Restricting unipotent characters in special orthogonal groups

Abstract

For all prime powers q we restrict the unipotent characters of the special orthogonal groups SO5(q) and SO7(q) to a maximal parabolic subgroup. We determine all irreducible constituents of these restrictions for SO5(q) and a large part of the irreducible constituents for SO7(q).

Decomposition numbers ofSO7(q)andSpHimstedt1, Noeske2 2014Journal of AlgebraSelf Cite180180View full textAdd to dashboardCite

“…The inertia subgroup in P of the linear character λ 0 is I 0 =P n−2 ⋉ U whereP n−2 ∼ = P n−2 . We identify Irr(P n−2 ) with Irr(P n−2 ) via the bijection in [HN,3.3] and extend λ 0 trivially toλ 0 ∈ Irr(I 0 ).…”

“…Let q be an arbitrary prime power. Each unipotent character of the groups SO 2m+1 (q), Sp 2m (q) and CSp 2m (q) is labeled by a symbol Λ or a triple [α, β, d] where (α, β) is a bipartition and d is the defect of Λ; see [Car85,Section 13.8] and [HN,Section 7]. Each of the groups SO 5 (q), Sp 4 (q) and CSp 4 (q) has six and each of the groups SO 7 (q), Sp 6 (q) and CSp 6 (q) has twelve unipotent characters.…”

“…If n = 5 we have L − 5 ∼ = GO − 2 (q), which is a dihedral group of order 2(q + 1); see [Tay92,Theorem 11.4]. For odd q, let ν − 1 be the non-trivial linear character of L − 5 with SO − 2 (q) ≤ ker(ν − 1 ); see [HN,Remark 7.1] for additional information. If q is even then L − 5 only possesses two linear characters, and in this case we write ν − 1 for the unique non-trivial linear character of L − 5 .…”

“…, Ψ 5 with the ordinary unipotent characters of G 5 are given in Table 6 and can be determined as follows: Since Harish-Chandra induction commutes with taking unipotent quotients (see [His93,Lemma 6.1]) it is straightforward to compute the scalar products of Ψ 1 , Ψ 3 , Ψ 4 with the ordinary unipotent characters of G 5 . The scalar products of Ψ 2 follow from [HN,Theorem 7.2] and Frobenius reciprocity. By [Car85, Section 12.1] the Steinberg character St G 5 is the only unipotent constituent of Ψ 5 and it has multiplicity one.…”

The irreducible characters of the Sylow $p$-subgroups of the Chevalley groups $\mathrm{D}_6(p^f)$ and $\mathrm{E}_6(p^f)$

“…More recently this has been taken further in [DM15] and [DM16] for some nonlinear classes of primes. For small rank groups the calculation of decomposition numbers in [Him11], [HN14] and [HN15] made strong use of the representation theory of parabolic subgroups along with induction/restriction methods to compute decomposition numbers. Most recently the third author [Pao17] was able to compute most decomposition numbers of the groups D 4 (2 f ) using the generic character table of UD 4 (2 f ) which was computed in [GLM17]; here and throughout UY r (q) denotes a Sylow p-subgroup of the group Y r (q) of type Y and rank r defined over F q .…”