On the Alperin-McKay conjecture for simple groups of type A

Brough, Julian; Späth, Britta

doi:10.1016/j.jalgebra.2020.02.010

Cited by 18 publications

(24 citation statements)

References 39 publications

(64 reference statements)

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“…This implies Theorem B and ensures Assumptions (i) and (ii) of Proposition 1.12. In [BS20,Thm. 4.3] and [CSS21,Prop.…”

Section: The Levi Subgroup and Its Normalizermentioning

confidence: 99%

“…Like in the proofs given in [BS20] and [CSS21], we essentially apply the following statement providing an extension map for non-linear characters.…”

Section: The Levi Subgroup and Its Normalizermentioning

confidence: 99%

“…Studying the group structure of N , our proof uses essentially the Steinberg relations for the structure of G, not its realization as spin group, making probably more uniform a case-by-case but effective proof of Geck-Lusztig's theorem for other quasi-simple groups of Lie type (see [BS20,4.3] and [CSS21,4.13] for types A and C).…”

Section: Introductionmentioning

confidence: 99%

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Extensions of characters in type D and the inductive McKay condition, I

Späth

2021

Preprint

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This is a contribution to the study of Irr(G) as an Aut(G)-set for G a finite quasi-simple group. Focusing on the last open case of groups of Lie type D and 2 D, a crucial property is the so-called condition A ′ (∞) expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in Irr(G). This is part of the stronger A(∞) condition introduced in the context of the reduction of the McKay conjecture to a question on quasi-simple groups. Our main theorem is that a minimal counter-example to condition A(∞) for groups of type D would still satisfy A ′ (∞). This will be used in a second paper to fully establish A(∞) for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of arbitrary standard Levi subgroups of G = D l,sc (q) extend to their stabilizers in the normalizer of that Levi subgroup. This allows to control the action of automorphisms on these extensions. From there Harish Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.

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“…This implies Theorem B and ensures Assumptions (i) and (ii) of Proposition 1.12. In [BS20,Thm. 4.3] and [CSS21,Prop.…”

Section: The Levi Subgroup and Its Normalizermentioning

confidence: 99%

“…Like in the proofs given in [BS20] and [CSS21], we essentially apply the following statement providing an extension map for non-linear characters.…”

Section: The Levi Subgroup and Its Normalizermentioning

confidence: 99%

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Extensions of characters in type D and the inductive McKay condition, I

Späth

2021

Preprint

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“…The situation that a simple group has abelian Sylow -subgroups was considered in [Mal14], and certain low-rank cases have been settled in [Mal14,SF14]. Further, [CS15,BS19] consider the case of groups of Lie type A.…”

Section: Introductionmentioning

confidence: 99%

On the inductive Alperin–McKay conditions in the maximally split case

2021

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The Alperin-McKay conjecture relates height zero characters of an -block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin-McKay conditions about quasi-simple groups by the third author. The validity of those conditions is still open for groups of Lie type. The present paper describes characters of height zero in -blocks of groups of Lie type over a field with q elements when divides q − 1. We also give information about -blocks and Brauer correspondents. As an application we show that quasi-simple groups of type C over F q satisfy the inductive Alperin-McKay conditions for primes ≥ 5 and dividing q − 1. Some methods to that end are adapted from [MS16].

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“…To the present, the (iBAW) condition has been verified for some cases, such as simple alternating groups, many of the sporadic groups, simple groups of Lie type in the defining characteristic, Suzuki groups and Ree groups, simple groups of type G 2 and 3 D 4 , some cases of blocks of groups of type A; see for instance [8], [13], [17], [21], [38], [42], [52] and [54]. Unfortunately, it still seems a far way to deal with the case of simple groups of Lie type in non-defining characteristic in general.…”

Section: Introductionmentioning

confidence: 99%