Computation of Kazhdan-Lusztig polynomials and some applications to finite groups

Lübeck, Frank

doi:10.1090/tran/8037

Cited by 10 publications

(4 citation statements)

References 30 publications

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“…In this direction, a well known conjecture of G.E. Wall from the early 1960s (see [30]) asserts that the bound |M(H)| |H| holds for every finite group H. Wall himself proved this for soluble groups, but it has recently been shown to be Date: July 12, 2021. false in general (see [23]). However, the conclusion is still expected to be valid when H is an almost simple group, but a proof remains far out of reach, even though there have been major advances in recent decades in our understanding of the subgroup structure of almost simple groups.…”

Section: Introductionmentioning

confidence: 99%

A note on extremely primitive affine groups

Burness

Thomas

2020

Arch. Math.

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Let G be a finite primitive permutation group on a set $$\Omega $$ Ω with non-trivial point stabilizer $$G_{\alpha }$$ G α . We say that G is extremely primitive if $$G_{\alpha }$$ G α acts primitively on each of its orbits in $$\Omega {\setminus } \{\alpha \}$$ Ω \ { α } . In earlier work, Mann, Praeger, and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall’s conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture.

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Section: Introductionmentioning

confidence: 99%

A note on extremely primitive affine groups

Burness

Thomas

2020

Arch. Math.

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“…Wall from 1961 asserts that |M| < |H|. Wall's conjecture was originally formulated for all finite groups, but counterexamples have recently been constructed, see [62]. However, the conjecture has been established for all sufficiently large alternating and symmetric groups (see [58]) and a theorem of Liebeck, Martin and Shalev [49] implies that |M| < |H| 1+o (1) for all almost simple groups H of Lie type.…”

Section: Remarkmentioning

confidence: 99%

The Classification of Extremely Primitive Groups

Burness

Thomas

2021

International Mathematics Research Notices

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Dedicated to the memory of Jan Saxl Let $G$ be a finite primitive permutation group on a set $\Omega $ with nontrivial point stabilizer $G_{\alpha }$. We say that $G$ is extremely primitive if $G_{\alpha }$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha \}$. These groups arise naturally in several different contexts, and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger, and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger, and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.

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“…Remark 3.2. It is shown in Section 4 of Lübeck [17] that for all large enough primes, taking S to be the trivial module Fp , there are simple modules T for P SL(n, Fp ) with dim Fq H 1 (P SL(n, Fp ), T ) 3 16 469 36672 and then for large enough powers q of p, the restriction of these T are irreducible modules for F q P SL(n, F q ) with these dimensions for H 1 (P SL(n, F q ), T ). It is not known whether these dimensions are unbounded for larger values of n, though that seems very likely.…”

Section: Examples Involving Three Dimensional Ext 1 Groupsmentioning

confidence: 99%

Centralisers of finite groups in locally finite simple groups

Benson

2022

Preprint

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We answer in the negative a question of Hartley about representations of finite groups, by constructing examples of finite simple groups with arbitrarily large representations whose endomorphism ring consists of just the scalars. We show as a consequence that there are finite simple groups of automorphisms of the locally finite simple group SL(∞, F q ) with trivial centraliser. The smallest of our examples is A 6 with q = 9.

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Computation of Kazhdan-Lusztig polynomials and some applications to finite groups

Cited by 10 publications

References 30 publications

A note on extremely primitive affine groups

A note on extremely primitive affine groups

The Classification of Extremely Primitive Groups

Centralisers of finite groups in locally finite simple groups

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