Decomposition matrices for groups of Lie type in non-defining characteristic

Abstract

Even-dimensional split orthogonal groups 2. Unipotent decomposition matrix of E 6 (q) 3. A non-principal block of E 8 (q) 4. Even-dimensional non-split orthogonal groups 5. Odd-dimensional orthogonal groups 6. Symplectic groups 7. Unipotent decomposition matrix of F 4 (q) Chapter 7. Decomposition matrices at d ℓ (q) =

Decomposition numbers for the principal $Φ_{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$

“…B2 G = −2 and R w ; .1 2n G = 1. On the other hand, if Φ 2n (q) ℓ > 4n then there exists by [11,Ex. 1.17] an ℓ-character of T w in general position.…”

“…• (linear prime case) q has order n in F × ℓ , in which case n is necessarily odd, and the representation theory of G behaves as a type A phenomenon and can be deduced from the representation theory of q-Schur algebras of symmetric groups [26]; • (unitary prime case) q has order 2n in F × ℓ . Explicit decomposition matrices in that case were obtained by Okuyama-Waki for m = 2 [32] and by Malle and the first author for m = 4 [10] and m = 6 [11].…”

“…When G is not of type A, less is understood about the decomposition matrix of the unipotent blocks of G. There are more cuspidal unipotent representations in characteristic 0 which give rise to multiple Hecke and quasihereditary algebras, all of which play a role in the unipotent blocks of G. However, in the general case, the knowledge of the decomposition numbers for these algebras is not enough to determine those of G. In [8] the first author initiated the use of Deligne-Lusztig characters to find the missing numbers. This proved successful in determining decomposition matrices for finite groups of Lie type in small rank, see [27], [9], [10], [11].…”