–Smith result is the final paper in the original proof of the classification of finite simple groups and is also part of the second generation proof. These modules and variations also come up in Meierfrankenfeld’s program to study finite simple groups.

∗The first author was partially supported by NSF grant DMS 0140578

In [6], we essentially found all examples of such modules with F^{∗}(G) quasisimple aside from the case that F^{∗}(G) is a group of Lie type and V is a module in the natural characteristic for G. In this paper, we consider this case and classify all but three cases. In particular, we classify all such modules that arise in [2]. We expect to complete the classification of the remaining situations in a future article. We feel that it is important to have the results needed for the quasithin classification in the literature in a timely manner.

Theorem A. The absolutely irreducible 2F-modules for quasi-simple groups of Lie type in their defining characteristic are among those given in Table 1.

We also shows that all modules in Table 1 except possibly those in the last three rows actually satisfy the 2F-condition.

Table 1: 2F-modules in defining characteristic

G  d  V  f  conditions
SL_{n}(ℓ^{a})  n  Y_{n}  a
SL_{n}(ℓ^{a})  \binom{n}{2}  ∧^{2}(Y_{n})  a  n ≥ 4
SL_{n}(ℓ^{a})  \binom{n+1}{2}  Σ^{2}(Y_{n})  a  ℓ ≥ 3
SL_{n}(ℓ^{2a})  n^{2}  Y_{n} ⊗ Y_{n}^{(a)}  a
SL_{6}(ℓ^{a})  20  ∧^{3}(Y_{6})  a
SU_{n}(ℓ^{a})  n  Y_{n}  2a
Sp_{2n}(ℓ^{a})  2n  Y_{2n}  a
Sp_{2n}(ℓ^{a})  \binom{2n}{2} − 1 − δ_{ℓ,n}  ˜∧^{2}(Y_{2n})  a  n = 2, 3

or ℓ = 2, n = 4

Sp_{4}(ℓ^{2a})  16  Y_{4} ⊗ Y_{4}^{(a)}  a
Spin_{n}^{(±)}(ℓ^{a})  n  Y_{n}  a  n ≥ 6
Spin_{2n+1}(ℓ^{a})  2^{n}  Z_{n}  a  n = 3, 4, 5
Spin_{2n}^{+}(ℓ^{a})  2^{n−1}  Z_{n−1}  a  n = 4, 5, 6
Spin_{2n}^{−}(ℓ^{a})  2^{n−1}  Z_{n−1}  2a  n = 4, 5
2B_{2}(2^{2a+1})  4  M(λ_{1}), M(λ_{2}) 2a + 1  a > 1
G_{2}(ℓ^{a})  7 − δ_{ℓ,2}  M(λ_{2})  a
F_{4}(2^{a})  26  M(λ_{1}), M(λ_{4})  a
E_{6}(ℓ^{a})_{sc}  27  M(λ_{1}), M(λ_{6})  a
F_{4}(ℓ^{a})  26 − δ_{ℓ,3}  M(λ_{4})  a  ℓ > 2
2E_{6}(ℓ^{a})_{sc}  27  M(λ_{1}), M(λ_{6})  2a
E_{7}(ℓ^{a})_{sc}  56  M(λ_{7})  a

In this table, the modules are labelled by their highest weights, and the fundamental weights λ_{i} are labelled as in Table 4, §3.1. Furthermore, Y_{n} denotes the natural module (of dimension n), Z_{n} is the 2^{n}-dimensional spin module of the spin group Spin_{2n+1}(q) with highest weight λ_{1} (this includes the case of symplectic groups in even characteristic), and Z_{n−1} denotes either one of the two 2^{n−1}-dimensional half-spin modules of spin groups Spin_{2n}^{(±)}(q) with highest weight λ_{1} or λ_{2}. Next, ∧^{2}, ∧^{3}, and Σ^{2} denote the alternating square, the alternating cube, and the symmetric square, respectively, and M^{(a)} is the ℓ^{a}-Frobenius twist of the module M. Also, recall that for a given type X of finite groups of Lie type there are several isogenous variants which share the same non-abelian composition factor and the same order but have slightly different structures. In particular, the adjoint group X_{ad} is almost simple, whereas the simply-connected type group X_{sc} is quasi-simple (for example, ^{2}E_{6}(q)_{sc} is quasi-simple with centre of order (3, q + 1)).

The three open cases are E_{7}(q)_{sc}, ^{2}E_{6}(q)_{sc} and F_{4}(q) on their smallest modules (the last only for odd characteristic). Note that in the first case f(A) = q^{55} = |V |/q for A an elementary abelian subgroup of order q^{27}. There are two other open cases remaining from the first part — the two largest Conway groups on their smallest modules in characteristic 2. Again, in those two cases there are elementary abelian subgroups A with f(A) = |V |/2.

Our approach is to classify the modules satisfying (1.1) and not impose other conditions (of course, one obtains corollaries to our main results by then imposing extra conditions).

Given the results in [6], it thus remains to consider groups of Lie type G in the same characteristic ℓ. By a basic lemma, we know log_{ℓ}(|G|) grows quadratically with the rank. The F G-modules of this dimension are known. We quote a recent paper of Lübeck [11] for this (which depends upon a result of Premet [15]). This follows from the beautiful theory of representations of Chevalley groups in the natural characteristic (see for example [5, Sec. 2.8] for background on this). Liebeck [10] has obtained some weaker bounds than those given in [11], but these results are not sufficient for our purposes.

We should point out that the background material that we use is a bit different than that of Aschbacher–Smith [2] or Gorenstein–Lyons–Solomon [5].

In the first part [6], we assumed the results from the modular Atlas [8] about lower bounds for the sporadic groups, of James about representations of alternating groups and of Landazuri–Seitz–Zalesskii for cross characteristic representations. As we remarked above, in the equicharacteristic case, we are using the results of Premet and Lübeck. One could produce independent proofs of these results for the cases of interest in [2] to make that result more self-contained. For the exceptional groups we also use results of Lawther [9] on Jordan blocks of exceptional groups on certain small modules and some information on the order relation on unipotent classes, see Spaltenstein [16]. This does not concern any of the modules considered in [2].

In the present paper we treat the absolutely irreducible representations of finite quasi-simple groups of Lie type G in their defining characteristic ℓ. In fact, we will not consider here the non-simple groups Sp_{4}(2) and G_{2}(2) in characteristic 2, and ^{2}G_{2}(3) in characteristic 3, since these were considered as the automorphism groups 𝔖_{6}, U_{3}(3).2 and L_{2}(8).3 of simple groups in non-defining characteristic in the first part [6, Props. 3.1, 4.3, 4.1]. On the other hand, we do treat here the following pairs (G, ℓ) left open in the first part:

(𝔄_{5} ≅ SL_{2}(4), 2),  (2.𝔄_{5} ≅ SL_{2}(5), 5), (2.𝔄_{6} ≅ SL_{2}(9), 3),
(𝔄_{8} ≅ SL_{4}(2), 2),  (L_{2}(7) ≅ SL_{3}(2), 2), (2.L_{3}(2) ≅ SL_{2}(7), 7)
(2.U_{4}(2) ≅ Sp_{4}(3), 3), (S_{4}(3) ≅ SU_{4}(2), 2), (G_{2}(2)′ ≅ SU_{3}(3), 3),

as well as the ‘sporadic group’ (^{2}F_{4}(2)′, 2). Note that exceptional Schur multipliers of groups of Lie type are always of order a power of the defining characteristic, so we need not worry about them and just have to consider quotients of simply connected groups of Lie type by central subgroups. By a slight abuse of notation, in the sequel we’ll weaken the faithfulness-requirement a little and say that V is a 2F-module for G if it is so for some quotient by a central subgroup contained in G′. The explicit description of representations in the defining characteristic via their highest weight makes it easy to work out the kernel of such a representation in any individual case.

Since F-modules are, by definition, also 2F-modules we also obtain a classification of F-modules of finite quasi-simple groups as a rather easy application of our results on 2F-modules:

Theorem B. The absolutely irreducible F-modules of finite quasi-simple groups are given in Table 2.

Table 2: F-modules for quasi-simple groups

G  d  V ℓ f conditions
SL_{n}(ℓ^{a})  n  Y_{n} ℓ a
SL_{n}(ℓ^{a})  \binom{n}{2} ∧^{2}(Y_{n}) ℓ a  n ≥ 4
SU_{n}(ℓ^{a})  n  Y_{n} ℓ 2a  n ≥ 4
Sp_{2n}(ℓ^{a})  2n  Y_{2n} ℓ a
Spin_{n}^{(±)}(ℓ^{a})  n  Y_{n} ℓ a  n ≥ 6
Spin_{7}(ℓ^{a})  8  Z_{3} ℓ a
Spin_{2n}^{+}(ℓ^{a}) 2^{n−1}  Z_{n−1} ℓ a  n = 4, 5
G_{2}(2^{a})  6  M(λ_{2}) ℓ a
𝔄_{2n}  2n − 2  2 1
3.𝔄_{6}  3  2 2
𝔄_{7}  4  2 1

This result is already in the literature. The earlier papers used very heavily the fact that by the Thompson replacement lemma, one can assume that the subgroup A acts quadratically. In particular, see Aschbacher [1], Meierfrankenfeld-Stroth [12], [13] for alternating, sporadic and odd characteristic Lie-type groups in characteristic 2, and Meixner [14] for odd characteristic Lie-type groups in odd characteristic. Our proof is independent of these references.

We now give an outline of the contents of the paper. In order to study modules in defining characteristic and determine precisely which of them satisfy (1.1), very frequently we need to deal with tensor products, symmetric squares and exterior powers of smaller representations. Upper bounds for centralizers on such modules are derived in Section 2.2. In Section 3.1 we recall some basic results about representations in defining characteristic. In 3.3 we determine the maximal size of a subgroup of a group of Lie type consisting only of long root elements, and in 3.4 we collect information on the ℓ-ranks of classical groups.

In Sections 4–8 we treat the families of linear and unitary groups, of symplectic groups, of orthogonal groups, and of exceptional groups of Lie type, with Section 5 containing some results on the particularly difficult case of spin representations.

Note that, in contrast to the situation in the first part [6], we don’t reach a complete classification of 2F-modules for non-perfect almost quasi-simple groups. Still, we prove the following partial result: if F^{∗}(G) is quasi-simple but not an orthogonal group in even dimension, then all composition factors of the restriction to F^{∗}(G) of a 2F-module for G are again 2F-modules. We hope to come back to the case of the orthogonal groups and more generally to the complete determination of 2F-modules for almost quasi-simple groups in a future paper.

2. Auxiliary results

2.1. 2F-modules.

Let K be a finite field of characteristic ℓ, G a finite group and V a finite-dimensional KG-module. For x ∈ G we call the codimension of C_{V} (x) the rank of x (on V) (i.e., the rank of x − 1). For a subgroup A of G write

f(A) := f^{V} (A) := |A|^{2} · |C_{V} (A)|.

V is called a 2F-module for G with offender A if dim(V) > 1 and |V | ≤ f(A). Similarly we define

f_{1}(A) := f_{1}^{V} (A) := |A| · |C_{V} (A)|

and say that V is an F-module for G with offender A if dim(V) > 1 and |V | ≤ f_{1}(A). We also say that V is an F^{+}-module for G if f_{1}(A) > |V | for some A ≤ G.

The following well-known observation will be useful (see [3], [6, Lemma 2.1]):

Lemma 2.1. Let A, B ≤ G with f(B) maximal among all subgroups of G and let C := ⟨A, B⟩. Then f(C) ≥ f(A).

In particular, if V is a 2F-module for G with offender A ≠ 1, and G is the normal closure of A then f(A) ≤ f(G), Moreover, if V is irreducible then f(A) ≤ |G|^{2}. Also it follows that:

Lemma 2.2. Suppose C ≤ G is an elementary abelian ℓ-subgroup containing A and B, with f(A) = f(B) maximal among all subgroups of C. Then f(AB) = f(A).

This shows that when searching for examples satisfying (1.1) we may sometimes assume that A is maximal elementary abelian.

The following can be used to obtain upper bounds on the centralizer of an ℓ-subgroup:

Lemma 2.3. Let B ≤ A ≤ G with dim C_{V} (A) = dim C_{V} (B). Then

C_{V} (⟨A^{g} | g ∈ C_{G}(B)⟩) = C_{V} (B)

Proof. Let g ∈ C_{G}(B). Then C_{V} (A) = C_{V} (B) = C_{V} (B^{g}) = C_{V} (A^{g}). ⊓ ⊔

We will often use this in the following form. Let B be an elementary abelian ℓ-subgroup, g an ℓ-element in C_{G}(B) \ B and A := ⟨B, g⟩. If ⟨A^{h} | h ∈ C_{G}(B)⟩ has smaller centralizer in V than B, then A has smaller centralizer than B. For example, if C_{G}(B) is an extension of B by a simple group H, then all classes of ℓ-elements in C_{G}(B) outside B generate H. But H contains ℓ′-elements and their centralizer can be read off from the Brauer character of V.

The following result is very useful for inductive arguments along parabolic subgroups (see [6, Prop. 2.4]):

Proposition 2.4. Let V be a 2F-module for G. Let H ≤ G such that H contains an offender. The either there exists an offender A ≤ U := O_{ℓ}(H), or C_{V} (U) is a 2F-module for H/U.

The next result is used for treating automorphism groups (see [6, Prop. 2.7]):

Proposition 2.5. Let G be a finite group with a normal subgroup H ≤ G of index ℓ. Let W be an absolutely irreducible 𝔽_{ℓ^{f}} H-module of dimension dim W ≥ 2 whose induction V = Ind_{H}^{G}W to G is again irreducible. If V is a 2F-module for G with offender A, then one of the following holds:

(i) W is a 2F-module for H with offender B := A ∩ H;

(ii) ℓ = f = 2, dim C_{W} (B) = 1, dim W ≤ r where |A| = 2^{r};

(iii) |A| = ℓ = 2, dim W = 2, f ≤ 2.

Note that mutatis mutandis all the results above remain true for F-modules and f_{1}(A).

The following result will be applied to parabolic subgroups.

Lemma 2.6. Let P = UL be a semidirect product with U a normal elementary abelian ℓ-subgroup of maximal size in P, with complement L. Then U is not an F^{+}-module for L. Moreover, the following are equivalent:

(i) U is not an F-module for L;

(ii) U is the unique elementary abelian subgroup of P with order |U|.

Proof. If A ≤ L is an elementary abelian unipotent subgroup, then |A||C_{U}(A)| ≤ |U| (because U is the maximal order of an elementary abelian unipotent subgroup), whence U is not an F^{+}-module.

The same argument shows that if U is an F-module, then AC_{U}(A) has order |U| for any offender A (offender in the sense of F-modules).

Thus, U an F-module implies that U is not unique in P. Conversely, if W is another elementary abelian subgroup of P of order |U|, then take A = W U/U ≠ 1 and B = A ∩ U ≤ C_{U}(W). So |A||C_{U}(A)| = |U| and A is an offender. ⊓ ⊔

2.2. Centralizers on tensor products.

It will become obvious from the description of the irreducible modules of groups of Lie type in defining characteristic in Section 3.1 that in order to identify possible 2F-modules we need some information on centralizer spaces of unipotent subgroups on tensor products and alternating and symmetric powers.

We start by considering indecomposable representations of cyclic groups C_{ℓ} of prime order ℓ over a field K of characteristic ℓ. Here the following result of Hughes and Kemper [7] will be useful to estimate dimensions of centralizer spaces on symmetric and alternating squares:

Lemma 2.7. Let G = C_{ℓ} be the cyclic group of order ℓ and V an indecomposable KG-module of dimension n (so n ≤ ℓ). Then

dim(∧^{k}(V))^{G} = \frac{1}{2ℓ} ∑_{ζ∈M_{2ℓ}}(1 + ζ) ∑_{0≤i_{1}<...<i_{k}≤n} ζ^{kn−2i_{1}−...−2i_{k}},

where the first sum runs over the set M_{2ℓ} of all 2ℓth complex roots of unity.

This follows for example from [7, Th. 2.5 and Prop. 3.2].

Lemma 2.8. Let G = C_{ℓ} be the cyclic group of order ℓ and V, W be indecomposable KG-modules of dimension n, m respectively. Then

(a) dim(V ⊗ W)^{G} = min{n, m},

(b) dim Σ^{2}(V)^{G} = ⌊\frac{n+1}{2}⌋ when ℓ > 2,

(c) dim ∧^{2}(V)^{G} = ⌊\frac{n}{2}⌋.

Proof. By assumption both V, W are uniserial. Thus the centralizer of G on V ⊗W has dimension

dim(V ⊗ W)^{G} = dim Hom_{G}(1, V ⊗ W) = dim Hom_{G}(V^{∗}, W) = min{n, m}.

The third part follows easily by evaluation of the formula in Lemma 2.7. If ℓ > 2 then V ⊗V is the direct sum of ∧^{2}(V) and Σ^{2}(V), so the second part is immediate from (a) and (c). ⊓ ⊔

Let’s next consider arbitrary modules for the cyclic group.

Lemma 2.9. Let G = C_{ℓ} and V, W be KG-modules of dimension n, m respectively, with z := dim V^{G}, y := dim W^{G}. Then we have

(a)  dim(V ⊗ W)^{G} ≤ min{ny, zm, (n − z)(m − y) + zy},

(b)  dim Σ^{2}(V)^{G} ≤ min \{\frac{z(n + 1)}{2}, \binom{n − z}{2} + \binom{z + 1}{2}\} for ℓ > 2,

(c)  dim ∧^{2}(V)^{G} ≤ min \{\frac{zn}{2}, \binom{n − z + 1}{2} + \binom{z}{2}\} .

Proof. Part (a) has been proved in Lemma 2.8 when V, W are indecomposable. Now assume for example that V = V_{1} ⊕ V_{2} as KG-module, so that

V ⊗ W = V_{1} ⊗ W ⊕ V_{2} ⊗ W .

Since both taking fixed points as well as our stipulated bounds are clearly additive in both components, we get the desired result by induction.

In part (b), our formula holds again for indecomposable V by Lemma 2.8. If V = V_{1} ⊕ V_{2} then

Σ^{2}(V) ≅ Σ^{2}(V_{1}) ⊕ V_{1} ⊗ V_{2} ⊕ Σ^{2}(V_{2}).

Writing n_{i} := dim V_{i}, z_{i} := dim V_{i}^{G}, this shows that the centralizer space has dimension at most equal to

z_{1}(n_{1} + 1)/2 + (z_{1}n_{2} + z_{2}n_{1})/2 + z_{2}(n_{2} + 1)/2 = (z_{1} + z_{2})(n_{1} + n_{2} + 1)/2

by induction respectively by part (a), and similarly for the second bound. In case (c), assume that V = V_{1} ⊕ V_{2}. Then we have

∧^{2}(V) ≅ ∧^{2}(V_{1}) ⊕ V_{1} ⊗ V_{2} ⊕ ∧^{2}(V_{2}).

In the notation introduced in the previous part this gives a centralizer space of dimension at most

z_{1}n_{1}/2 + (z_{1}n_{2} + z_{2}n_{1})/2 + z_{2}n_{2}/2 = (z_{1} + z_{2})(n_{1} + n_{2})/2.

The second bound is proved analogously by using the second type of inequality from part (a). ⊓ ⊔

We are now ready to deal with arbitrary elementary abelian ℓ-groups G. Assume that G is generated by g_{1}, …, g_{r} say. Let V be a KG-module. Since G is abelian, g_{i} stabilizes C_{V} (g_{j}) for all j, and this allows to apply induction on the number r of generators.

Lemma 2.10. Let G = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian ℓ-group and V, W finite dimensional KG-modules. Define V_{0} := V, V_{i} := V_{i−1}^{⟨g_{i}⟩} (1 ≤ i ≤ r), n_{i} := dim V_{i} (0 ≤ i ≤ r), and similarly W_{i} of dimension m_{i}. Then

(a)

dim(V ⊗W)^{G} ≤ ∑_{i=1}^{r} min{(n_{i−1}−n_{i})(m_{i−1}−m_{i}), (n_{i−1}−n_{i})m_{i}, n_{i}(m_{i−1}−m_{i})}+n_{r}m_{r},

(b)

dim Σ^{2}(V)^{G} ≤ ∑_{i=1}^{r} ⌊\frac{(n_{i−1} − n_{i})}{2} min\{n_{i}, n_{i−1} − n_{i} − 1\}⌋ + \binom{n_{r} + 1}{2} for ℓ > 2,

(c)  dim ∧^{2}(V)^{G} ≤ ∑_{i=1}^{r} ⌊\frac{(n_{i−1} − n_{i} + 1)}{2} min\{n_{i}, n_{i−1} − n_{i}\}⌋ + \binom{n_{r}}{2}.

Proof. The proof will be by induction on r. The case r = 1 is Lemma 2.9. First consider part (a). By induction, for G′ := ⟨g_{1}, …, g_{r−1}⟩ we have

dim(V ⊗ W)^{G′} ≤ ∑_{i=1}^{r−1}(n_{i−1} − n_{i})(m_{i−1} − m_{i}) + n_{r−1}m_{r−1}.

On the other hand, clearly V^{G′} ⊗W^{G′} ≤ (V ⊗W)^{G′}, and this subspace is stabilized by g_{r}. Thus we obtain an upper bound for dim(V ⊗ W)^{G} as

dim(V ⊗ W)^{G} ≤ ∑_{i=1}^{r−1}(n_{i−1} − n_{i})(m_{i−1} − m_{i}) + dim(V^{G′} ⊗ W^{G′})^{⟨g_{r}⟩}

≤ ∑_{i=1}^{r−1}(n_{i−1} − n_{i})(m_{i−1} − m_{i}) + (n_{r−1} − n_{r})(m_{r−1} − m_{r}) + n_{r}m_{r}

by the case r = 1 in Lemma 2.9, as claimed.

In (b) and (c) we may argue in precisely the same way, using that Σ^{2}(V^{G}) ≤ Σ^{2}(V)^{G} and ∧^{2}(V^{G}) ≤ ∧^{2}(V)^{G}. ⊓ ⊔

The previous result allows to conclude that Lemma 2.9 in fact holds for arbitrary elementary abelian ℓ-groups.

Corollary 2.11. Let G be an elementary abelian ℓ-group, V, W two KG-modules of dimension n, m, with centralizer spaces of dimension z := dim V^{G}, y := dim W^{G} respectively. Then

(a)  dim(V ⊗ W)^{G} ≤ min{ny, zm, (n − z)(m − y) + zy},

(b)  dim Σ^{2}(V)^{G} ≤ min\{\frac{z(n + 1)}{2}, \binom{n − z}{2} + \binom{z + 1}{2}\} for ℓ > 2,

(c)  dim ∧^{2}(V)^{G} ≤ min\{\frac{zn}{2}, \binom{n − z + 1}{2} + \binom{z}{2}\}.

Proof. We have

dim(V ⊗ W)^{G} = dim Hom_{G}(1, V ⊗ W) = dim Hom_{G}(V^{∗}, W).

Now any G-homomorphism from V^{∗} to W is determined by the image of the head of V^{∗}, isomorphic to the socle V^{G} of V. So dim Hom_{G}(V^{∗}, W) ≤ dim V^{G} dim W = zm. The second bound in (a) follows by symmetry, the third with Lemma 2.10. In the notation there we have n = n_{0}, m = m_{0}, z = n_{r}, y = m_{r}, and

dim(V ⊗ W)^{G} ≤ ∑_{i=1}^{r}(n_{i−1} − n_{i})(m_{i−1} − m_{i}) + n_{r}m_{r}

≤ ∑_{i=1}^{r}(n_{0} − n_{r})(m_{i−1} − m_{i}) + n_{r}m_{r} = (n − z)(m − y) + zy.

Parts (b) and (c) follow in precisely the same way from Lemma 2.10(b) and (c). ⊓ ⊔

For convenient reference, let’s reformulate this result in terms of the codimension c := n − z of V^{G} (resp. d := m − y of W^{G}):

codim(V ⊗ W)^{G} ≥ max{nd, mc, (n − c)d + (m − d)c},

codim Σ^{2}(V)^{G} ≥ max\{\frac{c}{2}(n + 1), c(n − c + 1)\} for ℓ > 2,

codim ∧^{2}(V)^{G} ≥ max\{\frac{n}{2}(c − 1), c(n − c − 1)\}.

(2.1)

It would be possible with a little bit more work to obtain similar results as above for centralizers on Σ^{3}(V) and ∧^{3}(V). But in the present work we need such results only for ∧^{3} of at most 10-dimensional modules, where ad-hoc arguments can be used.

For the convenience of the reader we have collected the ranks, (i.e., the codimension of the centralizer, as defined in Section 2.1) of some unipotent elements x with simple Jordan types, on exterior square and cube of low dimensional vector spaces V in Table 3. The first column describes the Jordan type J(x) of x on V, the next six columns give the rank on ∧^{2}(V) for 3 ≤ dim V ≤ 8, and the last seven columns give lower bounds for the rank on ∧^{3}(V), with 4 ≤ dim V ≤ 10.

Table 3: Ranks on exterior squares and cubes

dim V 3 4 5 6 7 8 4 5 6 7 8 9 10
J(x) ∧^{2}(V) ∧^{3}(V)
21^{n−2} 1 2 3 4 5 6 1 3 6 10 15 21 28
2^{2}1^{n−4} − 2 4 6 8 10 2 4 8 14 22 32 44
2^{3}1^{n−6} − − − 6 9 12 − − 10 16 25 37 52
31^{n−3} 2 4 6 8 10 12 2 6 12 20 30 42 56

The following result gives the centralizer dimension for elements interchanging the factors of a tensor product:

Lemma 2.12. Let G be a group, V a finite dimensional KG-module and σ ∈ Aut(G). Let V^{σ} := {vσ | v ∈ V } denote a copy of V with KG-module structure defined by vσ.g^{σ} := (v.g)σ for g ∈ G, so V ⊗ V^{σ} becomes a G⟨σ⟩-module in the obvious way. Then

dim(V ⊗ V^{σ})^{⟨σ⟩} = \binom{dim V + 1}{2} .

Proof. Choose a basis B of V. Then σ acts as a permutation on the basis {v⊗wσ | v, w ∈ B\} of V ⊗ V^{σ}, with precisely \binom{dim V +1}{2} orbits. ⊓ ⊔

3. Groups of Lie type in defining characteristic

In this section, we first recall the description of the representations of groups of Lie type in their defining characteristic. In the second part we use orbit closure arguments to show that most of the time long root elements have largest centralizer on a given module. We next determine the maximal size of subgroups consisting of long root elements only. In the fourth part we prove results on centralizers of subspaces in the natural representation of classical groups. Finally, we outline the approach to the classification of 2F-modules in the individual cases.

3.1. Representations in defining characteristic.

We start by recalling the well-known description of absolutely irreducible representations of groups of Lie type in their defining characteristic, due to Steinberg, as expounded for example in [5, 2.8]. Let first Ḡ be a simple simply-connected algebraic group over the algebraic closure K of a finite field 𝔽_{ℓ^{a}} and Λ its weight lattice. To any dominant weight λ ∈ Λ there is associated an irreducible highest weight module M(λ) for Ḡ. The modules M(λ), for λ running over the dominant weights, form a complete set of non-isomorphic irreducible Ḡ-modules over K (see [5, 1.14.6]). Now let F : Ḡ → Ḡ be a Frobenius morphism defining a rational structure over 𝔽_{ℓ^{a}} with corresponding finite group of fixed points G := Ḡ^{F}. Then M(λ) remains irreducible upon restriction to G if λ is a restricted weight for G, that is, all coefficients of λ when expressed as a linear combination of fundamental weights are less than ℓ^{a}. Moreover, the (restrictions of the) M(λ), where λ is restricted, constitute a complete set of non-isomorphic irreducible G-modules over K [5, 2.8.2].

These can be described in more detail by Steinberg’s tensor product theorem. A highest weight representation M(λ) is called basic if λ is a basic weight for G, that is, all coefficients of λ when expressed as a linear combination of fundamental weights are less than ℓ. If V is a Ḡ-module over K and F : Ḡ → Ḡ is any endomorphism, we adopt the following notation for the Ḡ-module obtained by composing F with the representation on V. If F is the untwisted Frobenius morphism defining a rational structure over 𝔽_{ℓ^{i}}, then we write V^{(i)}. If Ḡ is of type B_{2}, G_{2}, F_{4} in characteristic ℓ = 2, 3, 2 respectively, and F is an exceptional Frobenius morphism of order two with corresponding group of fixed points Ḡ^{F} defined over 𝔽_{ℓ^{2a+1}} then we write V^{(a+1/2)}. Now first assume that G = ^{d}G_{n}(ℓ^{a}) is not a Suzuki or Ree group. Let M(λ) be an irreducible KG-module with highest weight λ = ∑_{i=0}^{a−1} ℓ^{i}λ_{i}, where the λ_{i} are basic. Then M(λ) is isomorphic to the tensor product

V_{0} ⊗ V_{1}^{(1)} ⊗ … ⊗ V_{a−1}^{(a−1)}  (3.1)

of different Frobenius twists of basic modules V_{i} ≅ M(λ_{i}), 0 ≤ i ≤ a − 1, for G. If G = ^{2}G_{n}(ℓ^{2a+1}) is a Suzuki or Ree group then any irreducible KG-module is isomorphic to a tensor product

V_{0} ⊗ V_{1}^{(1/2)} ⊗ … ⊗ V_{2a}^{(a)}  (3.2)

of different Frobenius twists of basic modules V_{i}, 0 ≤ i ≤ 2a, for G. In both cases this tensor decomposition is unique [5, 2.8.6].

We next describe the fields of definition for these representations (see [5, 2.8.8]). First assume that G is untwisted. Then the field of definition of a tensor product as in (3.1) has degree s over 𝔽_{ℓ}, where s denotes the index of the centralizer of (V_{0}, V_{1}, …, V_{a−1}) in the cyclic permutation group generated by (0, 1, …, a − 1). In particular, it has index at most r in 𝔽_{ℓ^{a}} if the tensor product involves precisely r ≥ 1 nontrivial factors. Thus all non-trivial basic modules have field of definition 𝔽_{ℓ^{a}}. The corresponding statement, with a replaced by 2a + 1, holds for the Suzuki and Ree groups. If G is a Steinberg twist, with corresponding graph automorphism ρ of Ḡ of order t, then the preceding description remains true if all factors V_{i}, that is, their basic weights, are invariant under ρ. Otherwise the field of definition is the extension of degree t of the field described above.

An effective tool in the investigation of groups of Lie type in their defining characteristic will be Proposition 2.4 in conjunction with the following result of Smith and Timmesfeld (see [5, Thm. 2.8.11] and the subsequent remark):

Theorem 3.1. Let Ḡ be a simple algebraic group over the algebraic closure K of a finite field, G = Ḡ^{F} a corresponding finite group of Lie type, P ≤ G a parabolic subgroup with Levi decomposition P = UL. If V is an irreducible KG-module then L acts irreducibly on C_{V} (U). Moreover, if V has highest weight λ for Ḡ then the highest weight for L on C_{V} (U) is the restriction of λ.

We label the fundamental weights for classical groups according to the convention given in Table 4 below.

Table 4: Labeling of Dynkin diagrams of classical type

[figure: A_{n}; n≥1; 1; 2; 3; n; B_{n}; n≥3; 1; <; 2; 3; n; C_{n}; n≥2; 1; >; 2; 3; n; D_{n}; n≥4; 1; 2; 3; 4; n]

3.2. Orbit closures and centralizers.

In this section we show that root elements are the non-trivial unipotent elements with largest centralizer dimension on any given module.

Lemma 3.2. Let Ḡ be a connected reductive algebraic group over an algebraically closed field K. Let B̄ be a Borel subgroup of Ḡ, and T̄ ≤ B̄ a maximal torus. Then the closure of the T̄-orbit of any unipotent element 1 ≠ u ∈ B̄ contains a root element.

Proof. Let Φ be the root system of Ḡ with respect to (T̄, B̄) and Φ^{+} ⊂ Φ the set of positive roots. Then there exist t_{α} ∈ K (α ∈ Φ^{+}) with u = ∏_{α} u_{α}(t_{α}), where {u_{α}(t) | t ∈ K} are the root subgroups in B̄. Assume that t_{α}, t_{β} ≠ 0 for at least two roots α ≠ β. Since no two roots in Φ^{+} are proper multiples of each other, there exists a subtorus T̄′ of T̄ of co-dimension 1 centralizing u_{β}(t) but acting non-trivially on the root subgroup for α. In particular the closure of the T̄-orbit of u contains an element u′ whose product decomposition involves at least one factor less than the one for u, but which involves u_{β}(t_{β}). Induction now shows the claim. ⊓ ⊔

Corollary 3.3. Let Ḡ be a connected reductive algebraic group over an algebraically closed field K. Then the closure of any non-trivial unipotent class contains a long root element, respectively a long or short root element if G has a factor C_{m} or F_{4} in characteristic 2, or a factor G_{2} in characteristic 2 or 3.

Proof. If all roots of G have the same length, this follows immediately from the previous lemma. Now assume that G has two different root lengths, that is, G has a factor of type B_{m}, C_{m}, m ≥ 2, G_{2} or F_{4}. Direct computation with the commutator relations shows that the closure of the class of short root elements contains long root elements in B_{2} = C_{2}, G_{2} and F_{4}, except in the characteristics given in the statement. Thus we may conclude with Lemma 3.2. ⊓ ⊔

Corollary 3.4. Let G be a finite group of Lie type in characteristic ℓ and V an F G-module over a field F of characteristic ℓ. Then the centralizer dimension on V of any non-trivial unipotent element is bounded above by the centralizer dimension of a long root element (respectively of a long or short root element if G is of type C_{m} or F_{4} in characteristic 2 or of type G_{2} in characteristic 2 or 3).

Proof. Let Ḡ denote the algebraic group corresponding to G over the algebraic closure K of the ground field. Then the representation of G on V extends to a representation of Ḡ on V̄ := V ⊗ K, see Section 3.1. Thus it suffices to prove the assertion for Ḡ. Now note that the property of u having centralizer dimension z on V̄ is equivalent to the fact that all (n − z + 1) × (n − z + 1)-minors of a matrix representing u − 1 vanish, where n = dim V. This closed condition is clearly inherited by elements in the closure of the conjugacy class of u. Thus the assertion follows from Corollary 3.3. ⊓ ⊔

We extend the previous result to outer elements in extensions of algebraic groups by graph automorphisms. For this let Ḡ be a connected reductive algebraic group over an algebraically closed field of characteristic ℓ and σ a graph automorphism of Ḡ of order ℓ. More precisely, fix a Borel subgroup B̄ and a maximal torus T̄ ≤ B̄ of Ḡ and assume (as we may) that σ stabilizes the set of simple roots with respect to (T̄, B̄).

Lemma 3.5. Let Ḡ, σ be as above. Then σ lies in the closure of any unipotent class in the coset σḠ.

Proof. Since σ stabilizes B̄, any unipotent class in σḠ has a representative σu in B̄. Let T̄′ ≤ T̄ be the centralizer in T̄ of σ, a subtorus. If α ∈ Φ^{+} vanishes on all of T̄′, then

α(t) = − ∑_{i=1}^{o(σ)−1} α(t^{σ^{i}}) for all t ∈ T,

whence ∑_{i=0}^{o(σ)−1} α^{σ^{i}} = 0. But σ stabilizes the set of positive roots Φ^{+}, hence this is not possible.

We may now copy the argument from the proof of Lemma 3.2 to conclude that the T̄′-orbit of σu contains σ. ⊓ ⊔

Since representations of groups of Lie type extended by a graph automorphism extend to the algebraic group, we obtain as above:

Corollary 3.6. Let G be a finite group of Lie type in characteristic ℓ, σ a graphautomorphism of G of order ℓ and V an F ⟨G, σ⟩-module over a field F of characteristic ℓ. Then the centralizer dimension on V of any non-trivial unipotent element in σG is bounded above by the centralizer dimension of σ on V.

The following result is well known.

Proposition 3.7. Let G be a finite group of Lie-type of simply connected type and P < G a proper parabolic subgroup, P^{−} the opposite parabolic subgroup. Let U and U^{−} denote the unipotent radicals. Then G = ⟨U, U^{−}⟩.

Proof. Let L be the common Levi subgroup of P and P^{−}. Set H = ⟨U, U^{−}⟩. Then L normalizes U and U^{−} and so H. So P and P^{−} each normalize H. Clearly G = ⟨P, P^{−}⟩ (since they are contained in no common proper parabolic subgroup and the only overgroups of P are parabolic subgroups). If G is simple (modulo its center), the result follows (since the center is contained in the Frattini subgroup). Note that in the cases that G is not simple, it is still true that G is the normal closure of its Sylow ℓ-subgroup. In particular, this shows that H = G if G is simple or G is rank one (for then U is a Sylow ℓ-subgroup). This leaves only the cases ^{2}F_{4}(2), Sp_{4}(2) and G_{2}(2). One checks directly that the unipotent radical of a maximal parabolic subgroup is not contained in the derived subgroup (for example, see [4]). ⊓ ⊔

3.3. Long root elements.

We also need an upper bound for abelian subgroups consisting entirely of long root elements.

Lemma 3.8. Let G be a simple algebraic group over an algebraically closed field. Let α, γ be two different long roots (with respect to the choice of a maximal torus T of G) with α + γ not a root. If the product X_{α}X_{γ} of the corresponding root subgroups contains a long root element outside of X_{α} ∪ X_{γ}, then γ − α is a root.

Proof. After conjugation with a suitable element of the Weyl group we may assume that γ is the highest root. Then X_{γ} is normalized by the corresponding Borel subgroup B = UT. Let 1 ≠ x ∈ X_{α}, 1 ≠ y ∈ X_{γ} with v := xy a long root element. Since G is simple, all long root subgroups are conjugate and there exists g ∈ G with v^{g} ∈ X_{γ}. Write g = uwb with u ∈ U, w in the Weyl group and b ∈ B according to the Bruhat decomposition, then v^{uw} ∈ X_{γ}, so v^{u} ∈ Xγ̃ with γ̃^{w} = γ. On the other hand v ∈ X_{α}X_{γ}, so v^{u} ∈ X_{α}^{u}X_{γ} ⊂ U. By the Chevalley commutator relations all non-trivial elements in X_{α}^{u} have lowest term x_{α}(t) for some t ≠ 0. On the other hand v^{u} = xγ̃(t′) for some t′ ≠ 0, so γ̃ = α. Note that X_{α}^{u} ≠ X_{α}. But then again by the commutator relations there must exist a root β with α + β = γ. ⊓ ⊔ Proposition 3.9. Let Ḡ be a simple algebraic group defined over 𝔽_{q}, G the set of fixed points under the corresponding Frobenius map, not of Suzuki- or Ree-type. Let A ≤ G be an abelian subgroup all of whose non-identity elements are long root elements. Let X be a long root subgroup.

(a) Then log_{q} |A| is bounded above by the values given in Table 5; and

(b) A is conjugate to a subgroup of the unipotent radical of the parabolic subgroup N_{G}(X).

Proof. Let x, y be two commuting long root elements and assume that v = xy is again a long root element. Let X, Y be long root subgroups containing x, y respectively. Then [X, Y] = 1 by [5, Prop. 3.2.9]. Moreover, since both lie in a Borel subgroup, and any two Borel subgroups have a maximal torus in common, we may assume that both are standard root subgroups: X = X_{γ}, Y = X_{α}. If X ≠ Y then by Lemma 3.8 the difference γ − α is again a long root. Thus an upper bound for A is obtained by determining the maximal size of a set of long roots such that no sum of two of them is a root, but any difference is. This is easily done. It turns out that the maximal rank is realized by the maximal type A-subsystem consisting of long roots.

Moreover, we can conjugate and assume that X = X_{γ} with γ the highest root. Write N_{G}(X) = LQ with Q the unipotent radical and L the Levi complement. Suppose that Y = X_{α}. If Y is not contained in Q, then Y ≤ L and also X_{−α} ∈ L. Thus, X_{−α} commutes with X and so γ − α is not a root, a contradiction to Lemma 3.8. ⊓ ⊔

Table 5: Maximal ranks of subgroups of commuting long root elements

\begin{matrix}G & A_{n} & B_{n}(n ≥ 2) & C_{n} & D_{n}(n ≥ 4) & G_{2} & F_{4} & E_{6} & E_{7} & E_{8} \\ log_{q} |A| & n & n − 1 & 1 & n − 1 & 2 & 3 & 5 & 7 & 8 \end{matrix}

3.4. ℓ-ranks and centralizers.

For the investigation of possible 2F-modules we sometimes need quite precise information on the ℓ-ranks of subgroups of classical groups with given centralizer on the natural module. This is a straightforward generalization of results in [5, Thm. 3.3.3]:

Proposition 3.10. Let A ≤ SL_{n}(ℓ^{a}) be an elementary abelian ℓ-subgroup with centralizer dimension dim C_{Y_{n}}(A) ≥ n − c on the natural module Y_{n}. Then

|A| ≤ min{ℓ^{ac(n−c)}, ℓ^{a⌊n^{2}/4⌋}}.

Moreover, for each c there is a subgroup A for which this bound is attained. In particular the ℓ-rank of SL_{n}(ℓ^{a}) equals a⌊n^{2}/4⌋.

Proof. Since SL_{n}(ℓ^{a}) acts transitively on the set of subspaces of Y_{n} of fixed dimension we may assume that C_{Y_{n}}(A) is spanned by the first n − c standard basis vectors. Thus

A ≤ R := \{( \begin{matrix}I & 0 \\ ∗ & ∗ \end{matrix} )\} .

The subgroup R is generated by full root subgroups of G. The result now follows precisely as in the proof of [5, Thm. 3.3.3], where the case c = n was treated.

An example of an elementary abelian ℓ-subgroup of maximal rank is given by the group of upper unitriangular matrices with non-zero entries only in the lower left n/2 × n/2-block if n is even (respectively (n + 1)/2 × (n − 1)/2-block if n is odd). ⊓ ⊔

For the other classical subgroups of GL_{n}(q) we make use of the invariant form on the natural module Y_{n}:

Proposition 3.11. Let A ≤ SU_{n}(ℓ^{a}), n ≥ 2, be an elementary abelian ℓ-subgroup with centralizer dimension dim C_{Y_{n}}(A) ≥ n − c on the natural module Y_{n}. Then

|A| ≤ min\{ℓ^{ac^{2}}, ℓ^{ar_{n}}\}, with r_{n} := \{ \begin{matrix}(\frac{n}{2})^{2}, & n even, \\ (\frac{n−1}{2})^{2} + 1, & n odd. \end{matrix}

In particular the ℓ-rank of SU_{n}(ℓ^{a}) is at most ar_{n}.

Proof. The precise ℓ-rank of SU_{n}(ℓ^{a}) is given in [5, Table 3.3.1], and a short examination shows that it is bounded above by ar_{n}, proving the second bound.

If c > ⌊n/2⌋ then the first bound is not smaller than the second, so we may now assume c ≤ ⌊n/2⌋. We will argue by induction on n, the induction base being given by the trivial case n = 2. Let W = R ⊕ X be the decomposition of the centralizer of A into its radical and the nonsingular part. Then c′ := dim R ≤ c and X has dimension n − c − c′. Now A acts faithfully on X^{⊥} of dimension c + c′. Since X^{⊥} ∩ W = R has dimension c′ the centralizer of A on X^{⊥} has dimension at most n − c. If c + c′ < n then |A| ≤ ℓ^{ac^{2}} by induction. Thus it remains to consider the case c + c′ = n, so c = c′ = n/2. In particular, n is even. Then the second bound gives the desired conclusion. ⊓ ⊔

We next consider the maximal ℓ-rank of elementary abelian ℓ-subgroups of Sp_{2n}(q) with given centralizer on the natural module Y_{2n}:

Proposition 3.12. Let A ≤ Sp_{2n}(ℓ^{a}) be an elementary abelian ℓ-subgroup with centralizer dimension dim C_{Y_{2n}}(A) ≥ 2n − c on the natural module Y_{2n}. Then

|A| ≤ min\{ℓ^{a\binom{c+1}{2}}, ℓ^{a\binom{n+1}{2}}\},

and for each c there is a subgroup A for which this bound is attained. In particular the ℓ-rank of Sp_{2n}(ℓ^{a}) equals a\binom{n+1}{2}.

Moreover, if c = n and |A| > ℓ^{a\binom{n}{2}} then C_{Y_{2n}}(A) is maximal totally isotropic.

Proof. Let A be an elementary abelian unipotent subgroup of G := Sp_{2n}(q), q := ℓ^{a}, trivial on a space of dimension 2n−c. If c ≥ n, then already a maximal abelian unipotent subgroup of G fixes a subspace of dimension n ≥ 2n − c, so assume that c ≤ n.

Let W be the centralizer of A and R the radical of W. Then W = R ⊕ X as A-module with X nonsingular. Since A is trivial on X, it leaves X^{⊥} invariant. Suppose that R has dimension c′ – note c′ ≤ c (since R is contained in W^{⊥}). Now X has dimension 2n − c − c′ and so X^{⊥} has dimension c + c′. But A is faithful on X^{⊥} and so by the result [5, Thm. 3.3.3] on maximal abelian subgroups, |A| ≤ qℓ^{d(d+1)/2} where d = (c + c′)/2 ≤ c, whence |A| ≤ q^{c(c+1)/2} – of course we may have c = c′ and obtain equality. Finally, if c = n and W is not totally isotropic, then c′ < c, d ≤ n − 1, and thus |A| ≤ q^{n(n−1)/2}. ⊓ ⊔

Lemma 3.13. Assume that n ≥ 2. Let A be an elementary abelian unipotent subgroup of G := Sp_{2n}(q) of order q^{n(n+1)/2}. Then A is the unipotent radical of the stabilizer of some totally singular subspace of dimension n in the natural representation for G unless q is even and n = 2. In that case then the result is true up to a graph automorphism.

Proof. Let P be the stabilizer of a maximal totally singular subspace with Q the unipotent radical of P. We may assume that A ≤ P. By Lemma 2.6 it suffices to show that Q is not an F-module for L. For q odd, Q is the symmetric square of the natural module of GL_{n}(q), and the claim follows from (2.1) in conjunction with Proposition 3.10 (see also Theorem 9.5).

Suppose now that q is even. Then Q contains an L-submodule R isomorphic to the wedge square of the natural module for L. It follows from Proposition 3.14 that R is not an F^{+} module for n > 3, whence Q is not an F-module.

If n = 3, we still see that Q is not an F-module since its composition factors are the dual of the natural module and the Frobenius twist and the same argument applies. If n = 2, the result follows by an easy inspection (or by considering abelian sets of roots). ⊓ ⊔

Finally, the same investigation for orthogonal groups:

Proposition 3.14. Let A ≤ SO_{n}^{(±)}(ℓ^{a}), n ≥ 4, be an elementary abelian ℓsubgroup with centralizer dimension dim C_{Y_{n}}(A) ≥ n − c on the natural module Y_{n}. Then

|A| ≤ min\{ℓ^{a\binom{c}{2}}, ℓ^{ar_{n}}\}, with r_{n} := \{ \begin{matrix}\binom{\frac{n−1}{2}}{2} + 1, & n ≥ 9 odd, ℓ > 2, \\ n − 2, & n = 5, 7, ℓ > 2, \\ \binom{\frac{n}{2}}{2}, & n ≥ 8 even, \\ \binom{\frac{n}{2}}{2} + 1, & n = 4, 6. \end{matrix}

In particular the ℓ-rank of SO_{n}^{(±)}(ℓ^{a}) is at most ar_{n}.

Proof. The precise ℓ-rank of SO_{n}^{(±)}(ℓ^{a}) is given in [5, Table 3.3.1], and a short examination shows that it is bounded above by ar_{n}, proving the second bound.

If c > ⌊n/2⌋ then the first bound is not smaller than the second, so we may now assume c ≤ ⌊n/2⌋. We will argue by induction on n, the induction base being given by the trivial case n = 2. As above let W = R ⊕ X be the decomposition of the centralizer of A into its radical and the nonsingular part. Then c′ := dim R ≤ c and X has dimension n − c − c′. Our group A acts faithfully on X^{⊥} of dimension c + c′. Since X^{⊥} ∩ W = R has dimension c′ the centralizer of A on X^{⊥} has dimension at most n − c. If c + c′ < n then |A| ≤ ℓ^{a\binom{c}{2}} by induction. Thus it remains to consider the case c + c′ = n, so c = c′ = n/2, R totally isotropic. If n ≠ 4, 6 then again the second bound gives the desired conclusion. For n = 4, 6 it is easy to write out explicit matrices for the centralizer of a maximal isotropic subspace in the orthogonal group and verify (with the argument from loc.cit.) that its ℓ-rank is \binom{n/2}{2}. ⊓ ⊔

The following result will allow to deal with the 27-dimensional modules for E_{6}(q).

Lemma 3.15. Use the notation of 3.3.4 in [5]. Suppose that V has a unique tame subalgebra I of maximal order and moreover V = V (A) where A is the unipotent radical of a parabolic subgroup P. Then A is the unique elementary abelian subgroup of U of order |A|.

Proof. Suppose that B is an elementary abelian unipotent subgroup of U of order |A|. By [5, 3.3.4] it follows that V (B) has the same order and so by hypothesis, V (B) = V (A).

Let Π be the set of fundamental roots corresponding to U. Now A is just the product of the root subgroups, U_{λ} for λ ∈ Σ \ Σ′ where Σ′ is the set of roots corresponding to the Levi subgroup L of P. Let Π′ = Π ∩ Σ′.

Choose the linear functional f (as in loc. cit.) so that f(α) > f(γ) where α ∈ Π \ Π′ and γ ∈ Σ′. With this ordering, it is clear that V (B) ≤ V (A) implies that B ≤ A. In particular, V (B) = V (A) implies that B = A, whence the result. ⊓ ⊔

Lemma 3.16. Let A be an elementary abelian unipotent subgroup of Spin_{14}^{+}(q) of order |A| > q^{20}. Then A is contained in the unipotent radical of the stabilizer of a totally singular subspace of maximal dimension.

Proof. By Proposition 3.14 the tame abelian homogeneous subalgebra I(A) must belong to an abelian set of roots of maximal size. Direct computation shows that such sets correspond to unipotent radicals as in the claim, so I(A) ≤ I(Q) for such a radical Q. Since these roots correspond to a radical, we see that for some appropriate ordering of the roots, A ≤ U(I(Q)) = Q. ⊓ ⊔

3.5. The general strategy.

We can now describe the general approach to the determination of 2F-modules for groups of Lie type in their defining characteristic. Let G̃ be a finite group with G := F^{∗}(G̃) = ^{t}G(q), q = ℓ^{a}, a quasi-simple group of Lie type, and (G̃ : G) of ℓ-power order. Note that G is a central quotient of the group of fixed points under a Frobenius map of a simple simply connected algebraic group over the algebraic closure of 𝔽_{ℓ}, or G = ^{2}F_{4}(2)′. Let V be a faithful absolutely irreducible 𝔽̄_{ℓ}G̃-module. Thus, the restriction of V to G is a direct sum of absolutely irreducible 𝔽̄_{ℓ}G-modules as described in Section 3.1, or G = ^{2}F_{4}(2)′. The 2-modular irreducible representations of the latter are contained in [8].

We distinguish two cases. If V remains irreducible upon restriction to G, then by Steinberg’s tensor product theorem it is a product V = V_{1} ⊗ … ⊗ V_{r} of r ≤ a Frobenius twists of nontrivial basic representations. Furthermore, by the description of fields of definition, it is defined over a subfield of 𝔽_{q^{t}} of index at most rt. So by Lemma 2.1 we arrive at the necessary condition

∏_{i=1}^{r} dim V_{i} ≤ 2r log_{q} |Aut(G)|  (3.3)

for V to be a 2F-module. If V is basic this reduces to dim V ≤ 2 log_{q} |G|, and all such modules have been classified by Lübeck [11]. For any given type we can thus reduce to a short list of possibilities which then have to be investigated further.

Alternatively, if V is a product of at least two non-trivial factors, by Lemma 2.10 the centralizer on V of a possible subgroup A has dimension at most dim V/d_{1} where d_{1} is the minimum of the {d_{i} := dim V_{i} | 1 ≤ i ≤ r}. If b denotes the ℓ-rank of G, we arrive at the inequality

d_{2} ⋯ d_{r} ≤ 2rb/a.  (3.4)

For small values of d_{1} this may be better than (3.3). If A involves outer automorphisms then we may replace b by the ℓ-rank of the centralizer of outer ℓ-elements, which gives even stronger conditions. Similarly, if G is twisted, then the field of definition of V may be larger and we get better inequalities.

On the other hand, if V restricted to G is reducible, that is, if V is induced, then Proposition 2.5 applies. Moreover, any outer ℓ-element has centralizer space of dimension at most dim(V)(ℓ−1)/ℓ on V. Since dim V ≥ 3, A has to contain nontrivial inner elements as well, which have at least d_{2} ⋯ d_{r}-dimensional commutator on the centralizer of outer elements. Together this gives the condition

d_{2} ⋯ d_{r}(d_{1}ℓ − d_{1} + 1) ≤ 2(rb + \frac{rℓ}{a}).  (3.5)

4. Linear and unitary groups

4.1. The case SL_{n}(q).

In this subsection we consider the case of L_{n}(q), q = ℓ^{a}. Let Y_{n} denote the natural module for SL_{n}(q) of dimension n, with right SL_{n}(q)-action. Let λ_{1}, …, λ_{n−1} denote the fundamental weights for SL_{n}, ordered along the Dynkin diagram of type A_{n−1}.

Clearly, an SL_{n}(q)-module V is a 2F-module if and only if its dual V^{∗} is. Similarly, if V is a 2F-module, then so are all of its Frobenius-twists V^{(a)}. It thus suffices to consider representations up to taking duals and up to Frobenius-twists. Now if V has highest weight ∑_{i} a_{i}λ_{i}, then the contragredient representation has highest weight ∑_{i} a_{n−i}λ_{i}. We therefor only need to consider highest weights up to the symmetry induced by the graph automorphism.

It turns out to be convenient to treat the 2-dimensional linear groups separately. This will also serve as an induction base for the investigation of the other groups of Lie type.

Proposition 4.1. The absolutely irreducible 2F-modules in defining characteristic for quasi-simple groups G with F^{∗}(G) = SL_{2}(ℓ^{a}) are as given in Table 7. In particular, their highest weights are one of

ℓ^{i}λ_{1}, 2ℓ^{i}λ_{1} (ℓ > 2), (ℓ^{i} + ℓ^{i+a/2})λ_{1} (if 2|a)

where 0 ≤ i < a.

Proof. The basic SL_{2}(ℓ^{a})-modules, with highest weights tλ_{1}, 0 ≤ t ≤ ℓ − 1, are the symmetric powers Σ^{t}(Y_{2}) of the natural module Y_{2}. On these, ℓ-elements of SL_{2}(ℓ^{a}) act indecomposably.

1. We first consider G = SL_{2}(ℓ^{a})-modules V. Let A be an offender on V. Then A is contained in the unipotent radical U of a Borel subgroup B of G. Since the maximal torus of B has only gcd(ℓ−1, 2) non-trivial orbits on U, Lemma 2.1 shows that U itself is an offender. Hence we may assume A = U. By the Theorem 3.1 of Smith the torus then acts irreducibly on C_{V} (A), so dim C_{V} (A) = 1. If V is defined over the field with ℓ^{f} elements this gives the condition

|V | = ℓ^{df} ≤ |A|^{2}|C_{V} (A)| = ℓ^{2a}ℓ^{f},

where d = dim V, hence f(d − 1) ≤ 2a. By the description in Section 3.1 V is a tensor product V_{1} ⊗ … ⊗ V_{r} of Frobenius twists of basic modules V_{i} of dimension d_{i} = dim V_{i} ≥ 2, with field of definition of index at most r in 𝔽_{ℓ^{a}}. Thus we have f ≥ a/r and hence 2^{r} − 1 ≤ 2r. This immediately forces r ≤ 2. In the case r = 1, that is, V is a Frobenius twist of a basic module, our inequality shows that d = d_{1} ≤ 3. This leads to Examples 4.1 and 4.3 with highest weights λ_{1} and 2λ_{1}.

If r = 2 the assumption f = a violates the first inequality, since d ≥ 4. Hence f = a/2, the two factors are conjugate and we arrive at Example 4.4 with highest weight (1 + ℓ^{a/2})λ_{1}.

2. The only outer automorphisms of S = L_{2}(ℓ^{a}) of order ℓ are field automorphisms, and these exist if and only if ℓ|a. So for the rest of the proof let G = SL_{2}(ℓ^{a}).ℓ, where ℓ|a. The ℓ-rank of the centralizer in S of a field automorphism of order ℓ is a/ℓ. We first consider extendable representations V = V_{1} ⊗ … ⊗ V_{r} of G. Then necessarily ℓ|r and our standard inequality (3.4) reads

d_{1} ⋯ d_{r−1}(d_{r} − 1) ≤ 2r/ℓ + 2r/a.

This is satisfied only when d_{i} = 2 and either r = ℓ = a = 3 or r = ℓ = 2, a ≤ 2. But for SL_{2}(27) the 4-dimensional centralizer of the outer automorphism of order 3 is not centralized by an inner 3-element, so this case is out. The second possibility gives an example for the highest weight (1 + ℓ^{a/2})λ_{1}.

3. Finally assume that V is induced from a representation W of SL_{2}(q) which is not invariant under the field automorphism of order ℓ. The field automorphism has centralizer of dimension dim(W) on the induced module V = W ↑_{S}^{G}, and on this centralizer, non-trivial inner ℓ-elements have at least d_{1} ⋯ d_{r−1}(d_{r} − 1)- dimensional commutator when W = V_{1} ⊗ … ⊗ V_{r} with d_{i} = dim V_{i}. Since V is defined over a subfield of index at most rℓ of 𝔽_{ℓ^{a}}, we arrive at the condition

d_{1} ⋯ d_{r−1}(d_{r}ℓ − 1) ≤ 2r + 2rℓ/a.

This forces r ≤ 2, ℓ = 2, d_{i} = 2. The case r = 1 occurs only when a = 2, 4 and leads to an example from the natural module. When r = 2 then moreover a = 2. But the group L_{2}(4) does not have non-extendable representations of degree 4. ⊓ ⊔

Note that the conclusion of Proposition 4.1 even holds for the solvable groups SL_{2}(2) and SL_{2}(3).

For the remaining groups of Lie type we make use of the results of Liebeck [10] classifying the basic modules of Chevalley groups G in defining characteristic of small dimension (see also the more extensive lists of Lübeck [11]).

Lemma 4.2. Let n ≥ 3, M(λ) a basic highest weight module for SL_{n}(ℓ^{a}) of dimension dim M(λ) ≤ n^{2} + 1. Then M(λ) is (possibly the dual of) one of the modules in Table 6.

This is Theorem 2.2 in [10]. Thus generically (i.e., for n ≥ 9) only four types of modules occur. Moreover, the two smallest non-trivial dimensions are n and \binom{n}{2}.

We need an extension of Lemma 4.2 giving lower bounds for the dimension of modules with symmetric highest weight for small n, as well as for modules with weights supported on the end nodes only; both follow for example from [11, Thm. 4.4]:

Lemma 4.3. Let λ = t_{1}(λ_{1} + λ_{n−1}) + t_{2}(λ_{2} + λ_{n−2}) + … with 0 ≤ t_{i} < ℓ be a symmetric basic weight for SL_{n}(ℓ^{a}), n ≥ 3. Then dim M(λ) ≥ 2n^{2} unless

λ = λ_{1} + λ_{n−1}, or λ = λ_{n/2} for n = 4, 6, 8, or λ = 2λ_{2} for n = 4.

Lemma 4.4. (a) We have dim M(λ_{1} + 2λ_{n−1}) ≥ 2n^{2} for n ≥ 4.

Table 6: The basic SL_{n}(ℓ^{a})-modules of dimension at most n^{2} + 1

λ M(λ) dim M(λ)
λ_{1}  Y_{n}  n
λ_{2} ∧^{2}(Y_{n})  \binom{n}{2}  n ≥ 4
2λ_{1} Σ^{2}(Y_{n})  \binom{n+1}{2}  ℓ > 2
λ_{1} + λ_{n−1}  Ã(Y_{n}) n^{2} − 1 − δ_{ℓ,n}  n ≥ 3
λ_{3} ∧^{3}(Y_{n})  \binom{n}{3}  6 ≤ n ≤ 8
3λ_{1} Σ^{3}(Y_{3})  10  n = 3, ℓ > 3
λ_{1} + λ_{2}  16  n = 4, ℓ = 3

(b) We have dim M(2λ_{1} + 2λ_{n−1}) ≥ 2n^{2} for n ≥ 3.

Lemma 4.5. Let n ≥ 3, λ = t_{1}λ_{1} + t_{2}λ_{n−1} with

t_{1}, t_{2} ∈ {ℓ^{i}, 2ℓ^{i}, ℓ^{i} + ℓ^{a/2+i} | 0 ≤ i < a/2}.

Then the highest weight module M(λ) is not a 2F-module for SL_{n}(ℓ^{a}).

Proof. 1. For λ one of λ_{1}+ℓ^{i}tλ_{n−1}, 2λ_{1}+ℓ^{i}tλ_{n−1} or t(λ_{1}+ℓ^{i}λ_{n−1}) with t = 1+ℓ^{a/2} the corresponding highest weight module M(λ) is a tensor product of dimension at least n(n^{2} − 2), n^{3}, (n^{2} − 2)^{2}, respectively. Only in the latter case M(λ) is defined over a proper subfield of 𝔽_{q}, of index 2. In any case the dimension exceeds the bound (2n^{2} − 1)a/f. Thus we may assume that t_{1}, t_{2} ∈ {ℓ^{i}, 2ℓ^{i}}, and then, by taking a suitable Frobenius twist, that t_{1} ∈ {1, 2}, t_{2} = 2ℓ^{i} if t_{1} = 2.

2. Now let V = M(λ) for λ = λ_{1} + ℓ^{i}λ_{n−1}. Thus V is the tensor product Y_{n} ⊗ (Y_{n}^{∗})^{(i)}, except when i = 0 in which case V is the adjoint module Ã(Y_{n}) for SL_{n}(ℓ^{a}), a submodule of possibly a quotient by a 1-dimensional subspace of Y_{n} ⊗ Y_{n}^{∗}. Now first note that the assertion of Lemma 2.10(a) for the centralizer remains true for Ã(Y_{n}). Indeed, if A is cyclic, then all indecomposable modules are uniserial, so the centralizer on a quotient cannot increase. The general statement was derived by considering successive actions of cyclic groups on previous centralizers, so the proof remains valid. Returning to V, if the centralizer of A on the natural module or its dual has dimension less than (n−1)/2 then the first bound from Corollary 2.11(a) gives a contradiction. If A contains non-trivial elements with distinct centralizers, then Lemma 2.10(a) shows that the centralizer on V is too small. If A contains no transvections, then its rank is again too small. Thus A consists of transvections all fixing the same hyperplane in Y_{n}, and all fixing the same hyperplane in (Y_{n}^{∗})^{(i)}. But this forces |A| = q, and now Corollary 2.11(a) gives the final contradiction.

3. Thus, if t_{2} ≠ 2 then M(λ) is a tensor product defined over 𝔽_{q} of dimension at least n\binom{n+1}{2}, which is larger than 2n^{2} − 1. So we are left with (t_{1}, t_{2}) ∈ {(1, 2), (2, 2)}. For λ = 2λ_{1} +2λ_{n−1}, the corresponding highest weight module has dimension at least 2n^{2} by Lemma 4.4(b), hence this is also impossible. Also, the weight λ_{1} + 2λ_{n−1} for n ≥ 4 is excluded by Lemma 4.4(a).

4. So finally we consider λ = λ_{1} + 2λ_{2} in the case of G = SL_{3}(q). For this, we investigate possible offenders. Let Φ^{+} = {α, β, α + β} denote the set of positive roots for G with respect to the choice of a maximal torus and a Borel subgroup B containing it. For a root ν ∈ Φ^{+} let X_{ν} = {x_{ν}(t) | t ∈ 𝔽_{q}} be the corresponding root subgroup. If there exists an offender A in the unipotent radical U of a maximal parabolic subgroup, then we may assume A = U by Lemma 2.1. But U together with the unipotent radical of the opposite parabolic subgroup generate G, by Lemma 3.7, so the centralizer dimension of U on M(λ) is at most dim(M(λ))/2. This implies that dim(M(λ)) ≤ 8, which is not the case by Lemma 4.2. Hence A contains elements involving x_{α}(t) and x_{β}(u) for some non-zero t, u ∈ 𝔽_{q}. Then A also contains an element involving the product x_{α}(t)x_{β}(u). But the normal closure of such an element in the Borel subgroup B is the whole unipotent radical U of B. Thus again by Lemma 2.1 we may and will assume that A = U. We have |A| = ℓ^{3a}, dim C_{V} (A) = 1 by Theorem 3.1, and so dim(M(λ)) ≤ 6 if A is an offender on V. This is again a contradiction to Lemma 4.2. ⊓ ⊔

Lemma 4.6. The exterior cube ∧^{3}(Y_{7}) of the natural module, with highest weight λ_{3}, is not a 2F-module for SL_{7}(q).

Proof. 1. Let U be the unipotent radical of a Borel subgroup of SL_{7}(q) consisting of all lower unitriangular matrices. We may assume that a possible offender A on V = ∧^{3}(Y_{7}) is contained in U. By Table 3 any transvection has at most 25-dimensional centralizer on V, so by Lemma 3.4 the same is true for any nontrivial unipotent element. Moreover, if A contains more than q elements, then dim C_{V} (A) ≤ 21, by arguments similar to those in the proof of Lemma 2.10. Thus, |A| ≥ q^{(36−21)/2} > q^{7}.

Let P_{1} = U_{1}L_{1} be the end-node parabolic above U containing GL_{6}(q) on the first six basis vectors. Its unipotent radical U_{1} is elementary abelian of order q^{6}. Thus, by our previous argument, we can’t have A ≤ U_{1}.

2. Now assume that A ∩ U_{1} is trivial. Then A is isomorphic to its image B in the Levi subgroup of P_{1} of type GL_{6}(q). The composition factors of V for GL_{6}(q) are ∧^{3}(Y_{6}) and ∧^{2}(Y_{6}). Any unipotent element of GL_{6}(q) has rank at least 6 on ∧^{3}(Y_{6}) and at least 4 on ∧^{2}(Y_{6}). Moreover, any unipotent subgroup of order larger than q has centralizer codimension at least 9 on ∧^{3}(Y_{6}) and at least 7 on ∧^{2}(Y_{6}) by Lemma 2.10. Thus, |B| ≥ q^{8}. Then either B is contained in the unipotent radical of a GL_{4}(q)-parabolic of GL_{6}(q), which can be ruled out by the standard argument, or it contains elements with at least three different centralizers, which decreases both centralizer dimensions at least by 1, and |B| = q^{9}. Then by Proposition 3.10 B is the unipotent radical of a GL_{3}(q) × GL_{3}(q)-parabolic of GL_{6}(q), whose centralizer is too small by Theorem 3.1.

Hence A ∩ U_{1} is non-trivial, so the centralizer of A on ∧^{2}(Y_{6}) has codimension at least 10, and since A ≰ U_{1}, the centralizer on ∧^{3}(Y_{6}) has codimension at least 6. Thus |A| ≥ q^{8}.

3. Next assume that |A ∩ U_{1}| ≤ q. Since B := AU_{1}/U_{1} can have order at most q^{9} by Proposition 3.10, we have q^{8} < |A| ≤ q^{10}. Let P_{2} = U_{2}.GL_{5}(q) be the end-node parabolic of L_{1} containing GL_{5}(q) on the first five basis vectors. By Theorem 3.1, C_{V} (U_{1}) is ∧^{3}(Y_{6}) for GL_{6}(q), and on this the Levi factor GL_{5}(q) has composition factors ∧^{3}(Y_{5}) and ∧^{2}(Y_{5}). If |B ∩ U_{2}| ≤ q then the image of B in the Levi factor GL_{5}(q) has maximal possible order q^{6}. But then its centralizer is only 3-dimensional on ∧^{3}(Y_{5}) by Theorem 3.1, and we get a contradiction. On the other hand, if |B ∩ U_{2}| > q, then the centralizer of A ∩ U_{1}U_{2} has codimension at least 10+9 = 19 on V, whence |A| > q^{9}. Then the image of A in GL_{5}(q) has order larger than q^{3}, and thus a centralizer of codimension at least 5 on ∧^{3}(Y_{5}). Then |A| ≥ q^{12}, so A is the unipotent radical of a GL_{3}(q) ∘ GL_{4}(q)-parabolic, which has at most 2-dimensional centralizer by Theorem 3.1, impossible.

4. We may thus assume that |A ∩ U_{1}| > q. Then A ∩ U_{1} has centralizer of codimension at least 14 on V by direct computation, and the image B has centralizer of codimension at least 6 on C_{V} (U_{1}) = ∧^{3}(Y_{6}). Thus |A| ≥ q^{10}. But then necessarily B ∩ U_{2} ≠ 1. The case that A is contained in U_{1}U_{2} was ruled out before, so the image C of A in GL_{5}(q) is non-trivial. Then we obtain a centralizer of codimension at least 20 + 3 = 23 on V, whence |A| > q^{11}. First assume that |B ∩ U_{2}| ≤ q. Then |C| > q^{4}, which has centralizer of codimension at least 5 on ∧^{3}(Y_{5}), contradicting the bound. On the other hand, if |B ∩ U_{2}| > q, then the centralizer of A has codimension at least 14 + 9 + 3 = 26 on V, again impossible. ⊓ ⊔

As a first step we treat the case where G does not involve outer automorphisms of S = L_{n}(q).

Proposition 4.7. The absolutely irreducible 2F-modules in defining characteristic for SL_{n}(q), q = ℓ^{a}, n ≥ 3, are as given in Table 7. In particular, their highest weights are (up to Frobenius twists and taking the dual) one of

λ_{1}, 2λ_{1}, (1 + ℓ^{a/2})λ_{1}, λ_{2}, λ_{3} (for n = 6) .

Proof. We proceed by induction, the induction base being given by the case n = 2 in Proposition 4.1. Note that for basic 2F-modules, (3.3) shows that dim V ≤ 2n^{2} − 1.

1. Let V be a 2F-module for G = SL_{n}(q) of dimension d, defined over 𝔽_{ℓ^{f}}. Let P = U.L = q^{n−1}.GL_{n−1}(q) be a maximal parabolic subgroup of G corresponding to an end node on the Dynkin diagram. First assume that there exists an offender A ≤ U for V. Since U is the natural module for L, hence irreducible, Lemma 2.1 allows to conclude that in this case U itself is an offender. On the other hand, by Proposition 3.7 the centralizer dimension of U on V is at most d/2. Thus the 2F-condition gives df/2 ≤ 2(n − 1)a, hence d ≤ 4(n − 1)a/f. By Steinberg’s theorem, V is a tensor product V = V_{1} ⊗ … ⊗ V_{r} of Frobenius twists of non-trivial basic modules V_{i} of dimension d_{i} := dim V_{i} ≥ n, and defined over a subfield of 𝔽_{q} of index at most r. We obtain

n^{r} ≤ d ≤ 4(n − 1)a/f ≤ 4(n − 1)r.

Since n ≥ 3 this forces r ≤ 2. Moreover, if r = 2 then d_{i} = n, and V_{1}, V_{2} are Galois conjugate, so V = Y_{n} ⊗ Y_{n}^{(a/2)} up to Frobenius twists and taking the dual. This is Example 4.4. When r = 1 then d ≤ 4(n − 1), so by Lemma 4.2 we are in one of the cases of Example 4.1–4.3 or 4.5, or V = Ã(Y_{n}) with highest weight λ_{1} + λ_{n−1}, which was excluded in Lemma 4.5.

2. Thus, by Proposition 2.4 we may assume that C_{V} (U) is a 2F-module for L (hence for L′ = SL_{n−1}(q)) for both end node parabolics. Hence by induction the highest weight λ of V or its dual is one of

t_{1}λ_{1} + t_{2}λ_{n−1}  (t_{1}, t_{2} ∈ {0, ℓ^{i}, 2ℓ^{i}, ℓ^{i} + ℓ^{a/2+i}}),
ℓ^{i}λ_{2},
tλ_{2}  (t ∈ {2ℓ^{i}, ℓ^{i} + ℓ^{a/2+i}}, n = 4),
ℓ^{i}λ_{3}  (n = 6, 7).

Indeed, if λ has more than one non-zero coefficient, then there are precisely two, and these lie at the two ends, since by induction all 2F-weights have support only on one node. Moreover the entries are at most those for SL_{2}(q). Similarly for the cases with just one non-zero coefficient.

The weights of the form t_{1}λ_{1}+t_{2}λ_{n−1} not occurring in the conclusion were ruled out in Lemma 4.5; the possibility λ = λ_{3} for n = 7 was excluded in Lemma 4.6.

3. For n = 4, λ = (1 + ℓ^{a/2})λ_{2}, the module M(λ) is the tensor product of V_{1} with V_{1}^{(a/2)}, where V_{1} is the natural module for SO_{6}^{+}(q) (namely, the exterior square of the natural module for G). The precise bounds for the rank of a subgroup of SO_{6}^{+}(q) with given centralizer in Proposition 3.14 together with Corollary 2.11 show that this case does not occur.

4. Finally, the highest weight module M(λ) of G = SL_{4}(q) with highest weight λ = 2λ_{2} is the heart of Σ^{2}(∧^{2}(Y_{4})), the symmetric square of the alternating square of the natural module. By Lemma 2.10 and Corollary 3.4 any non-trivial unipotent element of G has centralizer of dimension at most 4 on ∧^{2}(Y_{4}). But then the centralizer on Σ^{2}(∧^{2}(Y_{4})) has dimension at most 11. As for Ã(Y_{4}) in the proof of Lemma 4.5 we conclude that the same is true on M(λ). Hence an offender has order at least q^{4}, so is equal to the unipotent radical of a GL_{2}(q)∘GL_{2}(q)-parabolic. But by Theorem 3.1 the latter only has centralizer dimension 3 on M(λ). Thus M(λ) is not an example. ⊓ ⊔

Proposition 4.8. Let V be an absolutely irreducible 2F-module in characteristic ℓ for a group G with F^{∗}(G) = SL_{n}(q), q = ℓ^{a}, n ≥ 3. Then all composition factors of the restriction V |_{F^{∗}(G)} are 2F-modules for F^{∗}(G).

Proof. Let V be a 2F-module for G.

1. First assume that the restriction of V to F^{∗}(G) is not irreducible. Let W be one of the irreducible constituents. By Proposition 2.5 either W itself is a 2F-module for a proper subgroup of G containing F^{∗}(G), or ℓ = 2, W is defined over 𝔽_{2^{2}} and dim W ≤ k, where k denotes the ℓ-rank of an offender A. If a ≤ 2 this forces dim W ≤ n^{2}/4 + 2, so W is a twist of a basic module, and Lemma 4.2 shows that W = Y_{n}, or n = 4, W = ∧^{2}(Y_{4}), up to twists and duals. These are Examples 4.1 and 4.2. If a ≥ 3 then W has to be a product of at least a/2 factors, of dimension at least n, which gives n^{a/2} ≤ an^{2}/8 + 2. This does not lead to new cases. Hence it remains to consider modules V with irreducible restriction to F^{∗}(G), that is, highest weight modules whose highest weight is invariant under G.

2. Assume that G = SL_{n}(ℓ^{a}).ℓ involves a field automorphism of order ℓ (so in particular ℓ|a). We may assume that A is not contained in F^{∗}(G), so neither of the unipotent radicals of end node parabolics contains an offender. Theorem 3.1 yields that W := C_{V} (U) is an example for SL_{n−1}(ℓ^{a}).ℓ. By induction (with induction base SL_{2}(q)) this forces V or its dual to have highest weight

λ ∈ {tλ_{1}, t(λ_{1} + λ_{n−1}), tλ_{2} (n = 4) | t = ℓ^{i} + ℓ^{i+a/2}} .

In particular ℓ = 2 and 2|a. The first case is Example 4.4. The second one is a Frobenius twist of Ã(Y_{n})⊗Ã(Y_{n}^{(a/2)}), hence has dimension at least (n^{2} −2)^{2}. This violates the upper bound 2(2n^{2} − 1). The third module is a Frobenius twist of the tensor product ∧^{2}(Y_{4}) ⊗ ∧^{2}(Y_{4}^{(a/2)}) for n = 4. In particular, the centralizer of an offender A has codimension at least 2\binom{4}{2}, so the 2F-condition reads 4(4 − 1) ≤ 2(4^{2}/4 + 1), contradiction.

3. Next consider the case that ℓ = 2 and G = SL_{n}(2^{a}).2 is obtained by adjoining the graph automorphism γ. The centralizer in SL_{n}(2^{a}) of a graph automorphism of order 2 is contained in a symplectic group Sp_{⌊ \frac{n}{2} ⌋}(2^{a}). Thus by Proposition 3.12 its 2-rank equals an(n + 2)/8 for n even, a(n^{2} − 1)/8 for n odd. By step 1 we may restrict to modules V with highest weight invariant under the graph automorphism.

3A. First consider modules with highest weight of the form t(λ_{1} + λ_{n−1}). Since ℓ = 2 these are tensor products of Frobenius twists of the adjoint module Ã(Y_{n}). If there are at least two factors, the dimension becomes too large. Now assume that V is a twist of Ã(Y_{n}). Let A be an offender on V for G. If B := A ∩ SL_{n}(q) has centralizer of codimension c on the natural module, then Corollary 2.11(a) and Lemma 2.12 show that the centralizer on V has codimension at least 2c(n − c) + \binom{n−c}{2}. This is minimal for c = 1, and comparing with the rank of A we deduce n ≤ 4. If n = 4 then necessarily B has rank at least 3a/2, thus contains elements with distinct centralizers. This gives a contradiction with Lemma 2.10(a). In the case n = 3 we have that Y_{3} ⊗ Y_{3}^{∗} = V ⊕ 1, that is, the centralizer on V is strictly smaller than on the tensor product Y_{3} ⊗ Y_{3}^{∗}. This leads to a stronger inequality and thus to a contradiction.

Since for n = 3 the only invariant weights are of the form t(λ_{1} + λ_{2}), we may conclude that n ≥ 4.

3B. By Proposition 2.4 we may assume that W := C_{V} (U) is an example for the Levi complement, extended by γ, of any γ-stable parabolic subgroup of G. For n odd, induction starting from the case n = 3 in step 3A shows that only weights of the form t(λ_{1} +λ_{n−1}) are possible, that is, tensor products of Frobenius twists of Ã(Y_{n}). But these were ruled out in step 3A. For n even we first study basic modules. For n = 4 by step 3A the only basic highest weight to consider is λ_{1} + λ_{2} + λ_{3}. But this belongs to the Steinberg module for SL_{4}(2) (by [5, Thm. 2.8.7]), so has dimension 64, bigger than 2n^{2} − 1. For n = 6, our induction argument together with step 3A only leaves the weight λ_{1} +λ_{3} +λ_{5}, which is ruled out by Lemma 4.3.

Now induction shows that there are no cases for even n ≥ 8.

Still for n even, if V is an invariant tensor product of r ≥ 2 factors, then necessarily all of its factors have to be invariant. We obtain the condition

d_{2} ⋯ d_{r} ≤ r \frac{n(n + 2)}{4} + \frac{2r}{a} .

This leads to r = 2, n = 4, d = \binom{4}{2}. Moreover, the two factors have to be Galois conjugate, so 2|a. The subgroup A ∩ SL_{4}(q) has centralizer of dimension at most 4 on ∧^{2}(Y_{4}), so its centralizer on V has dimension at most 20 by Corollary 2.11(a). This forces a = 1, but we just saw that a has to be even in this case. This completes the consideration of the extension by the graph automorphism.

4. Next consider the case that ℓ = 2 and G = SL_{n}(2^{a}).2 is obtained by adjoining a graph-field automorphism γ (so a is even). Here, the centralizer of γ of order 2 in SL_{n}(2^{a}) is SU_{n}(2^{a/2}). So by Proposition 3.11 it has 2-rank (\frac{n}{2})^{2}a/2 if n is even, respectively ((\frac{n−1}{2})^{2} +1)a/2 if n is odd. Basic representations do not extend, so V is a non-trivial tensor product. The only possibility for invariant tensor products is V = Y_{n} ⊗ Y_{n}^{∗}^{(a/2)}. But this is defined over 𝔽_{ℓ^{a}}, so Corollary 2.11(a) yields a contradiction.

5. Finally assume that ℓ = 2, 2|a, G = SL_{n}(2^{a}).2^{2} contains both the graph and the field automorphism of order 2. Here again basic representations are not invariant. The standard inequality (3.5) shows that the case of tensor products does not lead to examples in this situation. ⊓ ⊔

4.2. The case SU_{n}(q).

The reasoning for the unitary groups is very similar to the one for linear groups.

Proposition 4.9. Let G = SU_{6}(q) and V the exterior cube of the natural module. Let A be a nontrivial elementary abelian unipotent subgroup of G, then f(A) < |V |.

Proof. Let A be an elementary abelian unipotent subgroup with f(A) maximal among such subgroups and assume that f(A) ≥ |V |. We actually work in GU_{6}(q).

Let Q be the unipotent radical of the parabolic subgroup P fixing a totally singular three dimensional space in the natural module for G. So |Q| = q^{9}. Write P = QL where L ≅ GL_{3}(q^{2}) is the Levi complement.

We note that f(Q) = q^{19} by Theorem 3.1. In particular, since Q is an irreducible P -module, it follows that f(A) < |V | if A ≤ Q.

Set B = A∩Q. We note that if 1 ≠ x ∈ A, then C_{V} (A) is at most of dimension 14 and moreover if x is not a long root element, then C_{V} (A) has dimension at most 12, by Table 3. Since the only long root elements in A must be contained in Q, it follows that C_{V} (A) has dimension at most 12. In particular, |A| ≥ q^{4}.

Consider A/B (as a subgroup of L). Note that as a P -module, we have a filtration of V: 0 < C = C_{V} (P) < W < M < V where W/C and M/W are 9-dimensional 𝔽_{q}L-modules, with W/C isomorphic to the natural module for L tensor its Frobenius twist and M/W the dual of W/C.

First suppose that A/B contains a regular unipotent element of L (note that this implies that ℓ > 2). Choose x ∈ A with xB this regular unipotent element. It follows that C_{V} (A) is at most 8-dimensional, whence |A| ≥ q^{6}. In particular, |B| ≥ q^{2} and does not consist of long root elements. Thus, C_{V} (B)/C_{W} (B) has dimension at most 3. This implies that dim C_{V} (A) ≤ 7. In particular, |B| > q^{2} and is contained in C_{Q}(x). It follows by inspection that B contains a rank 3 element in Q. Thus, C_{V} (B)/C_{W} (B) is at most 1-dimensional. Thus, C_{V} (A) has dimension at most 1 + 3 + 1 = 5. So f(A) ≤ q^{19}, a contradiction.

So A/B consists of long root elements in L. Suppose that A/B embeds in a long root subgroup X of L (this is not a long root subgroup of G). If |A/B| > q, then |C_{S}(A/B)| ≤ q^{4} where S = Q or S is one of the two nontrivial L-composition factors on V. If B = 1, this implies that f(A) ≤ q^{14}. If B ≠ 1, then C_{V} (A) ≤ W and so C_{V} (A) has dimension at most 9. Then |A| > q^{5} and so |B| > q^{3}. This allows to improve the estimate on C_{V} (A) to be of dimension at most 8, whence |A| = q^{6} (since that is an upper bound). In particular, this implies that B has order q^{4} and so must be C_{Q}(X). A straightforward computation shows that C_{V} (B) is 4-dimensional. This implies that f(A) ≤ q^{16}.

The remaining possibility is that A/B is not contained in a long root subgroup but is contained in the unipotent radical of either the stabilizer of 1-space or 2-space in the natural representation of L. In the latter case, we observe that C_{W/C}(A/B) is 1-dimensional. If |B| ≤ q, then C_{V} (A) has dimension at most 2+5 = 7, a contradiction. If |B| > q, then C_{V} (A) has dimension at most 2+3 = 5 and |A| ≤ q^{2}|C_{Q}(A/B)| ≤ q^{6}, whence f(A) < |V |.

So A/B centralizes a 2-space in the natural representation for L but is not contained in a root subgroup. It follows that |C_{Q}(A/B)| = q^{4}, |C_{W/C}(A/B)| = q^{4} and C_{M/W} (A/B) is 1-dimensional. Thus, C_{V} (A) is at most 7-dimensional and so |A| ≥ q^{6}. This implies that B ≠ 1 and so C_{V} (A) is at most 6-dimensional. Thus, |A| = q^{7} (since it has order at most q^{7}). In particular, we see that B = C_{Q}(X) where X is the full unipotent radical containing A/B. We then compute that C_{V} (B) is 4-dimensional, whence f(A) ≤ q^{18}. ⊓ ⊔

Theorem 4.10. The absolutely irreducible 2F-modules in defining characteristic for SU_{n}(q), q = ℓ^{a}, n ≥ 3, are as given in Table 7.

Proof. 1. The unitary group SU_{n}(q) is a subgroup of the linear group SL_{n}(q^{2}) defined over the field of q^{2} elements. By the general results presented in Section 3.1, the irreducible representations of SU_{n}(q) are the restrictions of those of SL_{n}(q^{2}). We distinguish two cases.

2. First assume that the restriction V to SU_{n}(q) of the SL_{n}(q^{2})-representation is defined over a proper subfield (necessarily of index 2). Then its highest weight has to be invariant under the graph automorphism γ. In particular, each tensor factor is itself invariant under γ, so has dimension at least d_{0} := n^{2} −2 for n ≠ 4, 6, respectively d_{0} := \binom{n}{n/2} for n = 4, 6, by Lemma 4.2. We first show that in this case V is a twist of Ã(Y_{n}), or of ∧^{k}(Y_{n}) for n = 2k ∈ {4, 6, 8}.

2A. If V is a tensor product V_{1} ⊗ … ⊗ V_{r} of at least two factors, of dimensions d_{i} = dim(V_{i}) ≥ d_{0}, then our fundamental inequality (3.3) shows that d_{0}^{r} ≤ r(2n^{2} − 1). With the lower bound for d_{0} mentioned above this only leaves the case n = 4, r = 2, V_{i} Galois conjugate twists of ∧^{2}(Y_{n}). This is the natural module for O_{6}^{−}(q), and Proposition 3.14 in conjunction with Corollary 2.11(a) allows to exclude this.

2B. Hence we may now assume that V is (a twist of) a basic module, of dimension at most 2n^{2} − 1. By Lemma 4.3 the only possibility to consider is n = 4, λ = 2λ_{2}, with dim M(λ) ≥ 19. Let P_{1} = U_{1}.L_{1} be a maximal parabolic subgroup with Levi complement L_{1} of type GU_{2}(q), so U_{1} is special of order q^{1+4}. First suppose that there exists an offender A ≤ U_{1}. Since both Z(U_{1}) and U_{1}/Z(U_{1}) are irreducible L_{1}-modules, we may assume by Lemma 2.1 that either A = Z(U_{1}) or A = U_{1}. In the first case, A is a root subgroup, 7 conjugates of which generate G, so the 2F-condition gives d/7 ≤ 2, which is not the case. In the second case A is a unipotent radical, which by Theorem 3.1 has centralizer of dimension 3 on M(λ), too small.

So we may assume that A is not contained in U_{1}. Let P_{2} = U_{2}.L_{2} be the maximal parabolic subgroup with Levi complement L_{2} = GL_{2}(q^{2}). Then L_{2} acts irreducibly on the elementary abelian unipotent radical U_{2} of order |U_{2}| = q^{4}. Thus, if A is contained in U_{2}, we may assume that A = U_{2}. Since two conjugates of U_{2} generate SU_{4}(q) by Proposition 3.7, we get dim M(λ) ≤ 16, which is not the case.

We may thus assume that A is not contained in a conjugate of U_{1} or U_{2}. But then the normal closure of A in the unipotent radical U of a Borel subgroup is U itself, of order q^{6}, and with centralizer of dimension 1. This yields the final contradiction for M(2λ_{2}).

2C. It remains to deal with Ã(Y_{n}) and ∧^{k}(Y_{n}), n = 2k ≤ 8. The case k = 2 is Example 4.7. For V = Ã(Y_{n}) let A be an offender with centralizer of codimension c on the natural module Y_{n}. Then by Proposition 3.11 the 2F-condition becomes 2c(n − c) − 2 ≤ 2c^{2}. This shows that c ≥ (n − 1)/2, so the stronger inequalities from Lemma 2.10(a) apply and we obtain a contradiction.

The wedge cube ∧^{3}(Y_{6}) is not a 2F-module by Proposition 4.9. Let P = UL be a maximal parabolic subgroup of SU_{8}(q) with Levi complement of type GU_{6}(q) and U special of order q^{1+12}. If ∧^{4}(Y_{8}) is a 2F-module for SU_{8}(q), then the previous result implies that an offender must be contained in U. But this is ruled out immediately by Theorem 3.1 and Proposition 3.9.

3. Now assume that the representation of SL_{n}(q^{2}) is defined over the same field as the restriction to SU_{n}(q). Then any example for SU_{n}(q) is a fortiori an example for SL_{n}(q^{2}), hence one of the cases in Table 7. Of these, all except ∧^{2}(Y_{4}), ∧^{3}(Y_{6}), and the fourth example are defined over 𝔽_{q^{2}}. The first example, which is the natural module, is Example 4.6. For V = ∧^{2}(Y_{n}), n ≥ 5, let A be an offender with centralizer of codimension c on the natural module. With Proposition 3.11 and (2.1) we obtain the inequality 2c(n − c − 1) ≤ 2c^{2}, which forces c ≥ (n − 1)/2.

But this can be ruled out with the stronger assertion from Lemma 2.10(c). The symmetric square is excluded in the same way. ⊓ ⊔

Proposition 4.11. Let V be an absolutely irreducible 2F-module in characteristic ℓ for a group G with F^{∗}(G) = SU_{n}(q), q = ℓ^{a}, n ≥ 3. Then all composition factors of the restriction V |_{F^{∗}(G)} are 2F-modules for F^{∗}(G).

Proof. The case of 2F-modules V with reducible restriction to F^{∗}(G) may be dealt with precisely as in the proof of Proposition 4.8. So now assume that V is irreducible for SU_{n}(q).

1. First assume that ℓ is odd and G = SU_{n}(ℓ^{a}).ℓ involves a field automorphism of order ℓ. Since basic representations are not invariant under field automorphisms of odd order, V is a tensor product of at least 2 factors. If all factors V_{i} are already defined over 𝔽_{q}, then dim V_{i} ≥ n^{2} − 2 for n ≠ 4, 6, respectively dim V_{i} ≥ 6, 20 when n = 4, 6. Our standard inequality (3.3) shows that in this case V must be the product of two 6-dimensional representations for SU_{4}(q). But then the field of definition does not drop, and the inequality is not satisfied. If at least one factor is defined over 𝔽_{q^{2}} then the improved inequality again rules out this case.

2. It remains to consider the case that ℓ = 2 and G is obtained from SU_{n}(2^{a}) by adjoining the graph automorphism of order 2. Since V restricts irreducibly to SU_{n}(2^{a}), all factors must have symmetric highest weight. If V is basic, then all coefficients of the highest weight are at most 1, since ℓ = 2. For n = 3 this only leaves Ã(Y_{n}). For n = 4 we find Ã(Y_{n}), ∧^{2}(Y_{n}) and the Steinberg module with weight λ_{1} + λ_{2} + λ_{3}. The dimension 64 of the latter is too large. By Theorem 3.1 we may assume that C_{V} (U) is an example for the Levi complement of any stable parabolic subgroup P = U.L. Induction now proves that for n = 5 we only have to consider Ã(Y_{n}). For n ≥ 6 Lemma 4.3 only leaves Ã(Y_{n}), or ∧^{n/2}(Y_{n}) with n = 6, 8.

First note that V cannot be the adjoint module Ã(Y_{n}). Indeed, if A is an offender, with B = A ∩ SU_{n}(2^{a}) with centralizer of dimension n − c on the natural module, then

2c(n − c) ≤ 2\binom{c + 1}{2} + 2/a

by Proposition 3.12. This implies c ≥ 2/3n − 1, but then the upper bound for the right hand side from Proposition 3.12 eliminates this case. Let V := ∧^{3}(Y_{6}) for the extension G of SU_{6}(q) with the graph automorphism. The centralizers in SU_{6}(q) of both classes of outer involutions are contained in Sp_{6}(q), for which V has two natural composition factors Y_{6} (exchanged by outer elements of G) and one spin module Z_{3}. In particular the centralizer of B := A∩Sp_{6}(q) is contained in a module with composition factors Y_{6} and Z_{3}, and outer involutions have rank at least 6. Thus |A| ≥ q^{3}, and |B| ≥ q^{2}. By Propositions 3.12 and 3.14 such subgroups centralize at most a 4+6-dimensional subspace of Y_{6} ⊕Z_{3}, whence |A| ≥ q^{5}, |B| ≥ q^{4}. Again, such a subgroup has at most 3 + 4-dimensional centralizer on Y_{6} ⊕ Z_{3}, forcing |B| = q^{6} maximal possible. But then B is conjugate to the stabilizer of a totally isotropic subspace on the natural module, by Lemma 3.13, which has just 1-dimensional centralizer on Y_{6}. This would imply |B| > q^{6}, impossible. The case of ∧^{4}(Y_{8}) is now ruled out by induction.

If V is a product of at least two factors, then the usual argument only leaves the possibility n = 4, d_{1} = d_{2} = 6, that is, the V_{i} are natural modules for SO_{6}^{−}(q). This can be ruled out with Proposition 3.14 and (2.1). ⊓ ⊔

4.3. The examples for SL_{n}(q) and SU_{n}(q).

We proceed to verify that the cases left open in the previous proofs do in fact all lead to examples of 2F-modules. Table 7 contains the examples of 2F-modules V for SL_{n}(q) and SU_{n}(q) of dimension d defined over 𝔽_{ℓ^{f}}, up to Frobenius twists and taking duals, together with the size of one possible offender A.

Table 7: 2F-modules for linear and unitary groups in defining characteristic

G  d  V f conditions log_{ℓ} |A|
SL_{n}(ℓ^{a})  n  Y_{n} a  a
SL_{n}(ℓ^{a}) \binom{n}{2}  ∧^{2}(Y_{n}) a  n ≥ 4  (n − 1)a
SL_{n}(ℓ^{a}) \binom{n+1}{2}  Σ^{2}(Y_{n}) a  ℓ ≥ 3  (n − 1)a
SL_{n}(ℓ^{2a}) n^{2} Y_{n} ⊗ Y_{n}^{(a)} a  2⌊n^{2}/4⌋a
SL_{6}(ℓ^{a})  20  ∧^{3}(Y_{6}) a  5a
SU_{n}(ℓ^{a})  n  Y_{n} 2a  a
SU_{4}(ℓ^{a})  6  ∧^{2}(Y_{n}) a  a

Case 4.1: On its natural module Y_{n}, SL_{n}(q) acts as a transvection group, so we get an example with |A| = q.

Case 4.2: When n ≥ 4 the alternating square ∧^{2}(Y_{n}) of the natural module is irreducible for SL_{n}(q). The subgroup

A := ⟨( \begin{matrix}1 \\ & 1 \\ & & 1 \\ a_{1} & … & a_{n−1} & 1 \end{matrix} ) | a_{i} ∈ 𝔽_{q}⟩  (4.1)

of order q^{n−1} visibly has centralizer of dimension at least \binom{n−1}{2} on ∧^{2}(Y_{n}), so this yields an example.

Case 4.3: For ℓ > 2 the symmetric square Σ^{2}(Y_{n}) is irreducible. Again the subgroup A in (4.1) gives an example: its centralizer has dimension at least \binom{n}{2}.

Case 4.4: Let q = ℓ^{2a} and Y_{n}^{(a)} the a-th Frobenius twist of the natural module. Then Y_{n} ⊗ Y_{n}^{(a)} is irreducible for SL_{n}(q) and defined over the subfield 𝔽_{ℓ^{a}} of 𝔽_{q} of index 2. Choosing for A a maximal elementary abelian ℓ-subgroup, condition (1.1) reads

n^{2}a − za ≤ 2 · 2a ⌈\frac{n^{2} − 1}{4}⌉

where z ≥ 1 denotes the dimension of the centralizer of A. Thus this gives an example.

Case 4.5: For n = 6 the alternating cube ∧^{3}(Y_{6}) gives an example for SL_{n}(q) with the offending subgroup A from (4.1): its rank equals 5a, and it centralizes at least the alternating cube of a 5-dimensional subspace, of dimension 10.

Case 4.6: The special unitary group SU_{n}(q) acts as a transvection group on its natural module Y_{n} over 𝔽_{q^{2}}, thus we get an example with |A| = q.

Case 4.7: The 4-dimensional unitary group SU_{4}(q) is isomorphic to the 6dimensional orthogonal group of minus-type. The latter is a bi-transvection group, so we get an example with |A| = q.

5. Spin modules

In order to treat the symplectic and orthogonal groups we now collect some facts on the 2^{n}-dimensional spin module Z_{n} of the orthogonal group Spin_{2n+1}(q), with highest weight λ_{1} (this includes the case of symplectic groups in even characteristic), and of the two 2^{n−1}-dimensional half-spin modules Z_{n−1} of orthogonal groups Spin_{2n}^{(±)}(q) with highest weight λ_{1} or λ_{2}.

5.1. Ranks on spin modules.

For the following lemma we make use of the known classification and closure relation on the set of unipotent classes, which is described for example in the book of Spaltenstein [16]. Let K be the algebraic closure of a finite field 𝔽_{q}.

Lemma 5.1. Let n ≥ 5, 1 ≠ x ∈ G := Spin_{n}(K) be unipotent, V a (half) spin module for G. Then

(a) dim C_{V} (x) ≤ (3/4) dim V, and

(b) dim C_{V} (x) ≤ (5/8) dim V unless n = 8 or x is a long root element.

Proof. Since the spin module for Spin_{2k−1}(K) is the restriction of the half-spin module for Spin_{2k}(K) and the class of root elements of Spin_{2k}(K) intersects Spin_{2k−1}(K) in the class of long root elements, it is sufficient to prove the assertion for n even.

Clearly root elements x of Spin_{4}(K) ≅ SL_{4}(K) have dim C_{V} (x) = (3/4) dim V on the natural module V for SL_{4}(K). Now the half-spin module of Spin_{2k}(K) splits into the two different half-spin modules of Spin_{2k−2}(K) upon restriction, so the first claim follows.

For the remaining part let’s assume that q is even (the case of odd characteristic is similar). Then the unipotent classes of GO_{2k}(K) are parametrized by pairs (λ, ϵ) of a partition λ of 2k in which each even part occurs an even number of times (giving the Jordan blocks in the natural representation) and a map ϵ from the parts of λ to {0, 1}. Furthermore, if the class parametrized by (λ_{1}, ϵ_{1}) lies in the closure of the class parametrized by (λ_{2}, ϵ_{2}) then λ_{1} ≤ λ_{2} in the dominance order on partitions, plus a condition on the ϵ_{i} (see [16, I.2.5]). Now long root elements lie in the class parametrized by 2_{0}^{2}1^{2k−2}, and there are just two unipotent classes directly above this one, parametrized by 2^{2}1^{2k−4} and 2_{0}^{4}1^{2k−8}.

Let’s first consider k = 4. Then the class 2_{0}^{4} splits into two SO_{2k}(K)-classes, and all three classes are fused under triality. By our remarks above, the class 2^{2}1^{4} has rank 2 on the natural module while the other two classes have rank 4. Via triality this implies that 2^{2}1^{4} has rank 4 on the half-spin modules, while the other two classes have rank 2 on one of the half-spin modules, rank 4 on the other. Since all other unipotent classes of SO_{2k}(K) lie above all three of these classes, the assertion is proved for k = 4.

For k ≥ 5 the two classes above 2_{0}^{2}1^{2k−2} don’t split in SO_{2k}(K). The rank can be computed inductively by restricting to SO_{2k−2}(K), and we obtain that 2^{2}1^{2k−4} has rank (1/2) dim V while 2_{0}^{4}1^{2k−8} has rank (3/8) dim V. ⊓ ⊔

Lemma 5.2. Let V be a half-spin module for Spin_{2n}^{+}(q) with n ≥ 3. If |A| > q, then dim C_{V} (A) ≤ (5/8) dim V.

Proof. By Lemma 5.1 it suffices to consider A consisting of root elements. By Lemma 3.9 we see that A ≤ U_{1}, the unipotent radical of N_{G}(X) where X is a root subgroup (this is the stabilizer of a two dimensional totally singular subspace in the natural representation). We may assume that there is a nontrivial element x ∈ A ∩ X. Since |A| > q, we may choose y ∈ A \ X. Then ⟨x, y⟩ is trivial on a space of codimension 4 in the natural representation and so embed in Spin_{8}^{+}(q) × Spin_{2n−8}^{+}(q). Thus, it suffices to consider n = 4 (since the spin module for G is a direct sum of copies of the spin module when restricted to Spin_{8}^{+}(q)). We see by Proposition 3.14 that the fixed space of ⟨x, y⟩ on the spin module is at most 5 dimensional. ⊓ ⊔

The key to studying the spin modules is the existence of maximal parabolic subgroups whose unipotent radical has large centralizer space. Indeed, let G = Spin_{n}^{(±)}(q), and P = UL a maximal parabolic subgroup with U elementary abelian of order |U| = q^{n−2} and L of type Spin_{n−2}^{(±)}(q). Then upon restriction the (half-) spin module V for G splits into two (distinct half-)spin modules for L. Moreover, W := C_{V} (U) is a (half-)spin module for L. Let A ≤ P be a unipotent subgroup of G. Then we have

\begin{aligned}f_{1}(A) & = |A| |C_{V} (A)| ≤ |AU/U| |C_{W} (AU/U)| |A ∩ U| |C_{V} (A ∩ U)/W | \\ & = f_{1}^{W} (AU/U)|A ∩ U| |C_{V} (A ∩ U)/W |, \end{aligned}  (5.1)

and the corresponding equation for f(A).

5.2. Spin modules for even-dimensional orthogonal groups.

Proposition 5.3. On the half spin modules V of Spin_{10}^{+}(q) we have |A|^{\frac{7}{9}} |C_{V} (A)|< |V | for all unipotent elementary abelian subgroups A ≠ 1.

Proof. Let 1 ≠ A ≤ Spin_{10}^{+}(q) with f_{0}(A) := |A|^{\frac{7}{9}} |C_{V} (A)| maximal and assume that f_{0}(A) ≥ |V |. Root elements have rank 4 on V, so |A| ≥ q^{\frac{36}{7}} > q^{5}. In particular A contains non-root elements, of rank at least 6, whence |A| > q^{7}. Let P = UL be a maximal parabolic subgroup containing A with U abelian of order q^{8} and L of type Spin_{8}^{+}(q). Since C_{V} (U) has dimension 8 we have A ≰ U. Assume first that AU/U consists only of root elements, so |AU/U| ≤ q^{3} and |A ∩ U| > q^{4}. Then C_{V} (A) has codimension at least 8 by (5.1), forcing |A| ≥ q^{\frac{72}{7}} > q^{10}, larger than possible. Thus |AU/U| > q^{3}. Via triality we may interpret the spin module W := C_{V} (U) for L as the natural module, and then by Proposition 3.14 the centralizer of AU/U has codimension at least 4 on W. Since A ∩ U ≠ 1, its centralizer has codimension at least 4 as well, whence |A| > q^{10}, the final contradiction. ⊓ ⊔

Proposition 5.4. The half spin modules M(λ_{1}), M(λ_{2}) of Spin_{2n}^{+}(q) are not Fmodules for n ≥ 6.

Proof. Let V denote one of the two half-spin modules for Spin_{12}^{+}(q), A a possible offender on V, and P = UL a maximal parabolic subgroup containing A with U abelian of order q^{10}, L of type Spin_{10}^{+}(q). The ranks in Lemma 5.1 and Proposition 3.9 show that |A| ≥ q^{12}. Thus neither A ≤ U nor A ∩ U = 1. Assume first that |A ∩ U| > q^{5}. Then A ∩ U contains not only root elements, whence C_{V} (A) has codimension at least 4 + 12 = 16 by (5.1) and Lemma 5.1. Thus |A| ≥ q^{16}, too large. Hence we have that |A ∩ U| ≤ q^{5}, and |AU/U| ≥ q^{7}. Then (5.1) gives centralizer codimension at least 6 + 8 = 14. We conclude that |A| ≥ q^{14}, so |AU/U| ≥ q^{9}. But now Proposition 5.3 applied to AU/U ≤ L on W shows that |C_{W} (AU/U)| < q^{9}, so |A| > q^{15}, again too large. The case of n ≥ 7 follows by induction. ⊓ ⊔

Proposition 5.5. The half spin modules M(λ_{1}), M(λ_{2}) of Spin_{2n}^{+}(q) are not 2Fmodules for n ≥ 7.

Proof. Let V denote one of the two half-spin modules for Spin_{14}^{+}(q), A a possible offender on V, and P = UL the stabilizer of a singular vector containing A with U of order q^{12}, L of type Spin_{12}^{+}(q).

We first claim that for any subgroup B ≤ U with |B| > q we have f_{1}(B) ≤ q^{44}. Indeed, choose such a subgroup with f_{1}(B) maximal and subject to that with |B| maximal. Since q < |B| we have |C_{V} (B)| ≤ q^{40} by Lemma 5.2, so f_{1}(B) ≤ q^{43} if |B| ≤ q^{3}. By the maximality assumption, B is a vector space over 𝔽_{q} (this will not change C_{V} (B)), so |B| ≥ q^{4}. Note that either B contains a totally singular 2-space (considering B as an L-module) or B contains a nondegenerate 2-space. Let g ∈ L stabilize this 2-space. We see that f_{1}(BB^{g}) ≥ f_{1}(B) (since |B ∩ B^{g}| ≥ q^{2}) and so by the choice of B, B^{g} = B for any such g. Witt’s theorem implies that either B = U or the 2-space is the radical of B — now choose a different 2-space to see that B = U. Thus, f_{1}(B) = f_{1}(U) = q^{12+32} = q^{44}.

Now let W := C_{V} (U) and set B := A ∩ U. As usual we see from Lemma 5.1 that |A| ≥ q^{12}. Proposition 3.14 implies that B ≠ 1. If A = B, then f(A) = f(U) = q^{56} < |V |.

So B ≠ 1 and AU/U ≠ 1. Now

\begin{aligned}f(A) & ≤ |A| |AU/U| |C_{W} (AU/U)| |A ∩ U| |C_{V} (A ∩ U)/W | \\ & = |A|f_{1}^{W} (AU/U) f_{1}(B)/|W |, \end{aligned}

so Proposition 5.4 implies that f(A) < |A|f_{1}(B) ≤ |A|q^{44}. Thus, |A| > q^{20}. This implies by Lemma 3.16 that A ≤ Q where Q is the unipotent radical of the stabilizer of a maximal totally isotropic space. Thus, f(A) ≤ f(Q). However, C_{V} (Q) has dimension either 1 or 7 (depending upon which class of maximal totally isotropic subspaces we take). Thus, f(Q) ≤ q^{49}, a contradiction. The case of n ≥ 8 follows by induction. ⊓ ⊔

Proposition 5.6. The half spin modules M(λ_{1}), M(λ_{2}) of Spin_{2n}^{−}(q) are not Fmodules for n ≥ 4.

Proof. The half-spin module V for Spin_{6}^{−}(q) is the natural module for SU_{4}(q), defined over 𝔽_{q^{2}}. Proposition 3.11 shows that an offender A with f_{1}(A) ≥ |V | necessarily has |A| = q^{4} maximal possible and thus f_{1}(A) = |V |.

Now let G = Spin_{8}^{−}(q). Long root elements have rank 2 on the half-spin module V, so an offender A has |A| ≥ q^{4}. Application of Lemma 5.2 to the overgroup Spin_{8}^{+}(q^{2}) shows that dim C_{V} (A) ≤ 5, whence |A| = q^{6} is maximal possible. Consider a maximal parabolic subgroup P = UL containing A with U abelian of order q^{6} and Levi complement L of type Spin_{6}^{−}(q). If A ≤ U, then A = U, and Theorem 3.1 gives a contradiction to the F-condition. Otherwise, W := C_{V} (U) is a half-spin module for L, so from (5.1) and the first part of the proof we have

f_{1}(A) < q^{6}q^{2}f_{1}^{W} (AU/U) ≤ q^{16}.

The case of n ≥ 5 now follows by induction using Theorem 3.1. ⊓ ⊔

Proposition 5.7. The half spin modules M(λ_{1}), M(λ_{2}) of Spin_{2n}^{−}(q) are not 2Fmodules for n ≥ 6.

Proof. The half-spin modules V of dimension 32 for Spin_{12}^{−}(q) are defined over 𝔽_{q^{2}}. As usually we see from Lemma 5.1 that |A| ≥ q^{12}. Let U be the unipotent radical of a maximal parabolic subgroup containing A and with Levi subgroup of type Spin_{10}^{−}(q). Note that AU/U ≠ 1, so dim C_{W} (AU/U) ≤ 12. If A ∩ U contains not only long root elements then dim C_{V} (A ∩ U)/W ≤ 4 and |A| ≥ q^{16}, too large. Thus |A ∩ U| ≤ q^{4}, so |AU/U| ≥ q^{8},

dim C_{W} (AU/U) ≤ 10,  and  dim C_{V} (A ∩ U)/W ≤ 8,

whence |A| ≥ q^{14}. But then |AU/U| = q^{10} is maximal possible and only centralizes half of W. This forces |A| ≥ q^{16}, again too large. The case of n > 6 follows by induction using Theorem 3.1. ⊓ ⊔

5.3. Spin modules for odd-dimensional orthogonal groups.

Proposition 5.8. The spin module M(λ_{1}) is not an F-module for Spin_{2n+1}(q), n ≥ 4, q odd.

Proof. On the spin module for Spin_{9}(q), q = ℓ^{a} odd, long root elements have rank 4, and all other non-trivial elements have rank at least 6 by Lemma 5.1. Thus an offender A has |A| ≥ q^{6}. Consider a maximal parabolic subgroup P = UL with U abelian of order q^{7} and L of type Spin_{7}(q). The case that A ≤ U is excluded by Theorem 3.1. Let W := C_{V} (U), the spin module for L. If A ∩ U consists of long root elements only, then |AU/U| ≥ q^{3} by Table 5. But such a group has |C_{W} (AU)| ≤ q^{5} by Lemma 5.1, while |C_{V} (A∩U)/W | ≤ q^{4}. So (5.1) gives |A| ≥ q^{7}, whence |AU/U| ≥ q^{4}. This has centralizer codimension at most 4 on W, forcing |A| ≥ q^{8}, too large. Otherwise, |C_{W} (AU/U)| ≤ q^{6} and |C_{V} (A ∩ U)/W | ≤ q^{2}, which is again impossible. By induction this also excludes the spin module for Spin_{2n+1}(ℓ^{a}), n ≥ 5. ⊓ ⊔

Proposition 5.9. The spin module M(λ_{1}) is not a 2F-module for Spin_{13}(q).

Proof. The half-spin modules of Spin_{14}^{+}(q) restrict irreducibly to the spin module of the subgroup Spin_{13}(q). Thus the result follows from Proposition 5.5. ⊓ ⊔

5.4. Spin modules for symplectic groups.

Our Lemma 5.1 also applies to the spin module of Sp_{2n}(2^{a}), but note that long roots of Spin_{2n+1}(2^{a}) become short roots of Sp_{2n}(2^{a}).

Proposition 5.10. The spin module M(λ_{1}) is not an F-module for Sp_{2n}(q), q even, when n ≥ 4.

Proof. Let V = M(λ_{1}) be the spin module for Sp_{8}(q) of dimension 16. From Lemma 5.1 we find that |A| ≥ q^{4}. By Proposition 3.9 (viewing our group as a group of type B_{n} where the long roots are short roots in type C_{n}), A can’t consist of short root elements only, so there are elements of rank at least 6 and |A| ≥ q^{6}. Let P = UL be a maximal parabolic subgroup containing A with U abelian of order q^{7} and L of type Sp_{6}(q), the normalizer of a long root subgroup X ≤ U. If A ≤ U then f_{1}(A) ≤ f_{1}(U) = q^{7}q^{8} < |V |. On the other hand, if B := A ∩ U = 1 then |AU/U| = |A| = q^{6} is maximal possible. By Lemma 3.13 A is the centralizer in Sp_{6}(q) of a maximal isotropic subspace on Y_{6}, which has 1-dimensional centralizer on the spin module Z_{3}, a contradiction.

So B is a nontrivial proper subgroup of A. Let W := C_{V} (U), the spin module for L. This embeds Sp_{6}(q) into Spin_{8}^{+}(q) on W, which may be viewed as the natural module for Spin_{8}^{+}(q) by triality. In particular Proposition 3.14 gives upper bounds for unipotent subgroups of L with given centralizer on W.

Suppose that |B| ≤ q. Then |AU/U| ≥ q^{5} and so C_{W} (AU/U) has dimension at most 4 and C_{V} (A)/C_{W} (A) has dimension at most 4. Thus, |A| ≥ q^{8} and |A/B| ≥ q^{7}, too large.

So |B| > q. Then Lemma 5.2 implies that

|C_{W} (AU/U)| ≤ q^{6}  and  |C_{V} (A ∩ U)/W | ≤ q^{2},

so |A| ≥ q^{8} and |AU/U| ≥ q. But U/X is the natural module for L, so a subgroup of order q centralizes at most a subspace of size q^{6} of U. Thus |A ∩ U| ≤ q^{6}, implying |AU/U| ≥ q^{2}. But this only centralizes a 5-dimensional subspace, whence |AU/U| ≥ q^{3}. This in turn has at most 5-dimensional centralizer on W, so |A| ≥ q^{9}. This forces |AU/U| ≥ q^{4}, with at most 4-dimensional centralizer on W. Hence |A| ≥ q^{10} and |AU/U| ≥ q^{5}, but then |A ∩ U| ≤ q^{4}, a contradiction.

The case of n ≥ 5 follows by induction applying Theorem 3.1 to the maximal parabolic subgroup stabilizing a 1-space. ⊓ ⊔

6. Symplectic groups

Let Y_{2n} denote the natural 2n-dimensional symplectic module for Sp_{2n}(q) and Z_{n} the 2^{n}-dimensional spin module. We need some information on small basic modules from Lübeck [11, Thm. 4.4]:

Lemma 6.1. Let M(λ) be a basic highest weight module for Sp_{2n}(ℓ^{a}), n ≥ 2, of dimension dim M(λ) ≤ 4n^{2} +2n+1. Then M(λ) is one of the modules in Table 8.

Here the fundamental weights are numberered according to Table 4.

6.1. The case Sp_{4}(q).

It is convenient to handle the case of Sp_{4}(q) ≅ Spin_{5}(q) separately. (Note that the isomorphism interchanges the two fundamental weights λ_{1} and λ_{2}.)

Proposition 6.2. Let V be an absolutely irreducible 2F-modules in characteristic ℓ for a group G with F^{∗}(G) = Sp_{4}(q), q = ℓ^{a} ≠ 2. Then the composition factors of V |_{F^{∗}(G)} are as given in Table 9.

Proof. 1. We start by considering modules for G = Sp_{4}(q). The two end-node parabolic subgroups P_{1} = U_{1}.Sp_{2}(q)(q − 1) and P_{2} = U_{2}.GL_{2}(q) have special respectively abelian unipotent radical of order |U_{i}| = q^{3}. Let U_{0} denote the invariant 1-dimensional subspace of U_{1}.

Table 8: The basic Sp_{2n}(ℓ^{a})-modules of dimension at most 4n^{2} + 2n + 1

λ  M(λ)  dim M(λ)
λ_{n}  Y_{2n}  2n
λ_{n−1} ˜∧^{2}(Y_{2n}) \binom{2n}{2} − 1 − δ_{ℓ,n}
2λ_{n} Σ^{2}(Y_{2n})  \binom{2n+1}{2}  ℓ > 2
λ_{1}  Z_{n}  2^{n}  n ≤ 8, ℓ = 2
λ_{1} + λ_{2}  16 − 4δ_{ℓ,5}  n = 2
2λ_{1}  14 − δ_{ℓ,5}  n = 2, ℓ > 2
3λ_{2} Σ^{3}(Y_{2})  20  n = 2, ℓ > 3
λ_{1}  14  n = 3, ℓ > 2
λ_{1}  42 − δ_{ℓ,3}  n = 4, ℓ > 2
λ_{2}  48 − 8δ_{ℓ,3}  n = 4
λ_{3}  110 − 10δ_{ℓ,2} n = 5

1A. If there exists an offender A ≤ U_{0}, then by Lemma 2.1 we may assume that A = U_{0}. This is a full long root subgroup, five of which generate G. Hence the 2F-condition for a module V of dimension d defined over 𝔽_{ℓ^{f}} reads df/5 ≤ 2a, that is, d ≤ 10a/f. On the other hand, if A is contained in U_{1} but not in U_{0}, then we may choose A = U_{1}. Now two conjugates of A generate G, so we arrive at the condition df/2 ≤ 6a. The same argument applies when A ≤ U_{2}. Hence in any case d ≤ 12a/f. If V = V_{1} ⊗ … ⊗ V_{r} then using that d_{i} = dim(V_{i}) ≥ 4 and f ≥ a/r we get (4)^{r} ≤ 12r. This is only satisfied when r = 2, d_{1}d_{2} ≤ 24 and V_{1}, V_{2} are Galois conjugate, or when r = 1, d ≤ 12. In the first case necessarily d_{1} = d_{2} = 4, which is Example 6.4. If r = 1 and d = 12 then A ∈ {U_{1}, U_{2}} and dim C_{V} (A) = 6. But Lemma 6.1 shows that V = M(λ_{1} + λ_{2}), whence dim C_{V} (A) ≤ 4, too small. On the other hand, if d ≤ 11, then V is one of the modules in Lemma 6.1 which we will consider in 1B.

1B. If there exists an offender contained in none of U_{1}, U_{2}, then it must involve root elements corresponding to both simple roots. But the normal closure of such a subgroup inside the Borel subgroup is the whole unipotent radical. Then |A| = q^{4}, dim C_{V} (A) = 1, so (d − 1)f ≤ 8a. This forces d ≤ 9, and again we are in one of the cases of Lemma 6.1. So V has highest weight λ_{1}, λ_{2} or 2λ_{2}. The first two are Examples 6.1 and 6.2. The highest weight module for 2λ_{2}, ℓ ≠ 2, is the symmetric square of Y_{4}. Let A be an offender with centralizer of codimension c on Y_{4}. Then the centralizer on Σ^{2}(Y_{4}) has codimension at least c(5 − c) by (2.1). Since the rank of A is bounded above by \binom{c+1}{2}a by Proposition 3.12, we conclude that c = 2, |A| = ℓ^{3a}. But then the stronger inequality in Lemma 2.10(b) gives a contradiction.

2. Now assume that G = Sp_{4}(ℓ^{a}).ℓ involves a field automorphism, so ℓ|a. Let first V be a module with irreducible restriction to Sp_{4}(ℓ^{a}). Since basic representations are not invariant this implies that V is a product of r ≥ 2 factors. Then (3.4) yields that r = ℓ = 2 and d_{1} = d_{2} = 4. Let A be a candidate offender with centralizer of dimension 4 − c on the natural module. For these, by Lemma 2.10 and Proposition 3.12 we obtain c = a = 2. But in this case the centralizer of an outer involution is isomorphic to 𝔖_{6} × 2, so has 2-rank 4, and our inequality is violated.

If Ṽ is induced from V, then (3.5) yields that r = 1. If d = 4 then by Proposition 3.12 we have d(ℓ−1)+c ≤ c^{2}+c+2ℓ/a for a subgroup A with centralizer dimension 4 − c. This is satisfied if d = 4, the natural module. Otherwise A has at most d − 2-dimensional centralizer on Ṽ, and we have d(ℓ − 1) ≤ 6, which by Lemma 6.1 gives no further example.

3. Finally we assume that ℓ = 2, 2 |̸a and G = Sp_{4}(2^{a}).2 involves a graphautomorphism. Since we treated Sp_{4}(2) = 𝔖_{6} in the first part [6, Prop. 3.1], we may assume that a ≥ 3. Our standard inequalities allow to rule out this case immediately. ⊓ ⊔

6.2. Some fundamental representations.

We first recall properties of the fundamental representations of the symplectic group. In characteristic 0, the fundamental representations occur as direct summands of exterior powers of the natural representation as follows:

∧^{n+1−k}(Y_{2n}) = M(λ_{k}) ⊕ M(λ_{k+2}) for 1 ≤ k ≤ n

as Sp_{2n}(ℂ)-modules (where M(λ_{k+2}) is the zero-module for k + 2 > n). In positive characteristic, the fundamental representations still occur inside the exterior powers of the natural representation. If the dimensions of the M(λ_{k}) are known, then the composition factors of ∧^{k}(Y_{2n}) can be worked out inductively, using the restriction formula

∧^{k}(Y_{n})|_{GL_{n−1}} = ∧^{k}(Y_{n−1}) ⊕ ∧^{k−1}(Y_{n−1}).

Proposition 6.3. The highest weight module M(λ_{1}) of Sp_{2n}(q), q odd, is not a 2F-module for n ≥ 3.

Proof. Let V = M(λ_{1}) be the irreducible Sp_{6}(q)-module of dimension 14. Note that ∧^{3}(Y_{6}) = V ⊕ Y_{6}. Thus, by Table 3, long root elements have rank 5, elements in Sp_{4}(q) which are not long root elements have rank 6, and elements with 3 Jordan blocks of size 2 have rank 7.

Let P be the stabilizer of a totally isotropic 3-dimensional space in the natural module. So P = QL where L = GL_{3}(q) and |Q| = q^{6}. Consider an L-composition series of V: 1 < F < W < M < V where F and V/M are trivial 1-dimensional modules, W/F is the symmetric square of a natural module for L and M/W is its dual. Moreover, Q ≅ (W/F)^{∗} as L-modules. In particular, if 1 ≠ x ∈ L, then dim C_{W/F} (x) ≤ 3 by (2.1).

Let A be a nontrivial elementary abelian unipotent subgroup with f(A) maximal and assume that f(A) ≥ |V |. If A ≤ Q then f(A) ≤ f(Q) = q^{13} < |V |, impossible. Thus both B := A ∩ Q ≠ 1 and AU/U ≠ 1. Since Q ≅ (W/F)^{∗} this forces |B| ≤ q^{3}. Moreover, since C_{V} (B) ≤ M we find that dim C_{V} (A) ≤ 7, and |A| > q^{3}. In particular, |B| > q, and B contains short root elements. But then dim C_{V} (A)/C_{W} (A) = dim C_{M}(A)/C_{W} (A) ≤ 2, so dim C_{V} (A) ≤ 6 and |A| ≥ q^{4}.

We note also that we can view A as a vector space over 𝔽_{q} (since we may replace the nilpotent part of any element of A by a scalar multiple – these still generate an elementary abelian unipotent subgroup and centralize the same space). In particular, |A| is a power of q.

First assume that AU/U doesn’t consist of root elements of L only. Since C_{V} (B) ≤ M this implies that C_{V} (A) has dimension at most 5. Thus |A| > q^{4}, |B| > q^{2}, contradicting the fact that AU/U only centralizes a 2-dimensional subspace of Q.

So A/B consists of root elements of L. If |B| = q^{3}, we see that C_{V} (B) is at most 6-dimensional (because C_{V} (B)/C_{W} (B) has dimension at most 2 and C_{W} (B) has dimension at most 4).

Suppose that C_{V} (A) = C_{V} (B). Then B′ := ⟨A^{g}|g ∈ Q⟩ also centralizes C_{V} (A). Since [A, Q] ≠ C_{A}(Q) (since A does not act quadratically on Q), B′ ≠ B. The fact that the centralizers are the same on W implies that they generate the same 𝔽_{q}-subspace of Q and so C_{W} (B) has dimension less than 4. Thus, C_{V} (A) has dimension at most 5 and |A/B| = q^{2}. It follows that A/B is the full unipotent radical of a maximal parabolic subgroup of L (and the one that centralizes a 3dimensional subspace of Q). Thus, A/B centralizes a 1-dimensional subspace on W/F and so C_{V} (A) has dimension at most 4. Thus, C_{V} (A) has dimension 4 and |A| = q^{5}. We leave this case for the moment and return to it later.

So |B| < q^{3} and |A| < q^{5}, whence C_{V} (A) has dimension at least 6 and so C_{W} (A) has dimension at least 4. So |B| = q^{2} and |A/B| = q^{2}. It follows that C_{V} (A) = C_{V} (B) is six dimensional – a contradiction as above.

We go back to the remaining case and consider the other parabolic UJ = R = N_{G}(X) where X is a long root subgroup. We see that B does contain a long root subgroup and so we may take X ≤ A < R.

Let B_{1} = A ∩ U. If |B_{1}| > q^{2}, then A ≤ C(B_{1}) is conjugate to Q, a case we have already dealt with. Otherwise, |B_{1}| ≤ q^{2} and so since |A| = q^{5}, we have |B_{1}| = q^{2} and A/B_{1} of order q^{3} in J.

Taking a composition series for J consisting of the orthogonal module, the natural module and the orthogonal module, we see that since X ≤ A, the centralizer of A is contained in the first two. Since |A/B| = q^{3} this is the unipotent radical of the stabilizer of a totally singular 2-space in the natural module of J; this fixes a 2-space in the natural module and a 1-space in the orthogonal module. It follows that C_{V} (A) is 3-dimensional. Thus, f(A) = q^{13}. Thus, V is not an 2F-module. The case of n ≥ 4 follows by induction. ⊓ ⊔

Proposition 6.4. The highest weight module M(λ_{2}) of Sp_{8}(q) is not a 2F-module.

Proof. In characteristic different from 3 we have ∧^{3}(Y_{8}) = M(λ_{2})⊕Y_{8}. Thus from Table 3 we see that transvections have rank 14 on V, and bitransvections (and hence by [16, p.237] any other non-transvections) have rank at least 20. Since A can’t consist of transvections only by Proposition 3.9, this forces |A| = q^{10}, maximal possible. But this means that A is the unipotent radical of a GL_{4}(q)- parabolic, which leads to a contradiction with Theorem 3.1.

In characteristic 3 the wedge cube ∧^{3}(Y_{8}) has composition factors M(λ_{2}) and two copies of Y_{8}. Let P = UL be a maximal parabolic subgroup with Levi factor of type Sp_{6}(q), and special unipotent radical U of order q^{1+6}. Then the restriction of V to the Levi factor has composition factors M(λ_{1}) and two copies of M(λ_{2}) (the heart of the exterior square ∧^{2}(Y_{6})) for Sp_{6}(q). Clearly the centralizer of any element of Sp_{6}(q) on V is at most as large as on the semisimplification of V. For the latter, Table3 shows that transvections have rank at least 13. Hence |A| > q^{6} and A has to contain non-transvections. Any other non-trivial unipotent elements have rank at least 18, whence |A| ≥ q^{9}. Then |A ∩ U| ≥ q^{3}, so this group contains non-transvections. These contribute rank at least 18 on V. On the other hand the image of A in L, of order at least q^{5}, has centralizer codimension at least 4 on the Sp_{6}(q)-module C_{V} (U) = ˜∧^{2}(Y_{6}). This forces |A| ≥ q^{11}, impossible. ⊓ ⊔

Proposition 6.5. The highest weight module M(λ_{3}) of Sp_{10}(q) is not a 2F-module.

Proof. Let first q be odd. Then ∧^{3}(Y_{10}) has composition factors V := M(λ_{3}) and M(λ_{1}) = Y_{10}. As Y_{10} is self-dual for Sp_{10}(q) we have ∧^{3}(Y_{10}) = M(λ_{3})⊕Y_{10}. Thus from Table 3 and Corollary 3.4 we see that transvections have rank 27 on V, and any other non-trivial unipotent element has rank at least 42. With Proposition 3.12 this implies that an offender on V has to consist of long root elements, but this is a contradiction with Proposition 3.9.

In characteristic 2 the wedge cube ∧^{3}(Y_{10}) has composition factors V := M(λ_{3}) and two copies of Y_{10}. In order to determine the centralizers of transvections and bitransvections on V, we restrict to Sp_{8}(q). As an 𝔽_{q}Sp_{8}(q)-module, V has composition factors M(λ_{2}) and two copies of M(λ_{3}). Clearly the centralizer of any element of Sp_{8}(q) on V is at most as large as on the semisimplification. Using the results in the proof of Proposition 6.4 we find that transvections have rank at least 26 on V, and bitransvections (and hence by [16, p.237] any other nontransvections) have rank at least 40. We may now conclude as before. ⊓ ⊔

Let G := Sp_{2n}(q), n ≥ 2. Let E_{n} denote the codimension 1 submodule of the exterior square of the natural module for G. This is either an irreducible module or contains a 1-dimensional trivial module (the latter occurring precisely when ℓ|n). Let V be the nontrivial irreducible composition factor of E_{n}. In particular, dim V = 2n^{2} − n − δ where δ = 1 unless ℓ|n in which case δ = 2.

Let P be the parabolic subgroup of G fixing a 1-space in the natural representation. Then P = N_{G}(X) where X is a long root subgroup. Let U be the unipotent radical of P. Write P = LU where L is the Levi complement.

Note that U/X is the natural L-module and that U is special if ℓ is odd and is abelian if q is even.

Let W = C_{V} (U), W′ = C_{V} (U^{opp}) and M = C_{V} (X). We see that W and W′ are natural L-modules and that M/W is isomorphic to ∧^{2}(W) as an [L, L]-module if ℓ does not divide n and is isomorphic to E_{n−1} otherwise. By (2.1) any nontrivial unipotent element of G has rank at least 2n − 2 on V.

Lemma 6.6. Let B be a nontrivial subgroup of U. Then (C_{V} (B) + M)/M has dimension at most 1. If B ∩ X ≠ 1 or |B| > q, then C_{V} (B) ≤ M. In particular, if A is a subgroup of P and A ∩ U ≠ 1, then dim C_{V} (A)/C_{M}(A) ≤ 1.

Proof. Note that if 1 ≠ x ∈ X, then C_{V} (x) = C_{V} (X) = M and the result is clear. Note also that all elements in U \ X are conjugate in P. So we can choose any such element x.

Choose a basis for the natural module for G of the form e, f, v_{1}, …, v_{n−2} where e generates the space fixed by P and the v_{i} are in e^{⊥}. Now choose x ∈ U with xf = f + v_{n−2}. Let z ∈ C_{V} (B) \ M. So z = f ∧ v + m where m ∈ M. Thus, z = xz = xf ∧ v + m + e ∧ v′, where v′ ∈ e^{⊥}. Thus, z = (f + v_{n−2}) ∧ v + m. This implies that v_{n−2} ∧ v = 0. Thus, v is a multiple of v_{n−2} (modulo e) and the result follows. If |B| > q, the same argument shows that C_{V} (B) ≤ M.

This proves the first statement. The second statement follows for

C_{V} (A)/C_{M}(A) ≅ (C_{V} (A) + M)/M ⊆ (C_{V} (A ∩ U) + M)/M.
⊓ ⊔

Lemma 6.7. Let A be an elementary abelian subgroup of U. If |AX| ≥ q^{d+1}, then C_{M}(A)/W has dimension at most c(c − 1)/2 where c = 2n − 2 − d.

Proof. Fix y ∈ U \ X. Let W_{1} be the fixed hyperplane of y on W. Fix w′ ∈ W not in W_{1} with yw′ = w′ + e. Identifying M/W with the exterior square of W, we see that y − 1 maps w′ ∧ w to e ∧ w. Thus, the image of y − 1 is precisely e ∧ W_{1} and has dimension 2n − 3. Moreover, we see that the fixed space of y on M/W is the exterior square of W_{1}.

So we see that if y_{1}, …, y_{d} are linearly independent elements in U/X, then the centralizer of the subgroup generated by them (modulo W) is isomorphic to the exterior square of a c-dimensional subspace of W, whence the result. ⊓ ⊔

Lemma 6.8. Let A be an elementary abelian unipotent subgroup of G := Sp_{2n}(q). Assume that n ≥ 3 and |A| ≤ q^{n(n−1)/2}. Then f(A) < |V |.

Proof. The fundamental inequality from (2.1) and Proposition 3.12 for an offender A on V with centralizer codimension c on the natural module reads

c(2n − c − 1) ≤ 2\binom{c + 1}{2},

so c ≥ n − 1 and |A| ≥ q^{n(n−1)/2}. In particular A contains non-trivial elements with different centralizers, and the stronger inequality from Lemma 2.10(c) forces |A| > q^{n(n−1)/2}. ⊓ ⊔

Lemma 6.9. Let G = Sp_{6}(q). If V is the 14 dimensional irreducible (indecomposable in characteristic 3) with highest weight λ_{2}, then

1. V is not an F-module; and

2. f(A) ≤ q^{15} for any elementary abelian unipotent subgroup 1 ≠ A of G.

Proof. By the previous result, f(A) < |V | if |A| ≤ q^{3}. Note that f(Q) = q^{15}. So this inequality holds for any subgroup A of Q as well (since either |A| ≤ q^{3} or the weak closure of A in Q is Q and so f(A) ≤ f(Q)).

Similarly any subgroup A of U satisfies f(A) ≤ f(U) = q^{14}. Indeed, if q is odd, then |A| ≤ q^{3} and so f(A) < q^{14}. Moreover, we can replace A by the 𝔽_{q}-subspace it generates (this will not change the centralizer in V) and so f(A) ≤ q^{13} for q odd. The same argument applies if q is even and |A| ≤ q^{3}.

Let A be an abelian unipotent subgroup of G with f(A) maximal. So |A| > q^{3}. Set B = A ∩ U. If B = 1, then A embeds in L and so |A| ≤ q^{3}, a contradiction. So B ≠ 1. By Lemma 6.6 this implies that we may assume that X ≤ B (for

either A ∩ X = 1 and C_{V} (A)/C_{V} (AX) has dimension at most 1 or C_{V} (A) = C_{V} (AX)). Similarly, we may assume that B is an 𝔽_{q}-subspace of U.

If |B| > q^{3}, then q is even and C_{L}(B) has order at most q (the subspace generated by B will contain a nondegenerate space). Thus, f(A) ≤ q^{2}f(B). Now C_{W} (A) has dimension at most 3 and C_{W} (B) = W, so in fact f(A) ≤ qf(B) ≤ q^{15}. The same argument applies unless B/X is totally isotropic (which is always the case for q odd).

So B/X is totally isotropic. In particular, f(B) ≤ q^{13}. If |A| ≤ q, this implies that f(A) ≤ q^{2}f(B) ≤ q^{15}. So AU/U centralizes at most a 2-dimensional space on W. So f(A) ≤ |A/B|^{2}f(B)/q^{2} ≤ q^{15} unless |A/B| > q^{2}. In the latter case, we see that the 𝔽_{q}-subspace generated by A/B in L is the unipotent radical of the stabilizer of a totally singular 2-space (in the natural module of L). It follows that A/B centralizes only a 1-dimensional space in the orthogonal module M/W. So C_{V} (A) has dimension at most 3, whence f(A) ≤ q^{12}q^{3} = q^{15}.

This proves the second statement. Since f_{1}(A) = f(A)/|A|, the only possibility for f_{1}(A) ≥ |V | is that |A| ≤ q and centralizes a hyperplane. Since G contains no transvections on V, this cannot occur. ⊓ ⊔

Proposition 6.10. Let A be an elementary abelian unipotent subgroup of G := Sp_{2n}(q). Assume that n ≥ 4. Then f(A) < |V | unless possibly f(A) = |V | when n = 4 and q is even.

Proof. Choose A with f(A) maximal and assume the contrary. We keep notation as above. In particular, W and W′ are the spaces centralized by U and U^{opp} and M = C_{V} (X). Set V′ = M/W and as we have remarked V′ is either the exterior square or the irreducible constituent of the exterior square of the natural module for L.

We induct on n. We may assume that A ≤ P, and, by Lemma 6.8, that |A| > q^{n(n−1)/2}. Set B = A ∩ U.

Suppose that A centralizes some nondegenerate subspace of the natural module or more generally A∩U = 1. There is no loss of generality in assuming that A ≤ L. Thus, f(A) ≤ |A|^{2}|C_{V′}(A)||W |^{2}/q^{2}. This is less than |V | by induction if n > 4 and by the previous lemma if n = 4.

Suppose that A ≤ U. If A ≤ X, then f(A) ≤ f(X) < |V |, a contradiction. Otherwise, f(A) ≤ f(U) = |U|^{2}q^{2n−2} = q^{6n−4} < |V |, also a contradiction.

If |B| ≤ q, then C_{V} (A)/C_{M}(A) is at most 1-dimensional by Lemma 6.6. So |AU/U|^{2}|C_{V} (A)| ≤ q|AU/U|^{2}|C_{V′}(AU/U)||W |/q and so

f(A) ≤ |AU/U|^{2}|C_{V′}(AU/U)||W |q^{2} < |V |

(we are using induction if n > 4 and the previous lemma if n = 4).

So assume that |B| > q. In particular, C_{V} (A) = C_{M}(A).

If |B| > q^{n} or more generally the space generated by BX/X contains a nondegenerate subspace (of dimension 2), then |A/B| ≤ q^{(n−2)(n−3)/2}. Thus, |A| ≤ q^{n+1}q^{(n−2)(n−3)/2} ≤ q^{n(n−1)/2}, a contradiction (since n ≥ 4).

So BX/X is totally singular. On the other hand, |A| > q^{n(n−1)/2}, whence |A/B| > q^{n(n−3)/2}. It follows that the fixed space of A/B on W is totally singular and so has dimension at most n.

Plugging this into our formula above yields

f(A) ≤ |AU/U|^{2}|C_{V′}(AU/U)|q|B|^{2}q^{n−1} < |V |,

unless |B|^{2} ≥ q^{3n−4}. However, |B| ≤ q^{n}, a contradiction unless n = 4. If q is even, this still gives the desired conclusion that f(A) ≤ |V |.

The remaining case is n = 4, |B| = q^{4} and q is odd. So we see that B/X is a maximal totally isotropic subspace of U/X. Thus, A ≤ C_{P} (B) is contained in a conjugate of Q. Since f(Q) < |V |, the result follows. ⊓ ⊔

6.3. 2F-modules for Sp_{2n}(q).

Theorem 6.11. The absolutely irreducible 2F-modules in defining characteristic for Sp_{2n}(q), q = ℓ^{a}, n ≥ 3, are as given in Table 9.

Proof. Let P_{1} = U_{1}.Sp_{2n−2}(q)(q − 1) and P_{2} = U_{2}.GL_{n}(q) denote the two endnode parabolic subgroups, with unipotent radicals of order |U_{1}| = q^{2n−1} and |U_{2}| = q^{\binom{n+1}{2}}.

1. First assume that there exists an offender A ≤ U_{1}. If A lies in the invariant 1-dimensional Sp_{2n−2}(q)-subspace U_{0} of U_{1} (that is, in the center of U_{1} if ℓ is odd), then by Lemma 2.1 we may assume that A = U_{0}. This is a full long root subgroup. But 2n + 1 conjugates of it generate Sp_{2n}(q), so the 2F-condition for a module V of dimension d defined over 𝔽_{ℓ^{f}} reads df/(2n + 1) ≤ 2a, that is, d ≤ 2(2n + 1)a/f. Otherwise, if A is not contained in U_{0}, then again by Lemma 2.1 we may assume A = U_{1}. Now two conjugates of A generate Sp_{2n}(q), so we arrive at the condition df/2 ≤ 2(2n − 1)a. Hence in any case d ≤ 4(2n − 1)a/f. If V = V_{1} ⊗ … ⊗ V_{r} then using that d_{i} = dim(V_{i}) ≥ 2n and f ≥ a/r we get

(2n)^{r} ≤ 4(2n − 1)r.

This is only satisfied when either r = 2, n = 4, or r = 1. In the first case, we necessarily have f = a/r, so V_{1}, V_{2} are Galois conjugate, and d = 2n. Inspection shows that f(U_{0}) < |V | and f(U_{1}) < |V |. If r = 1 then we saw that d ≤ 4(2n − 1), so V is a twist of one of the modules in Table 8. The ones not occurring in the conclusion are the spin module for n = 6, which was already ruled out in Proposition 5.9, and the modules with highest weights 2λ_{n} or λ_{1} for n = 3, ℓ > 2. Explicit computations show that U_{0} is not a centralizer. Similarly, computing the centralizer of U_{1} via Theorem 3.1 shows that U_{1} is not an offender on M(2λ_{n}). On the other hand, the non-abelian group U_{1} is an offender on M(λ_{1}) for n = 3; this case was excluded in 6.3.

2. From now on we may assume that there is no offender in U_{1}, hence by Proposition 2.4 the centralizer C_{V} (U_{1}) is a 2F-module for Sp_{2n−2}(q). These are known by induction starting with the case of Sp_{4}(q) in Proposition 6.2. In particular, the restriction of the highest weight of V to the first n−1 nodes is known. If V is basic, then it occurs in Table 8. The symmetric square does not lead to an example, as can be seen by using Lemma 2.10(b), as in the proof of the previous proposition. The other modules are either excluded in Proposition 5.9, Propositions 6.3-6.5 or in Proposition 6.10 or occur in the conclusion.

The case of tensor products V = V_{1} ⊗ … ⊗ V_{r}, r ≥ 2, of twists V_{i} of basic modules of dimension d_{i} leads to the inequality (3.4)

d_{2} ⋯ d_{r} ≤ rn(n + 1).

Since d ≥ 2n this forces r ≤ 3. The case r = 3 can only hold when n = 3, d = 2n = 6, and the three modules form a Galois orbit. Moreover, A necessarily has d − 1-dimensional centralizer on V_{i} by Lemma 2.10. But then the rank of A is too small by Proposition 3.12.

3. Thus r = 2. First assume that V_{1}, V_{2} are Galois conjugate, hence of the same dimension d. If V_{i} are not twists of the natural module, then any non-trivial unipotent element has rank at least 2 on V_{i}, so we obtain the sharper inequality d ≤ n(n + 1). This is only satisfied when ℓ = 2, n ≤ 4, and V_{i} are spin modules of dimension d = 2^{n}. If n = 3 then (2.1) together with Lemma 5.1 shows that short and long root elements have rank at least 24 on V_{1} ⊗ V_{2}, so an offender has size |A| ≥ q^{6}, maximal possible. So A is the stabilizer of a maximal isotropic subspace, with centralizer of dimension 1, too small. The case n = 4 is ruled out by induction. Thus V_{1}, V_{2} are Galois conjugate twists of the natural module. By Proposition 3.12 and Corollary 2.11(a) we obtain the inequality

max\{2nc, 2c(2n − c)\} ≤ 4\binom{c + 1}{2}

for the codimension c of the centralizer on the natural module of a possible offender A on V := Y_{2n} ⊗ Y_{2n}^{(a)}. Thus c ≥ (2n − 1)/2 > n − 1 and |A| ≥ q^{n^{2}/2}. Moreover, if c > n then |A| = q^{\binom{n+1}{2}} is maximal, so A equals the unipotent radical U_{2} of P_{2}. Similarly, if c = n then Proposition 3.12 shows that C_{Y_{2n}}(A) must be maximally isotropic and again A ≤ U_{2}. But the centralizer of U_{2} on V is only of dimension n^{2}.

4. So we are left with the case that V is the tensor product of r = 2 factors which are not conjugate. Then d_{2} ≤ n(n + 1), so either d_{1} = d_{2} = 2n, or ℓ = 2, n ≤ 4 and d_{2} = 2^{n}. The first case is certainly not an example, since even in the Galois conjugate case it wasn’t. In the cases involving a spin module, dim V is larger than the bound (3.3). ⊓ ⊔

Proposition 6.12. Let V be an absolutely irreducible 2F-module in characteristic ℓ for a group G with F^{∗}(G) = Sp_{2n}(q), q = ℓ^{a}, n ≥ 3. Then all composition factors of the restriction V |_{F^{∗}(G)} are 2F-modules for F^{∗}(G).

Proof. Field automorphisms are the only outer automorphisms of Sp_{2n}(q), n ≥ 3, so assume that G = Sp_{2n}(ℓ^{a}).ℓ involves a field automorphism. If V has reducible restriction to F^{∗}(G) then the standard argument shows that the claim holds. So now let V be a module with irreducible restriction to Sp_{2n}(ℓ^{a}). Since basic representations are not invariant this implies that V is a product of r ≥ 2 factors. Then (3.4) yields that r = ℓ = 2 and d_{1} = d_{2} = 2n. Let A be a candidate subgroup with centralizer of dimension 2n − c on the natural module. For these, by Lemma 2.10 and Proposition 3.12 we obtain n = c = a = 2, which is excluded here. ⊓ ⊔

6.4. The examples for Sp_{2n}(q).

The conventions for the following table are as for Table 7.

Table 9: 2F-modules for symplectic groups in defining characteristic

G  d  V f  log_{ℓ} |A|
Sp_{2n}(ℓ^{a})  2n  Y_{2n} a  a
Sp_{2n}(ℓ^{a}) \binom{2n}{2} − 1 − δ_{ℓ,n}  ˜∧^{2}(Y_{2n}) a  n = 2, 3  \binom{n+1}{2}a
or ℓ = 2, n = 4  10a
Sp_{2n}(2^{a})  2^{n}  Z_{n} a  n = 3, 4, 5  (2n − 1)a
Sp_{4}(ℓ^{2a})  16  Y_{4} ⊗ Y_{4}^{(a)} a  3a

Case 6.1: The symplectic group Sp_{2n}(q) acts as a transvection group on its natural module Y_{2n} over 𝔽_{q}, thus we get an example with |A| = q.

Case 6.2: The irreducible representation with highest weight λ_{n−1} of Sp_{2n}(ℓ^{a}) is the heart of the exterior square of the natural representation. Consider the maximal parabolic subgroup of type GL_{n}. The centralizer of its unipotent radical U has weight λ_{n−2}, hence dimension \binom{n}{2}. The unipotent radical U is elementary abelian of ℓ-rank a\binom{n+1}{2}. Thus we find an example when n ≤ 3. In the case n = 2 we have an alternative interpretation: The 4-dimensional symplectic group is isomorphic to the 5-dimensional orthogonal group. The latter is a bi-transvection group on its natural module, so we get an example for Sp_{4}(q) in dimension 5, when ℓ ≠ 2. When ℓ = 2 then this representation is reducible, and its irreducible constituent of dimension 4 yields an example.

Case 6.3: The spin module Z_{n} for Sp_{2n}(2^{a}) has highest weight λ_{1}. Consider the maximal parabolic subgroup of type Sp_{2n−2}. The centralizer of its unipotent radical U is again the spin module for Sp_{2n−2}(2^{a}), hence has dimension 2^{n−1}. But U is elementary abelian of rank (2n − 1)a. Thus U is an offender on Z_{n} if 2^{n} − 2^{n−1} ≤ 2(2n − 1), so for n ≤ 5.

Case 6.4: The tensor product of the natural module Y_{4} for Sp_{4}(ℓ^{2a}) with its twist Y_{4}^{(a)} is defined over 𝔽_{ℓ^{a}}. A maximal elementary abelian ℓ-subgroup A of order ℓ^{3a} has 2-dimensional centralizer on Y_{4}, hence at least 4-dimensional centralizer on Y_{4} ⊗ Y_{4}^{(a)}. Thus A is an offender.

7. Orthogonal groups

In this section we consider orthogonal groups Spin_{n}^{(±)}(q) of dimension n ≥ 7, defined over fields of odd order if n is odd. By the results on linear, unitary and symplectic groups in the previous sections and well-known isomorphisms this suffices to cover all finite quasi-simple orthogonal groups.

7.1. The case Spin_{2n+1}(q).

We start by recording the basic modules of small dimension from [11, Thm. 4.4].

Lemma 7.1. Let n ≥ 3, ℓ > 2, and M(λ) be a basic highest weight module for Spin_{2n+1}(ℓ^{a}) of dimension dim M(λ) ≤ 4n^{2} + 2n + 1. Then M(λ) is one of the modules in Table 10.

Table 10: The basic Spin_{2n+1}(ℓ^{a})-modules, n ≥ 3, ℓ > 2, of dimension at most 4n^{2} + 2n + 1

λ  M(λ)  dim M(λ)
λ_{n}  Y_{2n+1}  2n + 1
λ_{n−1} ∧^{2}(Y_{2n+1})  \binom{2n+1}{2}
2λ_{n} Σ̃^{2}(Y_{2n+1}) \binom{2n+2}{2} − 1 − δ_{ℓ,2n+1}
λ_{1}  Z_{n}  2^{n}  n ≤ 8
2λ_{1}  35  n = 3
λ_{1} + λ_{3}  40  n = 3, ℓ = 7

Theorem 7.2. The absolutely irreducible 2F-modules in defining characteristic ℓ > 2 for Spin_{2n+1}(q), q = ℓ^{a}, n ≥ 3, are as given in Table 12.

Proof. Let V be a 2F-module for G = Spin_{2n+1}(q), with offender A.

1. Let P_{1} = U_{1}.L_{1} denote the end node parabolic subgroup of G with Levi factor L_{1} of type Spin_{2n−1}(q). Then U_{1} is abelian of order q^{2n−1} and irreducible under L_{1}. Thus, if A ≤ U_{1} then by Lemma 2.1 we may assume that A = U_{1}. By Proposition 3.7, the 2F-condition for V implies dim V ≤ 4(2n − 1)a/f if V is defined over 𝔽_{ℓ^{f}}. By Lemma 7.1 any basic representation of G has dimension at least 2n + 1. This shows that V cannot be a tensor product of at least two factors. But otherwise, V is (a Frobenius twist of) one of the modules in Table 10 of dimension at most 4(2n − 1), and all these occur in the conclusion.

2. Thus, by Proposition 2.4 we may assume that C_{V} (U_{1}) is a 2F-module for Spin_{2n−1}(q). First assume that V is a basic representation, hence its highest weight is contained in Table 10. For Σ̃^{2}(Y_{2n+1}) the basic inequality from (2.1) and Proposition 3.14 gives a contradiction. Similarly for V = ∧^{2}(Y_{2n+1}) it gives n = 3, and |A| ≥ q^{4}. But then A contains non-trivial elements with at least two different centralizers and Lemma 2.10(c) yields a contradiction. The spin module Z_{6} for n = 6 is not an example by Proposition 5.9. By step 1 this also rules out the spin modules for n ≥ 7. The module with highest weight 2λ_{1} for n = 3 is not an example since the centralizer of U_{1}, with the same highest weight, is not an example for Spin_{5}(q) ≅ Sp_{4}(q) by Proposition 6.2. Explicit construction of the 40-dimensional module with highest weight λ_{1} + λ_{3} as a submodule of the tensor product of the natural module with the spin module shows that long root elements have 18-dimensional commutator space, hence by Corollary 3.4 and Proposition 3.14 this is not an example.

3. Next assume that V = V_{1} ⊗ ⋯ ⊗ V_{r} is a tensor product of r ≥ 2 factors of dimensions d = d_{1} ≤ d_{2} ≤ … ≤ d_{r}. Since G is not a transvection group, the commutator of any non-trivial element in any non-trivial representation is at least 2-dimensional. Then (3.4) shows that

d^{r−1} ≤ \frac{1}{2}r(n^{2} − n + 4).

This forces r = 2, V_{1}, V_{2} are Galois conjugate, and either d_{1} = d_{2} = 2n + 1, or n = 3, d_{1} = d_{2} = 8. Both cases are eliminated by the inequalities from (2.1) together with Lemma 3.14. ⊓ ⊔

Proposition 7.3. Let V be an absolutely irreducible 2F-module in characteristic ℓ > 2 for a group G with F^{∗}(G) = Spin_{2n+1}(q), q = ℓ^{a}, n ≥ 3. Then all composition factors of the restriction V |_{F^{∗}(G)} are 2F-modules for F^{∗}(G).

Proof. We only have to care about field automorphisms. For induced modules the assertion is immediate from Proposition 2.5(i) since ℓ ≠ 2. If V has irreducible restriction to F^{∗}(G), then its highest weight is invariant under the field automorphism, in particular it is not basic. So V is a tensor product of r ≥ 2 factors. The standard inequality (3.4) then shows that r = 2 and ℓ = 2, which is excluded here. ⊓ ⊔

7.2. The case Spin_{2n}^{+}(q).

Again we first collect the basic modules of small dimension from [11, Thm. 4.4].

Lemma 7.4. Let n ≥ 4, let M(λ) be a basic highest weight module for Spin_{2n}^{+}(ℓ^{a}) of dimension dim M(λ) ≤ 4n^{2} − 2n + 4. Then M(λ) is one of the modules in Table 11.

Table 11: The basic Spin_{2n}^{+}(q)-modules of dimension at most 4n^{2} − 2n + 4

λ  M(λ)  dim M(λ)
λ_{n}  Y_{2n}  2n
λ_{n−1} ∧^{2}(Y_{2n}) \binom{2n}{2} − (n, 2)δ_{ℓ,2}
2λ_{n} Σ̃^{2}(Y_{2n}) \binom{2n+1}{2} − 1 − δ_{ℓ,n} ℓ > 2
λ_{1}, λ_{2}  Z_{n−1}  2^{n−1}  n ≤ 9
λ_{1} + λ_{2}, λ_{1} + λ_{4}, λ_{2} + λ_{4}  56 − 8δ_{ℓ,2}  n = 4

Theorem 7.5. The absolutely irreducible 2F-modules in defining characteristic for Spin_{2n}^{+}(q), q = ℓ^{a}, n ≥ 4, are as given in Table 12.

Proof. 1. Let’s first handle some particular cases. The natural module Y_{2n} has highest weight λ_{n} and gives Example 7.1. The highest weight λ = (1 + ℓ^{a/2})λ_{n} belongs to the tensor product Y_{2n} ⊗ Y_{2n}^{(a/2)}. The 2F-condition for an offender with centralizer of codimension c on Y_{2n} then reads 2c(2n − c)/2 ≤ 2\binom{c}{2} by (2.1) and Lemma 3.14, which implies c > n, but then the upper bound for the rank in Lemma 3.14 is violated. For the heart of the symmetric square Σ̃^{2}(Y_{2n}) with highest weight 2λ_{n} we similarly obtain c(2n−c+1)−2 ≤ 2\binom{c}{2}, again not possible. The highest weight module for λ = λ_{n−1} is the heart of the exterior square of Y_{2n}. Lemma 3.14 and (2.1) show that c(2n − c − 1) − 2 ≤ 2\binom{c}{2}. This forces c ≥ n, A of maximal rank. But then A contains elements with different centralizers, and Lemma 2.10(c) gives a contradiction. The half spin modules for n ≤ 6 with highest weight λ_{1} or λ_{2} are Example 7.3. The half-spin modules Z_{6} for Spin_{14}^{+}(q) were excluded in Proposition 5.5.

2. Let P_{1} = U_{1}.L_{1} be a maximal parabolic subgroup of G corresponding to the last node of the Dynkin diagram, that is, with Levi factor of type Spin_{2n−2}^{+}(q). Then U_{1} is the natural module for L_{1}, hence irreducible. If A is contained in U_{1}, we may take A = U_{1}. Two conjugates generate G, hence df/2 ≤ 2(2n − 2)a for a 2F-module V defined over 𝔽_{ℓ^{f}} with offender A. If V is the tensor product of r factors V_{i}, of dimensions d_{i} = dim V_{i} ≥ 2n, then we get (2n)^{r} ≤ 8(n − 1)r. This shows that r = 1, and thus d < 2n^{2}. Hence V is a twist of one of the basic modules in Table 11 all of which were treated in the first part.

3. If U_{1} does not contain an offender, we may assume by Proposition 2.4 that V is an example for the Levi factor L_{1}, hence for Spin_{2n−2}^{+}(q). Let first n = 4, so L_{1} contains the linear group Spin_{6}^{+}(q) ≅ SL_{4}(q). Then we may repeat the above argument for all three end nodes of the Dynkin diagram. By Proposition 4.7 this only leaves the possibilities

λ_{4}, λ_{3}, 2λ_{4}, (1 + ℓ^{a/2})λ_{4}

and their twists and images under the graph automorphisms for λ. These were investigated in the first part.

4. Now let n ≥ 5 and P_{2} = U_{2}.L_{2} a maximal parabolic subgroup with L_{2} = GL_{n}(q), U_{2} the exterior square of the natural module for L_{2}. If A ≤ U_{2}, then d ≤ 4\binom{n}{2}a/f by the standard argument. This forces V to be a twist of a basic module, of dimension d < 2n^{2}, hence one of the cases in step 1. Otherwise, V is an example for SL_{n}(q). This, together with induction starting from step 3 shows that λ is one of

tλ_{n} (t = 1, 2, 1 + ℓ^{a/2}), λ_{1} (n ≤ 7), λ_{n−1}

up to twists and the graph automorphism of order 2. These were considered in the first part of the proof. ⊓ ⊔

7.3. The case Spin_{2n}^{−}(q).

Theorem 7.6. The absolutely irreducible 2F-modules in defining characteristic for Spin_{2n}^{−}(q), q = ℓ^{a}, n ≥ 4, are as given in Table 12.

Proof. 1. The natural module is Example 7.1. The cases Y_{2n} ⊗Y_{2n}^{(a/2)} and Σ̃^{2}(Y_{2n}) are excluded precisely as for orthogonal groups of plus-type. The spin modules for n ≤ 5 are Example 7.4. The half-spin modules for Spin_{2n}^{−}(q), n ≥ 6, were excluded in Proposition 5.7. The adjoint module ˜∧^{2} is excluded using Proposition 3.14 in conjunction with (2.1).

2. Let V be a 2F-module for G = Spin_{2n}^{−}(q) and A an offender on V. Let P_{1} = U_{1}.L_{1} be the end node parabolic with Levi complement of type Spin_{2n−2}^{−}(q). Then arguing as in step 2 of the proof for Spin_{2n}^{+}(q) we find that either V is one of the modules treated in step 1, or C_{V} (U_{1}) is an example for L_{1}.

3. Assume that V is a basic module M(λ). Then dim V ≤ 4n^{2}−2n+4 by (3.3), and dim V ≤ 2n^{2} − n + 2 if the highest weight λ is not invariant under the graph automorphism exchanging nodes 1 and 2. Thus

λ ∈ {λ_{n}, λ_{n−1}, 2λ_{n}}, or λ = λ_{1}, λ_{2} for n ≤ 7, or λ = λ_{1} + λ_{2} for n = 4

by Proposition 7.4. All except the last case were considered in step 1. The weight λ_{1} + λ_{2} does not lead to an example by step 2, since C_{V} (U_{1}) is not an example for Spin_{6}^{−}(q) ≅ SU_{4}(q) by Theorem 4.10.

4. If V is a tensor product of r ≥ 2 factors, then (3.4) shows that r = 2, both factors have dimension 2n and are Galois conjugate. Exactly the same calculation as for Spin_{2n}^{+}(q) allows to rule out this case. ⊓ ⊔

7.4. The examples for Spin_{n}^{(±)}(q).

In the first line of the following table, ℓ is odd when n is odd.

Table 12: 2F-modules for orthogonal groups in defining characteristic

G  d  V f  log_{ℓ} |A|
Spin_{n}^{(±)}(ℓ^{a})  n  Y_{n} a n ≥ 7  a
Spin_{2n+1}(ℓ^{a}) 2^{n}  Z_{n} a n = 3, 4, 5 (2n − 1)a
Spin_{2n}^{+}(ℓ^{a})  2^{n−1} Z_{n−1} a n = 4, 5, 6 (2n − 2)a
Spin_{2n}^{−}(ℓ^{a})  2^{n−1} Z_{n−1} 2a n = 4, 5 (2n − 2)a

Case 7.1: The orthogonal group Spin_{n}^{(±)}(q) acts as a bi-transvection group on its natural module, hence we obtain an example with offender of order q.

Case 7.2: The spin module Z_{n} for Spin_{2n+1}(ℓ^{a}) has highest weight λ_{1}. Consider the maximal parabolic subgroup of type Spin_{2n−1}. The centralizer of its unipotent radical U is again the spin module for Spin_{2n−1}(ℓ^{a}), hence has dimension 2^{n−1}. But U is elementary abelian of rank (2n − 1)a. Thus U is an offender on Z_{n} if 2^{n} − 2^{n−1} ≤ 2(2n − 1), so for n ≤ 5.

Case 7.3: For the half-spin modules Z_{n−1} of dimension 2^{n−1} of G = Spin_{2n}^{+}(q) let A be the unipotent radical of the maximal parabolic subgroup with Levi complement of type Spin_{2n−2}^{+}(q). Then the centralizer C_{V} (A) is a half-spin module for Spin_{2n−2}^{+}(q), of dimension 2^{n−2}, while A is elementary abelian of order q^{2n−2}, so we find an example whenever n ≤ 6.

Case 7.4: The half-spin modules Z_{n−1} of dimension 2^{n−1} of G = Spin_{2n}^{−}(q) are defined over 𝔽_{q^{2}}. Let A be the unipotent radical of the maximal parabolic subgroup with Levi complement of type Spin_{2n−2}^{−}(q). Then the centralizer C_{V} (A) is a spin module for Spin_{2n−2}^{−}(q), of dimension 2^{n−2}, while A is elementary abelian of order q^{2n−2}, so we find an example whenever n ≤ 5.

8. Exceptional groups

For exceptional groups we make use of the table of ℓ-ranks m_{ℓ} in [5, Table 3.3.1], which we reproduce here as Table 13 for the convenience of the reader, and the determination of small basic modules in [11, Thm. 4.4]. Moreover, we employ the tables in Lawther [9] on Jordan shapes of unipotent elements of exceptional groups on small modules in conjunction with information on the closure relation on the set of unipotent classes as can be found in Spaltenstein’s book [16], for example.

Theorem 8.1. Let V be an absolutely irreducible 2F-modules in defining characteristic for a group G with F^{∗}(G) an exceptional group of Lie type. Then the composition factors of V |_{F^{∗}(G)} are as given in Table 17.

Table 13: ℓ-ranks of exceptional groups

\begin{matrix}G & m_{ℓ}(G) & G & m_{ℓ}(G) \\ ^{2}B_{2}(2^{2a+1}) & 2a + 1 & F_{4}(ℓ^{a}) & \begin{cases}9a & ℓ > 2 \\ 11a & ℓ = 2 \end{cases} \\ ^{2}G_{2}(3^{2a+1}) & 2(2a + 1) & E_{6}(ℓ^{a}) & 16a \\ G_{2}(ℓ^{a}) & \begin{cases}3a & ℓ ≠ 3 \\ 4a & ℓ = 3 \end{cases} & ^{2}E_{6}(ℓ^{a}) & \begin{cases}12a & ℓ > 2 \\ 13a & ℓ = 2 \end{cases} \\ ^{3}D_{4}(ℓ^{a}) & 5a & E_{7}(ℓ^{a}) & 27a \\ ^{2}F_{4}(2^{2a+1}) & 5(2a + 1) & E_{8}(ℓ^{a}) & 36a \end{matrix}

The proof will be given in the following subsections. We denote by G a finite group such that F^{∗}(G) is a quasi-simple exceptional group of Lie-type in characteristic ℓ, and let S be the unique non-abelian simple composition factor of G. By our general considerations we may assume that F^{∗}(G) is the universal covering group of S, quotiented out by the ℓ-part of the Schur multiplier. Thus, F^{∗}(G) is the group of fixed points under a Frobenius morphism of a simple algebraic group of simply-connected type. We indicate this by a subscript sc. The corresponding simple group is written without this subscript.

First we rule out the adjoint modules:

Proposition 8.2. Let G be one of G_{2}(q), F_{4}(q), E_{6}(q)_{sc}, ^{2}E_{6}(q)_{sc}, E_{7}(q)_{sc}, E_{8}(q). Then the adjoint representation of G does not give rise to a 2F-module for G.

Proof. Consider first the case of G = E_{8}(q), q = ℓ^{a}. Let V be the adjoint representation of G on its Lie algebra. By [9, Tab. 9] root elements have rank 58 on V, so by Corollary 3.4 any offender A has order at least q^{29}. Moreover, if x ≠ 1 is not a root element, then its rank is at least 92, too large compared to the ℓ-rank 36a. Thus A consists of long root elements only. But then Proposition 3.9 shows that A has size at most q^{8}, too small.

For G = E_{7}(q)_{sc}, an offender A on the adjoint representation V can only contain elements from classes labelled A_{1}, 2A_{1}, 3A′′_{1} by [9, Tab. 8]. (Using the table of the closure relation of unipotent classes in [16, p. 248] and the argument in the proof of Corollary 3.4 we only have to know the centralizer dimensions of classes 3A′_{1} and 4A_{1} to verify this.) First assume that A contains an element x from class 3A′′_{1}, with rank at least 53 on V. Then A is contained in the centralizer of x, with semisimple part of type F_{4}, which lies in the maximal parabolic subgroup P = U.L of type E_{6}, with U elementary abelian of order 27. By Theorem 3.1 we don’t have A ≤ U, and since E_{6}(q)_{sc} has ℓ-rank 16a by Table 13, we neither have U ∩ A = 1. But any unipotent element of F_{4}(q) has commutator of dimension at least 16 on the Lie algebra of F_{4}, so A has centralizer of codimension at least 69, which is much too large. The case of elements of type 2A_{1} can be excluded similarly. So finally assume that A only consists of root elements, of type A_{1}, with centralizer codimension 34 on the Lie algebra. Then Proposition 3.9 gives a contradiction.

For G = E_{6}(q)_{sc}, long root elements have rank 22 on the Lie algebra, by [9, Tab. 6], so |A| ≥ q^{11}. Thus A cannot consist of long root elements only by Proposition 3.9. By [9, Tab. 6] the only other non-trivial unipotent elements possible in A are those of type 2A_{1}, with rank 32. Their centralizer has semisimple part of type B_{3}, so is contained in the maximal parabolic subgroup P = U.L of type D_{5}, with elementary abelian radical of order q^{16}. As usually we see that A is not contained in U, but no non-trivial unipotent element from L centralizes the Lie algebra of B_{3}, thus the centralizer has codimension larger than 32, which is impossible by Table 13.

For G = ^{2}E_{6}(q)_{sc}, essentially the same argument applies.

Now let G = F_{4}(q). By [9, Tab. 4] root elements have rank at least 16 on the 52-dimensional adjoint module, thus an offender A has |A| ≥ q^{8}. Let P = UL denote a maximal parabolic subgroup containing A with Levi complement L of type Sp_{6}(q). Then the composition factors of U as L-module have dimensions 1, 14 for 2 |̸q, respectively 1, 6, 8 for q even by [5, p.99]. By the previous estimate, A is not contained in a proper L-submodule of U. Thus A ≰ U by Lemma 2.1 and Theorem 3.1. Since L has ℓ-rank 6a we can’t have A ∩ U = 1 either. Now adding up centralizer codimensions of a pair of elements 1 ≠ x ∈ A ∩ U and y ∈ A \ U gives a contradiction to the 2F-condition.

Finally consider the adjoint module V for G = G_{2}(ℓ^{a}), ℓ ≠ 3. Let first ℓ ≠ 2. Long root elements have rank 6 on V, and any other nontrivial unipotent element has rank at least 7, by [9, Tab. 2]. By Table 13, any elementary abelian subgroup of G has order at most q^{3}, so it follows that either f(A) < |V | or A consists of long root elements. By Lemma 3.9, this implies that |A| ≤ q^{2} and so f(A) ≤ q^{12} < V. Suppose that ℓ = 2. As above, any nontrivial x ∈ A must have a fixed space of dimension 8. The only such elements are long or short root elements. This implies that the centralizer of x in G is [P, P] where P is a maximal parabolic subgroup and the fixed space of x on V is the Lie algebra of [P, P]. So if |A| > q, dim C_{V} (A) < 8 and f(A) < |V |. If |A| ≤ q, then f(A) ≤ q^{10}. ⊓ ⊔

8.1. The Suzuki-groups.

Lemma 8.3. If F^{∗}(G) = ^{2}B_{2}(2^{2a+1}) then any 2F-module for G is as in the statement of Theorem 8.1.

Proof. Let F^{∗}(G) = ^{2}B_{2}(q), q = 2^{2a+1}. Since Suzuki groups have no outer automorphisms of even order or multiplier of odd order we in fact have G = ^{2}B_{2}(2^{2a+1}). The basic representations have highest weights λ_{1}, λ_{2}, λ_{1} + λ_{2}. The first two are the natural representations, Examples 8.1. The Steinberg representation has dimension 16, larger than 10 ≥ 2 log_{q} |G|, so cannot occur. For tensor products V of r ≥ 2 factors (of dimension at least 4) we reach the inequality 4^{r}/r ≤ 10. Hence r = 2 and both factors have dimension 4. But then V is defined over 𝔽_{q}, q = 2^{2a+1}, and the inequality is not satisfied. ⊓ ⊔

8.2. Groups of type G_{2}.

From [11, A.49] we find:

Lemma 8.4. Let M(λ) be a basic highest weight module for G_{2}(q) of dimension d = dim M(λ) ≤ 29. Then M(λ) is one of the modules in Table 14.

Table 14: The basic G_{2}(q)-modules of dimension at most 29

λ dim M(λ)
λ_{2}  7 − δ_{ℓ,2}
λ_{1} 14 − 7δ_{ℓ,3}
2λ_{2} 27 − δ_{ℓ,7} ℓ ≠ 2
2λ_{1}  27 ℓ = 3

Lemma 8.5. If F^{∗}(G) = ^{2}G_{2}(3^{2a+1}) or F^{∗}(G) = G_{2}(q) then any 2F-module for G is as in the statement of Theorem 8.1.

Proof. Let first F^{∗}(G) = ^{2}G_{2}(q), q = 3^{2a+1} and first consider the simple group. Here 2 log_{q} |G| ≤ 14. The basic representations of dimension at most 14 are those with highest weights λ_{1}, λ_{2}, of dimension 7 both, by Table 14. The group G has two classes of elements of order 3, both of which have representatives already inside 2G_{2}(3) = Aut(L_{2}(8)). For the latter group, such elements have Jordan blocks of sizes 3,2,2 respectively 3,3,1 on the 7-dimensional module. In particular, the centralizer of any offender is at most 3-dimensional. But then A must have order q^{2}, which is maximal possible. In this case A intersects both conjugacy classes non-trivially. Now the sizes of Jordan blocks show that the centralizer has at most dimension 2, so we get no example. The case of tensor products of at least two factors is ruled out by the standard inequality (3.3).

Now assume that 3|2a + 1 and G = ^{2}G_{2}(3^{2a+1}).3 is the extension by the corresponding field automorphism. Let W be a basic representation. Since basic representations are not invariant, the induction V of W to G is irreducible. Outer elements have at most d/3-dimensional centralizer, where d = dim V. The standard inequality shows that this does not lead to examples. Similarly, tensor products can be ruled out.

Let now F^{∗}(G) = G_{2}(ℓ^{a}), ℓ^{a} ≠ 2 (the case ℓ^{a} = 2 was treated in the first part [6, Prop. 4.3] as U_{3}(3)). First consider the simple group G = G_{2}(ℓ^{a}). The 6- and 7-dimensional basic modules are Examples 8.2 and 8.3. The adjoint module was handled in Proposition 8.2.

The 26- and 27-dimensional representations are the heart of the symmetric square of the 7-dimensional representation(s) V_{0}. Since G_{2}(q) is not a transvection group, non-trivial unipotent elements have at most 5-dimensional centralizer on V_{0}, hence at most 15-dimensional centralizer on Σ̃^{2}(V_{0}) by (2.1). Thus they are not examples by Table 13. The inequality (3.3) excludes non-trivial tensor products except for the case of two factors of dimension at most 7. Again, this does not lead to examples since non-trivial unipotent elements are not transvections.

Now assume that G is G_{2}(ℓ^{a}) extended by a field automorphism of order ℓ, so ℓ|a. Let V be a 2F-module for G. The basic representations of G_{2}(ℓ^{a}) do not extend to G. Then (3.5) shows that dim V = 12 (and ℓ = a = 2), so V |_{F^{∗}(G)} indeed consists of 2F-modules. The case of tensor products is ruled out by (3.3).

Finally note that graph-automorphisms (of order 2) only occur in characteristic 3, so need not be considered here. ⊓ ⊔

8.3. The groups ^{3}D_{4}(q).

Lemma 8.6. If F^{∗}(G) = ^{3}D_{4}(q) then any 2F-module for G is as in the statement of Theorem 8.1.

Proof. Let V be a 2F-module for G and A an elementary abelian unipotent subgroup of G with f(A) maximal.

We note as usual that |V | ≤ q^{56}. It follows by Table 11 that (up to quasiequivalence) V is either the natural 8-dimensional module Y_{8} (over 𝔽_{q^{3}}) or the adjoint module ˜∧^{2}(Y_{8}) for the 8-dimensional orthogonal group. The latter has dimension (over 𝔽_{q}) 28 if ℓ is odd and 26 if ℓ = 2.

Consider the second case. If ℓ is odd, then dim C_{V} (x) = 18 for x a long root element and dim C_{V} (y) < 18 for 1 ≠ y a unipotent element that is not a long root element by Table 3. If A consists of long root elements, then f(A) ≤ q^{4}q^{18} < |V | by Proposition 3.9. Otherwise f(A) ≤ q^{10}q^{17} < |V | by Table 13.

If ℓ = 2, then dim C_{V} (x) = 16 for x a long root element and dim C_{V} (y) ≤ 14 for any involution y not a root element. The argument of the previous paragraph applies.

Finally consider the case that V is 8-dimensional (over 𝔽_{q^{3}}). Let X be a long root subgroup centralizing A. Let P = N_{G}(X) be a maximal parabolic subgroup with radical Q. Note dim C_{V} (x) = 6 (over 𝔽_{q^{3}}) for x a long root element (and is no bigger for any other unipotent element). Thus, |A| ≥ q^{3}.

Note that the composition factors for the Levi complement L ≥ SL_{2}(q^{3}) have dimension 2, 2 and 4. The latter is a tensor product of two twists of the natural module. In particular, it follows that for any unipotent element y ∈ P \ Q, |C_{V} (y)| ≤ q^{12}. So if A is not contained in Q, then f(A) ≤ q^{12}q^{10} < |V |. Thus, we may assume that A ≤ Q.

Note that AX is the normal closure of A in Q. Then f(AX) ≥ f(A) and since AX is abelian, there is no harm in assuming that X < A (and is proper). If C_{V} (A) has dimension less than 5, then f(A) ≤ q^{22} < |V |. Since A properly contains X and Q/X acts faithfully on W := C_{V} (X) (a 6-dimensional space), it must be the case that C_{V} (A) is 5-dimensional. Since L acts irreducibly on Q/X, Q is the normal closure of A in P and so f(A) ≤ f(Q) = |V |. So the only possibility is that f(A) = |V | and f(B) ≤ |V | for any subgroup B of Q. In particular, this implies that |A| = q^{9/2} (and so q is a square). Choose g ∈ L with A^{g} ≠ A. Then A ∩ A^{g} is trivial on W and so A ∩ A^{g} = X. Thus, AA^{g} has order q^{8} and fixes a 4 dimensional subspace of W. So f(AA^{g}) = q^{28} > f(Q), a contradiction.

Now assume that G is obtained by extending ^{3}D_{4}(q) by a field automorphism of order ℓ. Basic invariant modules only exist when ℓ = 3, and the only one below the dimension bound is the adjoint module of dimension 28. But as before any inner element has rank at least 10. But the outer 3-rank is controlled by the centralizer G_{2}(ℓ^{a}), hence at most 4a/3 + 1. Thus we necessarily have a = 3, but then the centralizer of inner elements has dimension at most 16, and we obtain a contradiction. Induced basic modules can be ruled out by Proposition 2.5, tensor products can’t occur by (3.3). ⊓ ⊔

8.4. Groups of type F_{4}.

From [11, A.50] we find:

Lemma 8.7. Let M(λ) be a basic highest weight module for F_{4}(q) of dimension d = dim M(λ) ≤ 105. Then M(λ) is one of the modules in Table 15.

Table 15: The basic F_{4}(q)-modules of dimension at most 105

λ dim M(λ)
λ_{4} 26 − δ_{ℓ,3}
λ_{1}  26  ℓ = 2
λ_{1}  52  ℓ ≠ 2

Lemma 8.8. If F^{∗}(G) = ^{2}F_{4}(2^{2a+1})′ or F^{∗}(G) = F_{4}(q) then any 2F-module for G is as in the statement of Theorem 8.1.

Proof. First assume F^{∗}(G) = ^{2}F_{4}(q)′, q = 2^{2a+1}. We only have to consider G = 2F_{4}(2^{2a+1})′ since there is no odd multiplier and the outer automorphism of order 2 of^{2}F_{4}(2)′ does not have representatives of order 2. We have 2 log_{q} |G| ≤ 52(2a+1), so by Table 15 only the basic highest weight modules of dimension 26 might occur. The 2-rank of G is 5(2a + 1) by Table 13. Both classes of involutions 2A, 2B have representatives already inside ^{2}F_{4}(2)′. Since these invert elements of order 5 respectively 13 we may conclude that their centralizers on the 26-dimensional modules are at most 16- respectively 14-dimensional. Hence a possible offender A must be 2A-pure of order 2^{5(2a+1)}. But the center of the centralizer of a 2Aelement clearly has a smaller 2-rank. Thus we get a contradiction with Lemma 2.3. For a tensor product of r factors we need 26^{r} ≤ 52r, which does not hold when r ≥ 2.

Now let G = F_{4}(q), q = ℓ^{a}. The small basic modules were collected in Table 15. The adjoint module was excluded in Proposition 8.2. We have to leave open the case of the 25- respectively 26-dimensional module. For tensor products, the inequality (3.3) leads to a contradiction. For groups G obtained from F_{4}(q) by extension with a field automorphism, the standard estimates rule out 2F-modules. The 2-rank of the centralizer of an exceptional graph automorphism of order 2 in characteristic ℓ = 2 is 5a + 1, and again the standard estimates rule out this case. ⊓ ⊔

8.5. Groups of type E_{n}.

From [11, A.51] we find:

Lemma 8.9. Let M(λ) be a basic highest weight module for E_{6}(q)_{sc} of dimension d = dim M(λ) ≤ 158. Then M(λ) is one of the modules in Table 16.

Table 16: The basic E_{6}(q)_{sc}-modules of dimension at most 158

\begin{matrix}λ & dim M(λ) \\ λ_{1}, λ_{6} & 27 \\ λ_{2} & 78 − δ_{ℓ,3} \end{matrix}

Lemma 8.10. If F^{∗}(G) = E_{n}(q)_{sc}, n = 6, 7, 8, or F^{∗}(G) = ^{2}E_{6}(q)_{sc}, then any 2F-module for G is as in the statement of Theorem 8.1.

Proof. 1. First let G = E_{6}(q)_{sc}. The unipotent radical U of a maximal parabolic subgroup of type Spin_{10}^{+}(q) is a spin module for the Levi subgroup, hence irreducible. If A ≤ U is an offender on a 2F-module V defined over 𝔽_{ℓ^{f}} then our usual argument gives df/2 ≤ 32a. In particular, V must be a twist of a basic module, of dimension at most 64. Hence by Table 16 it is one of the 27-dimensional modules. On the other hand, if U contains no offender, then C_{V} (U) is an example for Spin_{10}^{+}(q). Applying this to both classes of D_{5}-parabolics and using the result for the orthogonal groups in Theorem 7.5 we see that up to twist V is one of the basic modules one Table 16. The 27-dimensional modules are Example 8.4, the adjoint modules were treated in Proposition 8.2.

Now assume that G = E_{6}(q)_{sc}.ℓ involves a field automorphism of order ℓ. If V is a tensor product, then 27^{r}a/r ≤ 2 log_{ℓ} |G|. This is only possible for r = 1. But basic modules are not invariant, so V is induced from a twist of a basic module W. Furthermore, a 2F-module for G is either already an example for the derived group, or every proper parabolic subgroup leads to an example. The results for SL_{6} and Spin_{10} show that this implies that W is (a twist of) the 27-dimensional or the adjoint module. Using that outer elements have centralizer at most dim W on V and the ℓ-rank 8a + 1 we arrive at ℓ = 2, dim W = 27, which occurs in Table 17.

Next assume that ℓ = 2 and G is obtained from E_{6}(q)_{sc} by extension with the graph automorphism of order 2. The 27-dimensional module is not invariant, hence can be excluded by Proposition 2.5. Since the graph automorphism centralizes an F_{4}(q), outer involutions don’t act as bi-transvections, so B := A ∩ G′ ≠ 1, and |A| ≤ 2q^{11} by Table 13. The table in [9] shows that B has to consist of long root elements, but then |B| ≤ q^{5} by Table 5, while long root elements have rank 22, contradiction. Non-trivial tensor products do not satisfy the basic inequality 3.4. If G involves a graph-field automorphism of order 2, then no basic representations is invariant, and no case arises. The same happens for the extension of E_{6}(2^{a})_{sc} by the full group of graph and field automorphisms of order 2.

2. Now let G = ^{2}E_{6}(q)_{sc}. Arguing as above, only the basic representations of dimensions 27 and 78 − δ_{ℓ,3} need to be considered. The adjoint one is ruled out by Proposition 8.2. At present, we can’t decide whether the 27-dimensional module gives an example. The basic representations do not extend to the extension of 2E_{6}(q)_{sc} by a field automorphism of order ℓ. Thus, as before, this case can be ruled out by the standard arguments.

3. Now let G = E_{7}(q)_{sc}. By [11] the only small basic modules are the one of dimension 56 and the adjoint one of dimension 133−δ_{ℓ,2}. The latter was ruled out in Proposition 8.2. We leave open the case of the 56-dimensional module. Tensor products do not lead to examples, using that G has ℓ-rank 27a (see Table 13). For extensions by field automorphisms of order ℓ the standard estimates allow to conclude.

4. Finally let G = E_{8}(q). Here [11] shows that the only possible basic module is the adjoint one of dimension 248. This was ruled out in Proposition 8.2. Tensor products do not give rise to examples by our basic inequality, using that the ℓrank of G is 36a (see Table 13). Similarly, the case of field automorphisms can be excluded by standard estimates.

⊓ ⊔

8.6. The examples for exceptional groups.

Case 8.1: Let A be the center of a Sylow 2-subgroup of the Suzuki group^{2}B_{2}(2^{2a+1}), of order 2^{2a+1}. Then it is well known that A has 2-dimensional centralizer on the 4-dimensional modules, hence we obtain an example.

Case 8.2: The group G_{2}(2^{a}) has 2-rank 3a, so we find an example on the 6-dimensional module.

Case 8.3: The group G_{2}(ℓ^{a}) has ℓ-rank at least 3a, so we find an example on the 7-dimensional module(s).

Case 8.4: The Chevalley group F_{4}(2^{a}) contains the classical group SO_{9}(2^{a}) ≅ Sp_{8}(2^{a}). The 26-dimensional module for F_{4}(2^{a}) restricts to the heart of the exterior square of the natural module ˜∧^{2}(Y_{8}) for Sp_{8}(2^{a}), as can be seen on the extended Dynkin diagram. Since the latter is an example for the symplectic group by Table 9, it is a fortiori an example for F_{4}(2^{a}).

Table 17: 2F-modules for exceptional groups in defining characteristic

G  d  λ  f  |A|
2B_{2}(2^{2a+1})  4  λ_{1}, λ_{2} 2a + 1 a > 1 2a + 1
G_{2}(2^{a})  6  λ_{2}  a  3a
G_{2}(ℓ^{a})  7  λ_{2}  a ℓ > 2  3a
F_{4}(2^{a})  26  λ_{1}, λ_{4}  a  10a
E_{6}(ℓ^{a})_{sc}  27  λ_{1}, λ_{6}  a  16a
F_{4}(ℓ^{a})  26 − δ_{ℓ,3}  λ_{4}  a ℓ > 2
2E_{6}(ℓ^{a})_{sc}  27  λ_{1}, λ_{6}  2a
E_{7}(ℓ^{a})_{sc}  56  λ_{7}  a

Case 8.5: Let U be the abelian unipotent radical, of order q^{16}, of a maximal parabolic subgroup of type D_{5} of E_{6}(q)_{sc}. Clearly this gives an example on the 27-dimensional modules.

9. F-modules for quasi-simple groups

In this section we obtain the list of all absolutely irreducible F-modules of finite quasi-simple groups, as an easy application of our classification of 2F-modules in this paper and the previous part [6].

Lemma 9.1. The 56-dimensional module for G := E_{7}(q)_{sc} is not an F-module.

Proof. Let P = UL where U is elementary abelian of order q^{27} and L is of type E_{6}. Then V has an L-composition series F < W < M < V where F and V/M are 1-dimensional and W/F ≅ U^{∗} and M/W ≅ U.

Let A be a nontrivial elementary abelian unipotent subgroup of G with f_{1}(A) maximal and assume that f_{1}(A) ≥ |V |. Set B = A ∩ U. If B = 1, the result follows since the 27-dimensional module U for E_{6}(q) is not an F-module.

If B does not consist solely of long root elements, then C_{V} (B) has dimension at most 36 by [9, Table 7]. Since C_{W} (y) is 27-dimensional (note that U acts trivially on W/F, whence [y, W] = F for each nontrivial y ∈ U), this implies that C_{V} (A)/C_{W} (A) has dimension at most 9. Thus,

f_{1}(A) ≤ q^{9}|A/B||C_{W} (A/B)||B| < q^{37}|B|

since W/F is not an F-module for E_{6}(q). Thus, |B| > q^{19}. By loc. cit. this implies that A/B consists of long root elements.

Now we observe that if |A/B| > q, then |C_{B}(A/B)| ≤ q^{18}. (sketch: a long root element centralizes a space B_{0} of dimension 21 on B. Now work in the centralizer of this long root subgroup X in C := C_{E_{6}}(X) = q^{1+20}SL_{6}(q). The module B has three composition factors of dimensions 6, 15, 6 for SL_{6}(q). Suppose that y is in the unipotent radical Q of C outside X. Then y^{−1}y^{g} is a nontrivial element of X for some g ∈ Q. Since an element of X has rank 6 on B_{0}, it follows that y has rank at least 3 on B_{0}, whence C_{B_{0}}(y) = C_{B}(y, x) has dimension at most 18. If y is not in U, then it fixes a 5-dimensional space at most on each of the 6-dimensional composition factors and a 13-dimensional space on the 15-dimensional composition factor).

So in this case, |A| ≤ q|B|. Note that the action of U on W gives an identification of U with a subspace of W^{∗} (y → y − 1 and y − 1 is a map from W to F). In particular, any subspace R of U of dimension r centralizes a space of dimension 28 − r on W. In particular, |B||C_{W} (B)| ≤ |W |. Thus,

|C_{V} (A)||A| ≤ q|W ||C_{V} (A)/C_{W} (A)| ≤ q^{10}|W | < |V |.

So B consists solely of long root elements. It follows from Table 5 that |B| ≤ q^{7}. Thus, |A| ≤ q^{7}|A/B|. Moreover, f_{1}(A) ≤ q^{7}|A/B||C_{W} (A)||C_{V} (A)/C_{W} (A)|. By the E_{6}-result, |A/B||C_{W} (A)| < |W |. Even for y ∈ B a long root element, we have C_{V} (y)/C_{W} (y) of dimension at most 18, whence f_{1}(A) < q^{53}, a contradiction. ⊓ ⊔

For the properties of M_{24}, Co_{1} and Co_{2} needed in the following proofs we refer to [5, Tab. 5.3e, 5.3k and 5.3l].

Lemma 9.2. Let V be an irreducible 𝔽_{2}-module of dimension 11 for G = M_{24}. Then V is not an F-module.

Proof. We first note that the centralizer dimensions for involutions on V are 7 for a 2-central involution and 6 for the other class of involutions.

Suppose that A is an offender with f_{1}(A) maximal.

Let z be an involution of class 2A. We work with the subgroup A ≤ P with U = O_{2}(P) elementary abelian of order 16 and P/U = 𝔄_{8}. Note that this contains D := C_{G}(z) (after conjugating) where Q = O_{2}(D) is extraspecial of order 2^{7} and D/Q = L_{3}(2).

We claim that A must contain a conjugate of z. If not, then A ∩ U = 1 (since U intersects only one class of involutions). Then A embeds in P/U and so has order at most 16, but C_{V} (A) has dimension at most 6, a contradiction.

So we may assume that z ∈ A ≤ D. Set W = C_{V} (z) of dimension 7 and M = [z, V] of dimension 4. Note that W/M is an irreducible D-module.

Suppose that A ∩ Q has order 2. Then |A| ≤ 8, a contradiction. So let z ≠ y ∈ A ∩ Q be an involution. Since z = y^{−1}y^{g} for some g ∈ Q, it follows that [y, W] has dimension at least 2. Thus, C_{V} (y) has dimension at most 5, whence exactly 5 and |A| = 2^{6}. Thus, A is not contained in Q and so does not centralize M. Then C_{V} (A) has dimension at most 4, a contradiction. ⊓ ⊔

Lemma 9.3. Let V be the 22-dimensional 𝔽_{2}-module for G = Co_{2}. Then V is not an F-module.

Proof. Let A be an offender with f_{1}(A) maximal. Note that C_{V} (x) has dimension at most 16 for any involution x. Thus, |A| ≥ 2^{6}.

Let z be a 2-central involution and set D = C_{G}(z). Set U = O_{2}(D). Then D/U = Sp_{6}(2). Since |A| ≥ 64, A is not contained in U (the largest abelian subgroup of U has order 32).

If A contains an involution not conjugate to z, then |C_{V} (A)| ≤ 2^{14} and so |A| ≥ 2^{8}.

We first show that A contains a conjugate of z. If not, then as we saw |A| ≥ 2^{8} and B := A ∩ U ≠ 1. Set W = C_{V} (z) and M = [z, V]. So dim W = 16 and dim M = 6. Now U centralizes M (since it is irreducible for D). If A/B has order 2, then f_{1}(A) ≤ f_{1}(B) < |V |. If A/B has order at least 4, then it follows that C_{V} (A) has codimension at least 2 in C_{V} (B) and so dim C_{V} (A) ≤ 12, whence |A| = 2^{10}. Since that is the maximal rank possible, it follows that A contains a 2-central involution.

So we may assume that z ∈ A and B is a nontrivial proper subgroup of A. Note that C_{V} (A) is properly contained in C_{V} (B) as B = C_{M}(A). If A/B has order 2, then f_{1}(A) ≤ f_{1}(B) < |V |. If A/B has order at least 4, it follows that C_{M}(A) has dimension at most 4 and so C_{V} (A) has codimension at least 2 in C_{V} (B). If y ∈ B is an involution other than z, we see that C_{W} (y) has codimension at least 3 in W. Thus, C_{V} (B) has dimension at most 13, whence by the remark above, C_{V} (A) has dimension at most 11 and f_{1}(A) < |V |.

So B must have order 2 and |A| ≤ 2^{7}. Then W = C_{V} (z) is 16-dimensional and so C_{V} (A) has dimension at most 14, whence f_{1}(A) ≤ 2^{21}. ⊓ ⊔

Lemma 9.4. Let V be the 24-dimensional 𝔽_{2}-module for G = Co_{1}. Then V is not an F-module.

Proof. Suppose that A is a nontrivial elementary abelian subgroup of G with f_{1}(A) ≥ |V | and maximal. Note that |A| ≤ 2^{11}.

Consider D = C_{G}(z) with z an involution in class 2A. Set U = O_{2}(D) and recall that D/U = O_{8}^{+}(2). Note that C_{V} (A) has dimension at least 13, whence all involutions in A are conjugate to z (by centralizer dimensions). We assume that z ∈ A.

Note that D has 3 composition factors on V, each of dimension 8. Set W = C_{V} (z) (of dimension 16). Moreover, if x ∈ A \ U, then C_{W} (x) has dimension at most 12 (since x does not induce a transvection on any composition factor). So if A is not contained in U, then f_{1}(A) ≤ |A|2^{12} < |V |. If A is contained in U, then |A| ≤ 2^{5} and so f_{1}(A) ≤ 2^{21}. ⊓ ⊔

Theorem 9.5. The absolutely irreducible F-modules of finite quasi-simple groups are given in Tables 18 and 19.

Proof. By definition, F-modules are in particular 2F-modules, so we just have to go through our list of 2F-modules and determine which of them satisfy the Fcondition. We start with the groups of Lie type in their defining characteristic whose 2F-modules were classified in this paper.

The symmetric square Σ^{2}(Y_{n}) of the natural module for SL_{n}(q) is not an Fmodule by (2.1) in conjunction with Proposition 3.10. On the wedge cube V := ∧^{3}(Y_{6}) of the natural module for SL_{6}(q), transvections have rank 6, so a possible offender A has |A| ≥ q^{6}. Consider an end-node parabolic P = UL containing A. If A ≤ L = GL_{5}(q), then it is maximal possible, centralizing a 2- or 3-space on the natural module. But then the centralizer on V |_{GL_{5}(q)} = ∧^{3}(Y_{5}) ⊕ ∧^{2}(Y_{5}) is just 4-dimensional. Thus A∩U ≠ 1, and the centralizer of A is at most 11-dimensional, whence |A| = q^{9} is maximal possible, the unipotent radical of a GL_{3}(q) × GL_{3}(q)- parabolic. This is ruled out by Theorem 3.1.

For the tensor product module V := Y_{n} ⊗ Y_{n}^{(a)} of SL_{n}(ℓ^{2a}) we use induction. For n = 2 an offender A is contained in the unipotent radical of a Borel subgroup, but the latter violates the F-condition. For n > 2, induction and Theorem 2.4 show that a possible offender must lie inside the unipotent radical U of a maximal parabolic subgroup q^{n−1}GL_{n−1}(q). But U has centralizer dimension (n − 1)^{2} on V, thus we don’t get an example.

For SU_{3}(q), the maximal size of ℓ-subgroups centralizing subspaces of the natural module of given dimension is bounded in Proposition 3.11. This shows that the F-condition is not satisfied.

The heart of the exterior square ˜∧^{2}(Y_{2n}) of the natural module for Sp_{2n}(q), n ≥ 2, is not an example by (2.1) in conjunction with Proposition 3.12. The case of Y_{4} ⊗ Y_{4}^{(a)} for Sp_{4}(ℓ^{2a}) is ruled out by the same argument. The spin modules for Sp_{2n}(2^{a}), n ≥ 4, are not F-modules by Proposition 5.10.

The spin module for Spin_{2n+1}(q), n ≥ 4, q = ℓ^{a} odd, was excluded in Proposition 5.8.

The half-spin modules for Spin_{2n}^{+}(q), n ≥ 6, respectively for Spin_{2n}^{−}(q), n ≥ 4, were excluded in Propositions 5.4 and 5.6.

The Suzuki-group ^{2}B_{2}(2^{2a+1}) has 2-rank 2a+1, but does not contain transvections, so the 4-dimensional module over 𝔽_{2^{2a+1}} is not an F-module.

On the 7-dimensional module for G_{2}(ℓ^{a}), ℓ > 2, long root elements have rank 2, all other non-trivial unipotent elements have rank at least 4. Thus this is not an Fmodule for ℓ > 3. For ℓ = 3 we obtain that the only possibility is that |A| = q^{4} and C_{V} (A) is 3-dimensional. Thus the corresponding tame abelian subalgebra I(A) has order q^{4}, so must correspond to four abelian roots. The only possibility for this is the set of all non-simple roots. These roots are the ones of maximal weight in any reasonable ordering on the roots, so A = U(I(A)) is the subgroup generated by the root subgroups corresponding to the four positive nonsimple roots. Let X be any short root subgroup in A. Since C_{V} (X) = C_{V} (A), it follows that C_{V} (A) is invariant under N_{G}(X). Since there are two different short root subgroups in A and since N_{G}(X) is a maximal parabolic subgroup of G, it follows that G leaves C_{V} (A) invariant, a contradiction.

For F_{4}(q) on the 25- or 26-dimensional module, long root elements have rank 6, all other non-trivial unipotent elements rank at least 10, by [9, Tab. 3]. So by Table 13 this is not an F-module when ℓ > 2. For ℓ = 2, the 26-dimensional module occurs as a direct summand in the restriction of the 27-dimensional module for E_{6}(q). Since the latter is not an F-module (see below), this case is excluded as well.

Let P = UL be a maximal parabolic subgroup of E_{7}(q)_{sc} with Levi factor L of type E_{6}. Then U is one of the 27-dimensional modules for E_{6}(q) ≤ L. Now U is the unique abelian subgroup of maximal order in a Sylow ℓ-subgroup of P by Lemma 3.15. Thus, by Lemma 2.6 the module U is not an F-module for E_{6}(q). The other 27-dimensional module for E_{6}(q) is the image under the graph automorphism of order 2, hence isn’t an F-module either.

On the 27-dimensional modules for ^{2}E_{6}(q) over 𝔽_{q^{2}} long root elements have rank 6, all other non-trivial unipotent elements have rank at least 10, by [9, Tab. 5]. Thus Table 13 shows that the F-condition cannot be satisfied.

In non-defining characteristic we just have to consider the 2F-modules determined in the first part [6] in Tables 6.3, 6.5, and 6.7.

Subgroups of order ℓ^{k} in 𝔖_{n} contain elements with at most n − kℓ fixed points, so of rank (ℓ−1)k on the permutation module. Passing to the deleted permutation module we see that this is not an example when ℓ > 2. For ℓ = 2, subgroups of order 2^{k} fixing n−2k points necessarily contain odd elements. Thus if A ≤ 𝔄_{n} has size |A| = 2^{k} then it contains elements of rank k + 1. If n is odd, the permutation module is a direct sum of the n−1-dimensional irreducible module with the trivial one. Thus the deleted permutation module is not an F-module for 𝔄_{n}, n odd, in characteristic ℓ = 2.

For the 8-dimensional representations of 𝔄_{9} over 𝔽_{2} not contained in the permutation module, explicit computation shows that elementary abelian subgroups of order 8 have at most 4-dimensional centralizer and those of order 16 have 2dimensional centralizer. The 2-dimensional representation of 2.𝔄_{5} over 𝔽_{9} does not give an example since the Sylow 3-subgroup has only order 3. On the 8dimensional representation of 2.𝔄_{9} over 𝔽_{3} elements of order 3 have rank 4. Since the 3-rank is 3, we don’t get an example.

The 6-dimensional 𝔽_{2}-representation of U_{3}(3) is the restriction from G_{2}(2). There long root elements have rank 2, so A has to contain other non-trivial unipotent elements by Table 5. But these have rank at least 3, while U_{3}(3) only has 2-rank 2, so there are no offenders.

The 6-dimensional 𝔽_{3}-module for G := 2.L_{3}(4) is the restriction of the natural module for SO_{6}^{−}(3). Since G only has 3-rank 2, this is not an F-module for G by Proposition 3.14.

The 7-dimensional 𝔽_{3}-module for S_{6}(2) is orthogonal. On restriction to the subgroup SO_{6}^{−}(2) ≅ SU_{4}(2) ≅ S_{4}(3) ≅ SO_{5}(3) of 3′-index we find a copy of the natural module for SO_{5}(3). This is the heart of the exterior square of the natural module for Sp_{4}(3), hence not an F-module by what we showed above.

The 8-dimensional 𝔽_{3}-module for 2.S_{6}(2) is the restriction of the spin module for Spin_{7}(3). On restriction to the 3′-index subgroup SO_{5}(3) we obtain two copies of the spin module, the natural module for Sp_{4}(3). Now Proposition 3.12 shows that this is not an example.

The 6-dimensional 𝔽_{4}-module V of 3_{1}.U_{4}(3) is of orthogonal type, so involutions have rank at least 2. Hence |A| ≥ 2^{4}, but then A centralizes at most a 3dimensional subspace by Proposition 3.14. This contradicts the fact that 3_{1}.U_{4}(3) has 2-rank 4.

On the 8-dimensional 𝔽_{3}-module for 2.O_{8}^{+}(2), elements of order 3 have rank 2 or 4 and subgroups of order 9 have centralizer dimension at most 4. So if there exists an offender, the maximal torus of order 3^{4} is also an offender. In particular we may assume that A contains elements from all classes of 3-elements. Now the Weyl group of type E_{8} contains the direct product of two Weyl groups of type A_{4} (visible on the extended Dynkin diagram), as well as the product of four Weyl groups of type A_{2} (visible in E_{6} × A_{2}). Thus there are 3-elements with Jordan type 3^{2}1^{2} and with Jordan type 2^{4}. Clearly no subgroup containing these two types can have centralizer dimension 4.

The 6-dimensional 𝔽_{4}-module V of 3.M_{22} is the restriction of the natural module for SU_{6}(2). Since 3.M_{22} does not contain transvections the centralizer of a 5-dimensional subspace is trivial, and the centralizer of a 4-dimensional space has size less than 2^{4} by Lemma 3.11. Since 3.M_{22} has 2-rank 4, this shows that V is not an F-module.

The 6-dimensional 𝔽_{4}-module V of J_{2} is the restriction of the 6-dimensional module for G_{2}(4). By [9, Table 1] long root elements have rank 2, the other involutions have rank 3. Since J_{2} has 2-rank 4, an offender would have to consist of long root elements only. But such subgroups in G_{2}(4) have size at most 4 by Table 3.9.

On the 10-dimensional 𝔽_{2}-module of M_{12}, involutions have rank 4, but the 2-rank equals 3.

On the 10-dimensional 𝔽_{2}-module of M_{22}, involutions have rank 4, Klein four groups have centralizer codimension at least 5, but the 2-rank equals 4.

On the 11-dimensional 𝔽_{2}-module of M_{23}, involutions have rank 4, Klein four groups have centralizer codimension at least 5, but the 2-rank equals 4.

The 11-dimensional 𝔽_{2}-module of M_{24}, the 22-dimensional 𝔽_{2}-module of Co_{2} and the 24-dimensional 𝔽_{2}-module of Co_{1} were discussed in Lemmas 9.2–9.4.

On the 5-dimensional 𝔽_{3}-module of M_{11}, elements of order 3 have rank 3, so this is no example. On the 6-dimensional 𝔽_{3}-module of M_{12}, elements of order 3 have rank 3 or 4, so this is not an example. ⊓ ⊔

9.1. The examples for F-modules.

The following two tables contain the F-modules V for finite quasi-simple groups. In the last column we give the size of one possible offender on V. For the natural module Y_{n} of Spin_{n}^{(±)}(ℓ^{a}) in the fifth line, ℓ has to be odd when n is odd.

Case 9.1: On its natural module Y_{n}, SL_{n}(q) acts as a transvection group, so we get an example with |A| = q.

Case 9.2: The subgroup of SL_{n}(q) of order q^{n−1} in (4.1) has centralizer of dimension at least \binom{n−1}{2} on ∧^{2}(Y_{n}), so this yields an F-module.

Case 9.3: For the special unitary group SU_{n}(q), n ≥ 4, on its natural module Y_{n} over 𝔽_{q^{2}} Proposition 3.11 shows that there exists a subspace of codimension 2 whose centralizer contains an elementary abelian subgroup of order q^{4}.

Case 9.4: The symplectic group Sp_{2n}(q) acts as a transvection group on its natural module Y_{2n}, thus we get an example with |A| = q.

Table 18: F-modules for groups of Lie type in defining characteristic

G  d  V f conditions log_{ℓ} |A|
SL_{n}(ℓ^{a})  n  Y_{n} a  a
SL_{n}(ℓ^{a})  \binom{n}{2} ∧^{2}(Y_{n}) a  n ≥ 4  (n − 1)a
SU_{n}(ℓ^{a})  n  Y_{n} 2a  n ≥ 4  4a
Sp_{2n}(ℓ^{a})  2n  Y_{2n} a  a
Spin_{n}^{(±)}(ℓ^{a}) n  Y_{n} a  n ≥ 6  3a
Spin_{7}(ℓ^{a})  8  Z_{3} a  5a
Spin_{2n}^{+}(ℓ^{a}) 2^{n−1} Z_{n−1} a n = 4, 5 (2n − 2)a
G_{2}(2^{a})  6 M(λ_{2}) a  3a

Case 9.5: For the orthogonal group Spin_{n}^{(±)}(q), n ≥ 6, on its natural module Proposition 3.14 shows that there exists a subspace of codimension 3 whose centralizer contains an elementary abelian subgroup of order q^{3}.

Case 9.6: The spin module Z_{3} for Spin_{7}(q) has highest weight λ_{1}. Consider the maximal parabolic subgroup of type Spin_{5}(q). The centralizer of its unipotent radical U is again the spin module for Spin_{5}(q), hence has dimension 4. But U is elementary abelian of order q^{5}. Thus U is an offender on Z_{3}.

Case 9.7: For the half-spin modules V := Z_{n−1} of dimension 2^{n−1} of Spin_{2n}^{+}(q) let A be the unipotent radical of a maximal parabolic subgroup with Levi complement of type Spin_{2n−2}^{+}(q). Then the centralizer C_{V} (A) is a half-spin module for Spin_{2n−2}^{+}(q), of dimension 2^{n−2}, while A is elementary abelian of order q^{2n−2}, so we find an example whenever n ≤ 5.

Case 9.8: The 6-dimensional module V embeds G_{2}(q), q = 2^{a}, into Sp_{6}(q). Let B be a Borel subgroup of G_{2}(q). Then B fixes a totally singular 3-space W of V. This defines a homomorphism into GL_{3}(q), with kernel the centralizer A := C_{B}(W) of W. Thus |A| ≥ q^{3}, and A is an offender.

Table 19: F-modules in non-defining characteristic

\begin{matrix}G & d & f & ℓ & |A| \\ 𝔄_{2n} & 2n − 2 & 1 & 2 & 2^{n−1} \\ 3.𝔄_{6} & 3 & 2 & 2 & 4 \\ 𝔄_{7} & 4 & 1 & 2 & 4 \end{matrix}

Case 9.9: Let V be the 2n − 2-dimensional heart of the permutation module of 𝔄_{2n} in characteristic 2. Let A be the intersection with 𝔄_{2n} of a maximal 2subgroup of 𝔖_{2n} generated by commuting transpositions. Then |A| = 2^{n−1}, A centralizes a subspace of dimension n on the permutation module, and by direct computation, still an n − 1-dimensional subspace of V. Thus A is an offender.

(For 𝔄_{6} ≅ Sp_{4}(2)′ this is the natural module, for 𝔄_{8} ≅ SL_{4}(2) it is the exterior square of the natural module.)

Case 9.10: The group 3.𝔄_{6} on its 3-dimensional 𝔽_{4}-module gives an example with |A| = 4.

Case 9.11: The group 𝔄_{7} on its 4-dimensional 𝔽_{2}-module (the natural module for SL_{4}(2) ≅ 𝔄_{8}) contains 𝔄_{6} ≅ Sp_{4}(2)′ in its natural representation. The latter already is Example 9.9.

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R.M. G.: Department of Mathematics, University of Southern California, Los Angeles CA 90089 - 1113, USA

Email: guralnic@math.usc.edu

G.M.: FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany

Email: malle@mathematik.uni-kassel.de