On Blocks with Abelian Defect Groups of Small Rank

Sambale, Benjamin

doi:10.1007/s00025-016-0557-4

Cited by 6 publications

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“…For G = E 7 (q) by the table in [11, p. 354], the only relevant blocks are for ℓ = 2 the blocks lying above the d-cuspidal unipotent characters of Levi subgroups of type E 6 , and for ℓ = 3 a block lying above the d-cuspidal unipotent character of a Levi subgroup of type D 4 . In the case ℓ = 2 the defect groups are dihedral (see [11, p. 357]) in which case our claim is known to hold, see [29,Cor. 8.2].…”

Section: Exceptional Groups In Non-defining Characteristicmentioning

confidence: 90%

On a minimal counterexample to Brauer’s k(B)-conjecture

Malle

2018

Isr. J. Math.

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We study Brauer's long-standing k(B)-conjecture on the number of characters in p-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for p ≥ 5 nor in the case of abelian defect. For p = 3 we obtain that the principal 3-blocks do not provide minimal counterexamples. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes. Date: April 3, 2018. 2010 Mathematics Subject Classification. 20C15, 20C33. Key words and phrases. Number of simple modules, Brauer's k(B)-conjecture, blocks of simple groups. The author gratefully acknowledges financial support by SFB TRR 195.1 2 GUNTER MALLE Theorem 2. Let p be a prime and B a p-block of a finite quasi-simple group G with abelian defect groups. Then B is not a minimal counterexample to Brauer's k(B)-conjecture.We also obtain strong restrictions for the primes p = 2, 3:Theorem 3. Assume that a p-block B of a finite quasi-simple group G is a minimal counterexample to Brauer's k(B)-conjecture. Then p ≤ 3, G is of Lie type in characteristic not p, B is an isolated block of G, the defect groups of B are non-abelian, and either p = 2 or B is not unipotent. In particular, the principal 3-block is not a minimal counterexample.The proofs will be given in the subsequent sections, using the classification of finite simple groups in conjunction with Lusztig's theory of characters of finite reductive groups. While our methods fall short of verifying the k(B)-conjecture for all blocks of quasi-simple groups, since they partly rely on Bonnafé-Rouquier type reduction arguments to rule out a minimal counterexample, in the most interesting case of unipotent blocks with p > 2 and non-abelian defect our arguments actually show that these do satisfy the k(B)-conjecture.Remark 1.1. Let us comment on the cases left open by our results. For p = 2, 3 the isolated blocks of all quasi-simple groups of Lie type remain to be considered. For p = 2, already the case of unipotent blocks of SL n (q) seems hard. Remark 1.2. If B is a block with abelian defect group D, then according to (the proven direction of) Brauer's height zero conjecture all characters in Irr(B) are of height zero. By the Alperin-McKay conjecture they should be in bijection with the height zero characters of the Brauer corresponding block b of N G (D) (which also has defect group D), whence k(B) = k(b). Thus, assuming the validity of the Alperin-McKay conjecture, a block B with non-normal abelian defect group cannot be a minimal counterexample to the k(B)conjecture. The interesting situation hence rather seems to be the one of non-abelian defect groups; see also the recent result of Sambale recalled in Theorem 2.1.No reduction of the general conjecture to the case of (quasi-)simple groups has been found so far; see Navarro's article [24] for some thoughts and ideas in that direction.The paper is built up as follows: In Section 2 we settle the case of quasi-simple groups not of Lie type, mostly by collecting results from the l...

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