Alperin Weight Conjecture Research Articles

Weights for 𝜋-partial characters of 𝜋-separable groups

Abstract The aim of this paper is to confirm a conjecture of Isaacs and Navarro from 1995, which asserts that, for any π ′ \pi^{\prime} -subgroup 𝑄 of a 𝜋-separable group 𝐺, the number of π ′ \pi^{\prime} -weights of 𝐺 with 𝑄 as the first component is larger than or equal to the number of irreducible 𝜋-partial characters of 𝐺 with 𝑄 as their vertex. We also give a sufficient condition to guarantee that these two numbers are equal, and thereby strengthen their main theorem on the 𝜋-version of the Alperin weight conjecture.

Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

AbstractLet k(B0) and l(B0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B0 of a finite group G. We prove that, if k(B0)−l(B0) = 1, then l(B0) ≥ p − 1 or else p = 11 and l(B0) = 9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B0) ≥ p − 1 or else p = 11 and $$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ G / O p ′ ( G ) ≅ C 11 2 ⋊ SL ( 2 , 5 ) . These results are useful in the study of principal blocks with few characters.We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least $$2\sqrt {p - 1} + 1 - {k_p}(G)$$ 2 p − 1 + 1 − k p ( G ) , where kp(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.

Modular representations of finite groups and Lie theory

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent.The expected behavior of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broué's conjecture, and we raise a new question related to the change of central character.We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two-variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas.

Weights for ℓ-local compact groups

In this note, we initiate the study of F-weights for an ℓ-local compact group F over a discrete ℓ-toral group S with discrete torus T. Motivated by Alperin's Weight Conjecture for simple groups of Lie-type, we conjecture that when T is the unique maximal abelian subgroup of S up to F-conjugacy and every element of S is F-fused into T, the number of weights of F is bounded above by the number of ordinary irreducible characters of its Weyl group. By combining the structure theory of F with the theory of blocks with cyclic defect group, we are able to give a proof of this conjecture in the case when F is simple and |S:T|=ℓ. We also propose and give evidence for an analogue of the height zero case of Robinson's Ordinary Weight conjecture in this setting.

Morita equivalences and the inductive blockwise Alperin weight condition for type 𝖠

As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro–Tiep and Späth for simple groups of Lie type A {\mathsf {A}} , split or twisted. Key to the proofs is to reduce the verification of the inductive condition to the isolated (that means unipotent) blocks, using the Jordan decomposition for blocks of finite reductive groups given by Bonnafé, Dat and Rouquier.

On the inductive blockwise Alperin weight condition for type [formula omitted

In this paper we prove the blockwise Alperin weight conjecture for finite special linear and unitary groups, for finite groups with abelian Sylow 3-subgroups, and verify the inductive blockwise Alperin weight condition for certain cases of groups of type A. We also give a classification for the 2-blocks of special linear and unitary groups.

Alternating sums over π-subgroups

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p -defect is determined by local subgroups. In this paper we replace p by a set of primes π and prove a π -version of Dade's conjecture for π -separable groups. This extends the (known) p -solvable case of the original conjecture and relates to a π -version of Alperin's weight conjecture previously established by the authors.

Alperin weight conjecture and related developments

The Alperin weight conjecture is central to the modern representation theory of finite groups, and it is still open, despite many different approaches from different points of view. This paper surveys methods and results relating the Alperin weight conjecture, especially the recent developments in the inductive investigation.

Jordan decomposition for weights and the blockwise Alperin weight conjecture

The Alperin weight conjecture was reduced to simple groups by the work of Navarro–Tiep and Späth. To prove Alperin weight conjecture, it suffices to show that all finite non-abelian simple groups are BAW-good. We reduce the verification of the inductive conditions for groups of Lie type in non-defining characteristic to quasi-isolated blocks.

ON ALPERIN’S LOWER BOUND FOR THE NUMBER OF BRAUER CHARACTERS

We prove that the number of conjugacy classes of a finite group G consisting of elements of odd order, is larger than or equal to that number for the normaliser of a Sylow 2-subgroup of G. This is predicted by the Alperin Weight Conjecture.

Inductive blockwise Alperin weight condition for type [formula omitted] and odd primes

By the reduction theorems of Navarro–Tiep and Späth, a way to prove the Alperin weight conjecture and its blockwise version is to verify the so-called inductive Alperin weight condition and inductive blockwise Alperin weight condition for all finite simple groups respectively. In this paper, we establish the inductive blockwise Alperin weight condition for simple groups of type B and odd primes, using a criterion given by Brough and Späth recently.

A criterion for the inductive Alperin weight condition

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438, 2.12) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro–Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5).

Equivariant correspondences and the inductive Alperin weight condition for type 𝖠

In this paper, we establish the inductive Alperin weight condition for the finite simple groups of Lie type A \mathsf A , contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups.

Some results on a variant of the Green correspondence with applications to Alperin's weight conjecture for blocks

Some results on a variant of the Green correspondence with applications to Alperin's weight conjecture for blocks

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

In this paper characters of the normalizer of d-split Levi subgroups in SLn(q) and SUn(q) are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer. These results are applied to verify the Alperin-McKay conjecture for primes ℓ with ℓ∤6(q2−1) and the Alperin weight conjecture for ℓ-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of SLn(q) and SUn(q).

Weights and Nilpotent Subgroups

Abstract In a finite group $G$, we consider nilpotent weights and prove a $\pi $-version of the Alperin Weight Conjecture for certain $\pi $-separable groups. This widely generalizes an earlier result by I. M. Isaacs and the 1st author.

An equivariant bijection between irreducible Brauer characters and weights for Sp(2n,q)

The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Späth and Koshitani. Thus, to prove this conjecture, it suffices to verify the corresponding inductive condition for all finite simple groups. The first is to establish an equivariant bijection between irreducible Brauer characters and weights for the universal covering groups of simple groups. Assume q is a power of some odd prime p. We first prove the blockwise Alperin weight conjecture for Sp2n(q) and odd non-defining characteristics. If the decomposition matrix of Sp2n(q) is unitriangular with respect to an Aut(Sp2n(q))-stable basic set (this assumption holds for linear primes), we can establish an equivariant bijection between the irreducible Brauer characters and weights.

The Inductive Blockwise Alperin Weight Condition for PSp4(q)

Let G be a finite group and ℓ be any prime dividing [Formula: see text]. The blockwise Alperin weight conjecture states that the number of the irreducible Brauer characters in an ℓ-block B of G equals the number of the G-conjugacy classes of ℓ-weights belonging to B. Recently, this conjecture has been reduced to the simple groups, which means that to prove the blockwise Alperin weight conjecture, it suffices to prove that all non-abelian simple groups satisfy the inductive blockwise Alperin weight condition. In this paper, we verify this inductive condition for the finite simple groups [Formula: see text] and non-defining characteristic, where q is a power of an odd prime.

On the inductive blockwise Alperin weight condition for classical groups

Recently, there has been substantial progress on the Alperin weight conjecture. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for simple groups of classical type under some additional assumption.

The Alperin weight conjecture for symmetric and general linear groups revisited

Explicit bijections are given between the weights associated to a block of a symmetric group and the irreducible Brauer characters in the block. Similar bijections are given between the weights associated to a block of a general linear group and a basic set of the block.