APPENDIX: AN EXOTIC FOURIER TRANSFORM FOR H_{4}

Gunter Malle

Universität Heidelberg

0. Introduction

In [5], Lusztig gives the definition of a fusion datum. Such fusion data naturally occur in connection with two-sided cells in finite Coxeter groups. In that situation, they were called Fourier transforms. The Fourier transforms for all cells for all finite Weyl groups have been determined by Lusztig [3]. For the cells in H_{3} and the small cells in H_{4}, the matrix is easy to find and given in [2]. For the dihedral groups I_{2}(p), Lusztig determines a Fourier transform matrix in [5]. Here we complete the picture by giving a 74 × 74-matrix connected to the big two-sided cell in the Weyl group of type H_{4}, which was the last missing case.

1. Fusion data

For the convenience of the reader we recall the properties of a certain type of fusion datum from [5]. Let S = (s_{k,l})_{k,l} ∈ ℝ^{n×n} be a real symmetric matrix, and t = (t_{k}) ∈ ℂ^{n} a vector of complex roots of unity. Let further π be an involutory permutation of {1, …, n} with a fixed point k_{0}, 1 ≤ k_{0} ≤ n. If

(1) s_{k,l} = s_{π(k),π(l)} for all 1 ≤ k, l ≤ n,

(2) t_{k} = t̄_{π(k)} for all 1 ≤ k ≤ n,

(3) ∑_{j=1}^{n} s_{k,j}s_{j,l} = δ_{k,l} for all 1 ≤ k, l ≤ n,

(4) s_{k,k_{0}} > 0 for all 1 ≤ k ≤ n,

(5) t_{k_{0}} = 1,

(6) ∑_{j=1}^{n} \frac{s_{k,j}s_{l,j}s_{m,j}}{s_{k_{0},j}} ∈ ℕ for all 1 ≤ k, l, m ≤ n,

(7) ∑_{j=1}^{n} t_{j}s_{j,k}s_{l,j} = t_{k}t_{l}s_{π(k),l} for all 1 ≤ k, l ≤ n,

then by the definition in loc. cit. the tuple ({1, …, n}, k_{0}, π, π, S, t) is a fusion datum.

The interest in such fusion data stems from the fact that the base change matrix between unipotent character degrees and the degrees of fake characters for families of unipotent characters in finite groups of Lie type are examples of matrices S of the above type, with the vector t given by the eigenvalues of Frobenius attached to the occurring unipotent characters (see loc. cit.). In this case the matrix S is called a non-abelian Fourier transform.

Typeset by AMS-TEX

The author gratefully acknowledges financial support by the deutsche Forschungsgemeinschaft

The Fourier transforms for all families of unipotent characters for all finite groups of Lie type have been determined by Lusztig [3]. There it is also shown that every such matrix is connected to a finite group Γ, either an elementary abelian 2-group or one of the symmetric groups S_{3}, S_{4} or S_{5}, in such a way that the rows and columns of S may be indexed by pairs (σ, χ), where σ is an element of Γ and χ an irreducible character of C_{Γ}(σ), up to conjugation, and the entries are given by a simple formula in terms of the indices.

Now it turns out that one can also associate ‘unipotent character degrees’ to noncristallographic finite Coxeter groups. These degrees can be found in the paper [4] for the remaining cases I_{2}(p), H_{3} and H_{4}. Again, the unipotent characters (character degrees) can be partitioned into families, according to the two-sided cells of the corresponding Coxeter group. As we also have fake degrees, the concept of Fourier transform matrices also makes sense for these groups. For the group H_{3}, all families either have one or four elements, and the corresponding matrix is easily found (see for example [2]). For the dihedral groups I_{2}(p), Lusztig determines a Fourier transform matrix in [5]. The only remaining case is that of a family in H_{4} consisting of 74 elements.

We here give a corresponding matrix S over ℚ(\sqrt{5}). To keep its size reasonable, we have omitted one of every two columns/rows algebraically conjugate over ℚ(\sqrt{5})/ℚ. To obtain the complete matrix, for every row/column with irrational entries one has to add another one containing the algebraically conjugate values, possibly the whole row/column multiplied by −1 so as to make the ninth entry positive.

Proposition. The matrix S ∈ ℚ(\sqrt{5})^{74×74} together with the vector t and the permutation π which interchanges the characters with equal degree pairwise, and k_{0} = 9, satisfy conditions (1)–(7) above, so they form a fusion datum. Moreover, the vector of unipotent character degrees of H_{4}, multiplied with S, gives the vector of fake degrees belonging to the big two-sided cell of H_{4}.

Moreover the roots of unity in t are the eigenvalues attached to the unipotent characters via the d-Harish-Chandra theory in [2], and which are a replacement for the eigenvalues of Frobenius.

From the method of construction it is not clear whether S, t are uniquely determined by the properties stated in the Proposition.

Proof. Conditions (1), (2), (4) and (5) are obvious by inspection. Properties (3), (6) and (7) are best verified by a computer (for the 64824 assertions in (6) this takes 29 hours with the computer algebra system Maple). □

The vector t is given by

t = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ζ_{5}, ζ_{5}, ζ_{5}, ζ_{5}, ζ_{5}^{4}, ζ_{5}^{4}, ζ_{5}^{4}, ζ_{5}^{4},
1, −1, −1, −1, 1, 1, −1, −1, 1, −1, ζ_{5}, ζ_{5}^{4}, −ζ_{5}, −ζ_{5}^{4}, ζ_{5}, ζ_{5}^{4}, −ζ_{5}, −ζ_{5}^{4},
I, −I, ζ_{3}, ζ_{3}^{2}, −ζ_{3}^{2}, −ζ_{3}, −ζ_{5}^{3}, −ζ_{5}^{2}, −ζ_{5}^{3}, −ζ_{5}^{2}, ζ_{5}^{3}, ζ_{5}^{2}, ζ_{5}^{3}, ζ_{5}^{2}, ζ_{3}, ζ_{3}^{2}, ζ_{3}, ζ_{3}^{2},
ζ_{5}^{3}, ζ_{5}^{2}, ζ_{5}^{3}, ζ_{5}^{2}, ζ_{5}^{3}, ζ_{5}^{2}, ζ_{5}^{3}, ζ_{5}^{2}, ζ_{15}^{7}, ζ_{15}^{8}, ζ_{15}^{13}, ζ_{15}^{2}, −1, 1)^{t}

(here ζ_{d} := e^{2πi/d} are primitive d-th roots of unity.) As explained above, only 46 lines

APPENDIX: AN EXOTIC FOURIER TRANSFORM FOR H_{4} 3

and columns of S are presented on the following four pages (here s = \sqrt{5}, λ = (1+s)/2 and λ̄ = (1 − s)/2):

\frac{1}{120} ( \begin{matrix}36 & 0 & −36 & −12 & 36 & 12 & 12 & −12 & −12 & 24 & 0 \\ 0 & 45 & 0 & 0 & 0 & −30 & 15 & 0 & 0 & −15 & 30 \\ −36 & 0 & 36 & −12 & 36 & −12 & 12 & 12 & 12 & −24 & 0 \\ −12 & 0 & −12 & 24 − 4 s & −12 & −12 λ & 4 λ̄^{2} & −4 & 24 + 4 s & 12 & −20 \\ 36 & 0 & 36 & −12 & 36 & −12 & 12 & 12 & −12 & 24 & 0 \\ 12 & −30 & −12 & −12 λ & −12 & 24 & 6 & 12 λ̄ & 12 λ̄ & 18 & 0 \\ 12 & 15 & 12 & 4 λ̄^{2} & 12 & 6 & λ̄^{6} & 4 λ̄^{4} & 4 λ^{2} & 3 & 10 \\ −12 & 0 & 12 & −4 & 12 & 12 λ̄ & 4 λ̄^{4} & 4 & 4 & 12 & 20 \\ −12 & 0 & 12 & 24 + 4 s & −12 & 12 λ̄ & 4 λ^{2} & 4 & 24 − 4 s & 12 & 20 \\ 24 & −15 & −24 & 12 & 24 & 18 & 3 & 12 & 12 & 21 & 30 \\ 0 & 30 & 0 & −20 & 0 & 0 & 10 & 20 & 20 & 30 & 40 \\ 12 & 0 & 12 & 12 λ & 12 & 12 λ & −12 λ̄ & −12 λ̄ & 12 λ̄ & −12 & 0 \\ −12 & 0 & 12 & 12 λ & 12 & 12 λ & −12 λ̄ & −12 λ̄ & −12 λ̄ & 12 & 0 \\ −12 & 0 & −12 & −12 λ̄ & −12 & −12 λ̄ & 12 λ & 12 λ & −12 λ & 12 & 0 \\ 12 & 0 & −12 & −12 λ̄ & −12 & −12 λ̄ & 12 λ & 12 λ & 12 λ & −12 & 0 \\ 0 & 45 & 0 & 0 & 0 & 30 & 15 & 0 & 0 & −15 & −30 \\ −36 & 0 & −36 & −12 & 36 & 12 & 12 & −12 & 12 & −24 & 0 \\ 12 & −30 & 12 & −12 λ & −12 & −24 & 6 & −12 λ̄ & 12 λ̄ & 18 & 0 \\ 12 & 15 & −12 & 4 λ̄^{2} & 12 & −6 & λ̄^{6} & −4 λ̄^{4} & 4 λ^{2} & 3 & −10 \\ −12 & 0 & −12 & −4 & 12 & −12 λ̄ & 4 λ̄^{4} & −4 & 4 & 12 & −20 \\ 24 & −15 & 24 & 12 & 24 & −18 & 3 & −12 & 12 & 21 & −30 \\ 0 & 30 & 0 & −20 & 0 & 0 & 10 & −20 & 20 & 30 & −40 \\ 12 & 0 & −12 & 12 λ & 12 & −12 λ & −12 λ̄ & 12 λ̄ & 12 λ̄ & −12 & 0 \\ 12 & 0 & −12 & 12 λ & 12 & −12 λ & −12 λ̄ & 12 λ̄ & 12 λ̄ & −12 & 0 \\ −12 & 0 & −12 & 12 λ & 12 & −12 λ & −12 λ̄ & 12 λ̄ & −12 λ̄ & 12 & 0 \\ −12 & 0 & −12 & 12 λ & 12 & −12 λ & −12 λ̄ & 12 λ̄ & −12 λ̄ & 12 & 0 \\ 0 & −30 & 0 & 0 & 0 & 0 & 30 & 0 & 0 & −30 & 0 \\ 0 & −30 & 0 & 0 & 0 & 0 & 30 & 0 & 0 & −30 & 0 \\ 0 & 0 & 0 & 20 & 0 & 0 & 20 & −20 & −20 & 0 & 20 \\ 0 & 0 & 0 & 20 & 0 & 0 & 20 & −20 & −20 & 0 & 20 \\ 0 & 0 & 0 & 20 & 0 & 0 & 20 & 20 & −20 & 0 & −20 \\ 0 & 0 & 0 & 20 & 0 & 0 & 20 & 20 & −20 & 0 & −20 \\ 12 & 0 & 12 & −12 λ & −12 & 12 λ̄ & 12 λ & −12 λ̄ & 12 λ̄ & −12 & 0 \\ 12 & 0 & 12 & −12 λ & −12 & 12 λ̄ & 12 λ & −12 λ̄ & 12 λ̄ & −12 & 0 \\ 12 & 0 & −12 & −12 λ & −12 & −12 λ̄ & 12 λ & 12 λ̄ & 12 λ̄ & −12 & 0 \\ 12 & 0 & −12 & −12 λ & −12 & −12 λ̄ & 12 λ & 12 λ̄ & 12 λ̄ & −12 & 0 \\ 0 & 0 & 0 & −4 s & 0 & 0 & 4 s & −20 & 4 s & 0 & 20 \\ 0 & 0 & 0 & −4 s & 0 & 0 & 4 s & −20 & 4 s & 0 & 20 \\ −12 & 0 & −12 & −4 λ̄^{2} & −12 & −12 λ̄ & 4 λ̄^{2} & 12 λ̄ & −4 λ^{2} & 12 & 0 \\ −12 & 0 & −12 & −4 λ̄^{2} & −12 & −12 λ̄ & 4 λ̄^{2} & 12 λ̄ & −4 λ^{2} & 12 & 0 \\ −12 & 0 & 12 & −4 λ̄^{2} & −12 & 12 λ̄ & 4 λ̄^{2} & −12 λ̄ & −4 λ^{2} & 12 & 0 \\ −12 & 0 & 12 & −4 λ̄^{2} & −12 & 12 λ̄ & 4 λ̄^{2} & −12 λ̄ & −4 λ^{2} & 12 & 0 \\ 0 & 0 & 0 & −8 s & 0 & 0 & 8 s & 0 & 8 s & 0 & 0 \\ 0 & 0 & 0 & −8 s & 0 & 0 & 8 s & 0 & 8 s & 0 & 0 \\ 24 & 30 & 24 & 8 & −24 & 12 & 2 & 8 & −8 & 6 & −20 \\ 24 & 30 & −24 & 8 & −24 & −12 & 2 & −8 & −8 & 6 & 20 \end{matrix}

\begin{matrix}12 & −12 & −12 & 12 & 0 & −36 & 12 & 12 & −12 & 24 & 0 & 12 \\ 0 & 0 & 0 & 0 & 45 & 0 & −30 & 15 & 0 & −15 & 30 & 0 \\ 12 & 12 & −12 & −12 & 0 & −36 & 12 & −12 & −12 & 24 & 0 & −12 \\ 12 λ & 12 λ & −12 λ̄ & −12 λ̄ & 0 & −12 & −12 λ & 4 λ̄^{2} & −4 & 12 & −20 & 12 λ \\ 12 & 12 & −12 & −12 & 0 & 36 & −12 & 12 & 12 & 24 & 0 & 12 \\ 12 λ & 12 λ & −12 λ̄ & −12 λ̄ & 30 & 12 & −24 & −6 & −12 λ̄ & −18 & 0 & −12 λ \\ −12 λ̄ & −12 λ̄ & 12 λ & 12 λ & 15 & 12 & 6 & λ̄^{6} & 4 λ̄^{4} & 3 & 10 & −12 λ̄ \\ −12 λ̄ & −12 λ̄ & 12 λ & 12 λ & 0 & −12 & −12 λ̄ & −4 λ̄^{4} & −4 & −12 & −20 & 12 λ̄ \\ 12 λ̄ & −12 λ̄ & −12 λ & 12 λ & 0 & 12 & 12 λ̄ & 4 λ^{2} & 4 & 12 & 20 & 12 λ̄ \\ −12 & 12 & 12 & −12 & −15 & −24 & 18 & 3 & 12 & 21 & 30 & −12 \\ 0 & 0 & 0 & 0 & −30 & 0 & 0 & −10 & −20 & −30 & −40 & 0 \\ 24 & 24 & 12 λ & 12 λ & 0 & 12 & 12 λ & −12 λ̄ & −12 λ̄ & −12 & 0 & 24 \\ 24 & 24 & 12 λ & 12 λ & 0 & −12 & −12 λ & 12 λ̄ & 12 λ̄ & −12 & 0 & −24 \\ 12 λ & 12 λ & 24 & 24 & 0 & −12 & −12 λ̄ & 12 λ & 12 λ & 12 & 0 & 12 λ \\ 12 λ & 12 λ & 24 & 24 & 0 & 12 & 12 λ̄ & −12 λ & −12 λ & 12 & 0 & −12 λ \\ 0 & 0 & 0 & 0 & 45 & 0 & 30 & 15 & 0 & −15 & −30 & 0 \\ 12 & −12 & −12 & 12 & 0 & 36 & −12 & −12 & 12 & 24 & 0 & −12 \\ 12 λ & −12 λ & −12 λ̄ & 12 λ̄ & 30 & −12 & 24 & −6 & 12 λ̄ & −18 & 0 & −12 λ \\ −12 λ̄ & 12 λ̄ & 12 λ & −12 λ & 15 & −12 & −6 & λ̄^{6} & −4 λ̄^{4} & 3 & −10 & −12 λ̄ \\ −12 λ̄ & 12 λ̄ & 12 λ & −12 λ & 0 & 12 & 12 λ̄ & −4 λ̄^{4} & 4 & −12 & 20 & 12 λ̄ \\ −12 & −12 & 12 & 12 & −15 & 24 & −18 & 3 & −12 & 21 & −30 & −12 \\ 0 & 0 & 0 & 0 & −30 & 0 & 0 & −10 & 20 & −30 & 40 & 0 \\ 24 & −24 & 12 λ & −12 λ & 0 & −12 & −12 λ & −12 λ̄ & 12 λ̄ & −12 & 0 & 24 \\ −12 λ̄ & 12 λ̄ & −24 & 24 & 0 & −12 & −12 λ & −12 λ̄ & 12 λ̄ & −12 & 0 & −12 λ̄ \\ 24 & −24 & 12 λ & −12 λ & 0 & 12 & 12 λ & 12 λ̄ & −12 λ̄ & −12 & 0 & −24 \\ −12 λ̄ & 12 λ̄ & −24 & 24 & 0 & 12 & 12 λ & 12 λ̄ & −12 λ̄ & −12 & 0 & 12 λ̄ \\ 0 & 0 & 0 & 0 & −30 & 0 & 0 & 30 & 0 & −30 & 0 & 0 \\ 0 & 0 & 0 & 0 & −30 & 0 & 0 & 30 & 0 & −30 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −20 & 20 & 0 & −20 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −20 & 20 & 0 & −20 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −20 & −20 & 0 & 20 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −20 & −20 & 0 & 20 & 0 \\ 12 λ̄ & −12 λ̄ & −12 λ̄ & 12 λ̄ & 0 & −12 & −12 λ̄ & −12 λ & 12 λ̄ & 12 & 0 & −12 λ̄ \\ 12 λ & −12 λ & −12 λ & 12 λ & 0 & −12 & −12 λ̄ & −12 λ & 12 λ̄ & 12 & 0 & −12 λ \\ 12 λ̄ & 12 λ̄ & −12 λ̄ & −12 λ̄ & 0 & 12 & 12 λ̄ & −12 λ & −12 λ̄ & 12 & 0 & −12 λ̄ \\ 12 λ & 12 λ & −12 λ & −12 λ & 0 & 12 & 12 λ̄ & −12 λ & −12 λ̄ & 12 & 0 & −12 λ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 s & −20 & 0 & 20 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 s & −20 & 0 & 20 & 0 \\ 12 λ̄ & 12 λ̄ & −12 λ̄ & −12 λ̄ & 0 & −12 & −12 λ̄ & 4 λ̄^{2} & 12 λ̄ & 12 & 0 & 12 λ̄ \\ 12 λ & 12 λ & −12 λ & −12 λ & 0 & −12 & −12 λ̄ & 4 λ̄^{2} & 12 λ̄ & 12 & 0 & 12 λ \\ 12 λ̄ & −12 λ̄ & −12 λ̄ & 12 λ̄ & 0 & 12 & 12 λ̄ & 4 λ̄^{2} & −12 λ̄ & 12 & 0 & 12 λ̄ \\ 12 λ & −12 λ & −12 λ & 12 λ & 0 & 12 & 12 λ̄ & 4 λ̄^{2} & −12 λ̄ & 12 & 0 & 12 λ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 s & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 s & 0 & 0 & 0 & 0 \\ 12 & −12 & −12 & 12 & −30 & −24 & −12 & −2 & −8 & −6 & 20 & −12 \\ 12 & 12 & −12 & −12 & −30 & 24 & 12 & −2 & 8 & −6 & −20 & −12 \end{matrix}

\begin{matrix}12 & −12 & −12 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 12 & 12 \\ 0 & 0 & 0 & −30 & −30 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ −12 & −12 & −12 & 0 & 0 & 0 & 0 & 0 & 0 & 12 & 12 & −12 \\ 12 λ & 12 λ & 12 λ & 0 & 0 & 20 & 20 & 20 & 20 & −12 λ & −12 λ & −12 λ \\ 12 & 12 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & −12 & −12 & −12 \\ −12 λ & −12 λ & −12 λ & 0 & 0 & 0 & 0 & 0 & 0 & 12 λ̄ & 12 λ̄ & −12 λ̄ \\ −12 λ̄ & −12 λ̄ & −12 λ̄ & 30 & 30 & 20 & 20 & 20 & 20 & 12 λ & 12 λ & 12 λ \\ 12 λ̄ & 12 λ̄ & 12 λ̄ & 0 & 0 & −20 & −20 & 20 & 20 & −12 λ̄ & −12 λ̄ & 12 λ̄ \\ 12 λ̄ & −12 λ̄ & −12 λ̄ & 0 & 0 & −20 & −20 & −20 & −20 & 12 λ̄ & 12 λ̄ & 12 λ̄ \\ −12 & 12 & 12 & −30 & −30 & 0 & 0 & 0 & 0 & −12 & −12 & −12 \\ 0 & 0 & 0 & 0 & 0 & 20 & 20 & −20 & −20 & 0 & 0 & 0 \\ −12 λ̄ & 24 & −12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & 12 λ̄ & 12 λ & 12 λ̄ \\ 12 λ̄ & −24 & 12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & −12 λ̄ & −12 λ & 12 λ̄ \\ −24 & 12 λ & −24 & 0 & 0 & 0 & 0 & 0 & 0 & −12 λ̄ & −12 λ & −12 λ̄ \\ 24 & −12 λ & 24 & 0 & 0 & 0 & 0 & 0 & 0 & 12 λ̄ & 12 λ & −12 λ̄ \\ 0 & 0 & 0 & −30 & −30 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ −12 & 12 & 12 & 0 & 0 & 0 & 0 & 0 & 0 & −12 & −12 & 12 \\ −12 λ & 12 λ & 12 λ & 0 & 0 & 0 & 0 & 0 & 0 & −12 λ̄ & −12 λ̄ & 12 λ̄ \\ −12 λ̄ & 12 λ̄ & 12 λ̄ & 30 & 30 & −20 & −20 & −20 & −20 & −12 λ & −12 λ & −12 λ \\ 12 λ̄ & −12 λ̄ & −12 λ̄ & 0 & 0 & 20 & 20 & −20 & −20 & 12 λ̄ & 12 λ̄ & −12 λ̄ \\ −12 & −12 & −12 & −30 & −30 & 0 & 0 & 0 & 0 & 12 & 12 & 12 \\ 0 & 0 & 0 & 0 & 0 & −20 & −20 & 20 & 20 & 0 & 0 & 0 \\ −12 λ̄ & −24 & 12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & −12 λ̄ & −12 λ & −12 λ̄ \\ 24 & 12 λ̄ & −24 & 0 & 0 & 0 & 0 & 0 & 0 & −12 λ & −12 λ̄ & −12 λ \\ 12 λ̄ & 24 & −12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & 12 λ̄ & 12 λ & −12 λ̄ \\ −24 & −12 λ̄ & 24 & 0 & 0 & 0 & 0 & 0 & 0 & 12 λ & 12 λ̄ & −12 λ \\ 0 & 0 & 0 & 60 & −60 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −60 & 60 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 40 & −20 & 20 & −40 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & −20 & 40 & −40 & 20 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 20 & −40 & 40 & −20 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & −40 & 20 & −20 & 40 & 0 & 0 & 0 \\ −12 λ & 12 λ̄ & 12 λ & 0 & 0 & 0 & 0 & 0 & 0 & 24 & −12 λ & −24 \\ −12 λ̄ & 12 λ & 12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & −12 λ & 24 & 12 λ \\ −12 λ & −12 λ̄ & −12 λ & 0 & 0 & 0 & 0 & 0 & 0 & −24 & 12 λ & 24 \\ −12 λ̄ & −12 λ & −12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & 12 λ & −24 & −12 λ \\ 0 & 0 & 0 & 0 & 0 & 40 & −20 & −20 & 40 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & −20 & 40 & 40 & −20 & 0 & 0 & 0 \\ 12 λ & 12 λ̄ & 12 λ & 0 & 0 & 0 & 0 & 0 & 0 & 24 & −12 λ & 24 \\ 12 λ̄ & 12 λ & 12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & −12 λ & 24 & −12 λ \\ 12 λ & −12 λ̄ & −12 λ & 0 & 0 & 0 & 0 & 0 & 0 & −24 & 12 λ & −24 \\ 12 λ̄ & −12 λ & −12 λ̄ & 0 & 0 & 0 & 0 & 0 & 0 & 12 λ & −24 & 12 λ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ −12 & 12 & 12 & 0 & 0 & 20 & 20 & −20 & −20 & −12 & −12 & 12 \\ −12 & −12 & −12 & 0 & 0 & −20 & −20 & 20 & 20 & 12 & 12 & −12 \end{matrix}

\begin{matrix}12 & 0 & 0 & −12 & −12 & −12 & −12 & 0 & 0 & 24 & 24 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 30 & 30 \\ −12 & 0 & 0 & −12 & −12 & 12 & 12 & 0 & 0 & 24 & −24 \\ −12 λ & −4 s & −4 s & −4 λ̄^{2} & −4 λ̄^{2} & −4 λ̄^{2} & −4 λ̄^{2} & −8 s & −8 s & 8 & 8 \\ −12 & 0 & 0 & −12 & −12 & −12 & −12 & 0 & 0 & −24 & −24 \\ −12 λ̄ & 0 & 0 & −12 λ̄ & −12 λ̄ & 12 λ̄ & 12 λ̄ & 0 & 0 & 12 & −12 \\ 12 λ & 4 s & 4 s & 4 λ̄^{2} & 4 λ̄^{2} & 4 λ̄^{2} & 4 λ̄^{2} & 8 s & 8 s & 2 & 2 \\ 12 λ̄ & −20 & −20 & 12 λ̄ & 12 λ̄ & −12 λ̄ & −12 λ̄ & 0 & 0 & 8 & −8 \\ 12 λ̄ & 4 s & 4 s & −4 λ^{2} & −4 λ^{2} & −4 λ^{2} & −4 λ^{2} & 8 s & 8 s & −8 & −8 \\ −12 & 0 & 0 & 12 & 12 & 12 & 12 & 0 & 0 & 6 & 6 \\ 0 & 20 & 20 & 0 & 0 & 0 & 0 & 0 & 0 & −20 & 20 \\ 12 λ & 0 & 0 & 12 λ̄ & 12 λ & 12 λ̄ & 12 λ & 0 & 0 & 12 & 12 \\ 12 λ & 0 & 0 & 12 λ̄ & 12 λ & −12 λ̄ & −12 λ & 0 & 0 & −12 & 12 \\ −12 λ & 0 & 0 & −12 λ̄ & −12 λ & −12 λ̄ & −12 λ & 0 & 0 & −12 & −12 \\ −12 λ & 0 & 0 & −12 λ̄ & −12 λ & 12 λ̄ & 12 λ & 0 & 0 & 12 & −12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −30 & −30 \\ 12 & 0 & 0 & −12 & −12 & 12 & 12 & 0 & 0 & −24 & 24 \\ 12 λ̄ & 0 & 0 & −12 λ̄ & −12 λ̄ & 12 λ̄ & 12 λ̄ & 0 & 0 & −12 & 12 \\ −12 λ & 4 s & 4 s & 4 λ̄^{2} & 4 λ̄^{2} & 4 λ̄^{2} & 4 λ̄^{2} & 8 s & 8 s & −2 & −2 \\ −12 λ̄ & −20 & −20 & 12 λ̄ & 12 λ̄ & −12 λ̄ & −12 λ̄ & 0 & 0 & −8 & 8 \\ 12 & 0 & 0 & 12 & 12 & 12 & 12 & 0 & 0 & −6 & −6 \\ 0 & 20 & 20 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & −20 \\ −12 λ & 0 & 0 & 12 λ̄ & 12 λ & 12 λ̄ & 12 λ & 0 & 0 & −12 & −12 \\ −12 λ̄ & 0 & 0 & 12 λ & 12 λ̄ & 12 λ & 12 λ̄ & 0 & 0 & −12 & −12 \\ −12 λ & 0 & 0 & 12 λ̄ & 12 λ & −12 λ̄ & −12 λ & 0 & 0 & 12 & −12 \\ −12 λ̄ & 0 & 0 & 12 λ & 12 λ̄ & −12 λ & −12 λ̄ & 0 & 0 & 12 & −12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 40 & −20 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & −20 \\ 0 & −20 & 40 & 0 & 0 & 0 & 0 & 0 & 0 & 20 & −20 \\ 0 & −20 & 40 & 0 & 0 & 0 & 0 & 0 & 0 & −20 & 20 \\ 0 & 40 & −20 & 0 & 0 & 0 & 0 & 0 & 0 & −20 & 20 \\ 12 λ & 0 & 0 & 24 & −12 λ & −24 & 12 λ & 0 & 0 & −12 & 12 \\ −24 & 0 & 0 & −12 λ & 24 & 12 λ & −24 & 0 & 0 & −12 & 12 \\ −12 λ & 0 & 0 & 24 & −12 λ & −24 & 12 λ & 0 & 0 & 12 & −12 \\ 24 & 0 & 0 & −12 λ & 24 & 12 λ & −24 & 0 & 0 & 12 & −12 \\ 0 & 8 s & −4 s & 8 s & 8 s & 8 s & 8 s & 16 s & −8 s & −20 & −20 \\ 0 & −4 s & 8 s & 8 s & 8 s & 8 s & 8 s & −8 s & 16 s & −20 & −20 \\ −12 λ & 8 s & 8 s & 8 λ & 14 − 2 s & 8 λ & 14 − 2 s & −8 λs & 8 λ̄ s & −12 & −12 \\ 24 & 8 s & 8 s & 14 − 2 s & 8 λ & 14 − 2 s & 8 λ & 8 λ̄ s & −8 λs & −12 & −12 \\ 12 λ & 8 s & 8 s & 8 λ & 14 − 2 s & 8 λ & 14 − 2 s & −8 λs & 8 λ̄ s & 12 & 12 \\ −24 & 8 s & 8 s & 14 − 2 s & 8 λ & 14 − 2 s & 8 λ & 8 λ̄ s & −8 λs & 12 & 12 \\ 0 & 16 s & −8 s & −8 λs & 8 λ̄ s & −8 λs & 8 λ̄ s & 16 λ̄ s & 8 λs & 0 & 0 \\ 0 & −8 s & 16 s & 8 λ̄ s & −8 λs & 8 λ̄ s & −8 λs & 8 λs & 16 λ̄ s & 0 & 0 \\ 12 & −20 & −20 & −12 & −12 & 12 & 12 & 0 & 0 & 16 & −16 \\ −12 & −20 & −20 & −12 & −12 & 12 & 12 & 0 & 0 & −16 & 16 \end{matrix} )

2. Remarks on finding S and t

We now describe the method used to find the matrix S and the vector t. First one observes that the 16 fake degrees attached to the big cell of H_{4} (see [1]) are linearly independent regarded as polynomials in q. Hence the sixteen first columns (and by symmetry also the first 16 rows) of S are uniquely determined by the requirement that S transforms the vector of fake degrees into the vector of degrees of unipotent characters. Both the set of fake degrees and the set of principal series unipotent characters in the family are indexed by irreducible characters of the Weyl group of type H_{4}. If one chooses the same ordering on these two sets, the corresponding 16 × 16-submatrix of S turns out to be symmetric.

Condition (7) now already gives a set of linear equations on the entries of t. If we choose the first 16 entries of t equal to 1 (which makes sense since they should be the eigenvalues of Frobenius on principal series characters), these equations already determine the traces of all t_{k} in ℚ(\sqrt{5}).

The set of unipotent character degrees is invariant under the replacement of q by −q in the degree formulas. We now assume that this involution, applied to the rows and columns of S, sends zeroes to zeroes and preserves the irrationality properties of the entries. Then this together with orthogonality to the first 16 columns (condition (3)) allows to uniquely determine the ten columns corresponding to the characters obtained from principal series ones by replacing q by −q.

There exist two more rational character degrees which occur only once, so their entries should be rational and invariant under the permutation on the rows and columns induced by algebraic conjugation. Using orthogonality again and moreover property (6), we may solve for the columns corresponding to these.

At this point we assume in addition that the denominators of entries are divisors of the denominators of the leading terms of the corresponding unipotent character degrees. In particular, if the denominators of the leading terms of two character degrees are coprime, the corresponding entry should vanish. Applied to the six columns belonging to the character degrees with leading term 1/4 or 1/6, this determines several zeroes. Again from (4) and (6), and taking into account the equations (7) that can be derived from the known traces of the t_{k}, we get the six relevant columns.

The remaining unipotent character degrees all involve \sqrt{5}. By the symmetry of S and the previous computations, 34 of the entries in each column are already known. Now orthogonality to the first 34 columns, the positivity property (6) and also the equations in (7) give conditions to narrow down the possibilities for every column. Finally, the assumption on the connection between entries and the corresponding leading terms allows to uniquely solve for all the remaining unknowns.

Remarks. 1) Several properties of the matrix S seem to point to a connection with the group SL_{2}(5). On the other hand it is obvious that the usual non-abelian Fourier transform attached to that group, which also has 74 rows and columns, is different from S.

2) If there exists a tensor category attached to the above fusion datum (see [5]), the vector of dimensions for its simple objects is

dim = (12 λ^{6}, 15 λ^{6}, 12 λ^{6}, 4 λ^{4}, 4 λ^{8}, 12 λ^{6}, 6 λ^{6}, 6 λ^{6}, 1, λ^{12}, 4 λ^{2}, 4 λ^{10}, 4 λ^{8}, 4 λ^{4},
3 λ^{6}, 10 λ^{6}, 12 λ^{5}, 12 λ^{5}, 12 λ^{7}, 12 λ^{7}, 12 λ^{5}, 12 λ^{5}, 12 λ^{7}, 12 λ^{7}, 15 λ^{6}, 12 λ^{6},
6 λ^{6}, 6 λ^{6}, 1, λ^{12}, 4 λ^{2}, 4 λ^{10}, 3 λ^{6}, 10 λ^{6}, 12 λ^{5}, 12 λ^{5}, 12 λ^{5}, 12 λ^{5}, 12 λ^{7}, 12 λ^{7},

12 λ^{7}, 12 λ^{7}, 30 λ^{6}, 30 λ^{6}, 20 λ^{6}, 20 λ^{6}, 20 λ^{6}, 20 λ^{6}, 12 λ^{7}, 12 λ^{7}, 12 λ^{5}, 12 λ^{5}, 12 λ^{7},

12 λ^{7}, 12 λ^{5}, 12 λ^{5}, 9 λ^{6} − 1, 9 λ^{6} − 1, 9 λ^{6} − 1, 9 λ^{6} − 1, 4 λ^{4}, 4 λ^{4}, 4 λ^{8},
4 λ^{8}, 4 λ^{4}, 4 λ^{4}, 4 λ^{8}, 4 λ^{8}, 18 λ^{6} − 2, 18 λ^{6} − 2, 18 λ^{6} − 2, 18 λ^{6} − 2, 2 λ^{6}, 2 λ^{6})^{t}.

Such a category remains to be found. Its structure constants would be given by the numbers occurring in property (6).

References

[1] D. Alvis and G. Lusztig, The representations and generic degrees of the Hecke algebra of type H_{4}, J. reine angew. Math. 336 (1982), 201–212.

[2] M. Broué and G. Malle, Zyklotomische Heckealgebren (1993) (to appear in Astérisque).

[3] G. Lusztig, Characters of reductive groups over a finite field, Annals of Mathematical Studies, no 107, Princeton University Press, Princeton, New Jersey, 1984.

[4] G. Lusztig, Coxeter groups and unipotent representations (appendix to [2]) (1993) (to appear in Astérisque).

[5] G. Lusztig, Exotic Fourier transform, This issue (1993).

I.W.R., Im Neuenheimer Feld 368, D–6900 Heidelberg, Germany

   E-mail address: malle@kalliope.iwr.uni-heidelberg.de