Alperin Weight Conjecture Research Articles

Finite groups with odd Sylow automizers

Let p be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd p-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture, and the Alperin–McKay conjecture hold for such a group.

The blocks and weights of finite special linear and unitary groups

This paper has two main parts. First, we give a classification of the ℓ-blocks of finite special linear and unitary groups SLn(ϵq) in the non-defining characteristic ℓ≥3. Second, we describe how the ℓ-weights of SLn(ϵq) can be obtained from the ℓ-weights of GLn(ϵq) when ℓ∤gcd(n,q−ϵ), and verify the Alperin weight conjecture for SLn(ϵq) under the condition ℓ∤gcd(n,q−ϵ). As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent ℓ-block of SLn(ϵq) if ℓ∤gcd(n,q−ϵ).

Blocks with the hyperfocal subgroup [formula omitted

In this paper, we calculate the numbers of irreducible ordinary characters and irreducible Brauer characters in a block of a finite group G, whose associated fusion system over a 2-subgroup P of G (which is a defect group of the block) has hyperfocal subgroup Z2n×Z2n for a positive integer number n, when the block is controlled by the normalizer NG(P) and the hyperfocal subgroup is contained in the center of P, or when the block is not controlled by NG(P) and the hyperfocal subgroup is contained in the center of the unique essential subgroup in the fusion system and has order at most 16. In particular, Alperin's weight conjecture holds in the considered cases.

The Inductive Blockwise Alperin Weight Condition for PSL(3, q)

The blockwise Alperin weight conjecture assets that for any finite group G and any prime l, the number of the Brauer characters in an l-block B equals the number of the G-conjugacy classes of l-weights belonging to B. Recently, the inductive blockwise Alperin weight condition has been introduced such that the blockwise Alperin weight conjecture holds if all non-abelian simple groups satisfy these conditions. We will verify the inductive blockwise Alperin weight condition for the finite simple groups PSL(3, q) in this paper.

The inductive blockwise Alperin weight condition for G2(q) and [formula omitted

The inductive blockwise Alperin weight condition is a system of conditions whose verification for all non-abelian finite simple groups would imply the blockwise Alperin weight conjecture. We establish this condition for the groups G2(q), q⩾5, and D43(q) for all primes dividing their order.

Completion of A Bijective Proof of the Alperin Weight Conjecture for S n when p = 2

In this paper, we prove the Alperin weight conjecture for the symmetric group Sn in a bijective way when p = 2. More precisely, we provide a bijection between the set of conjugacy classes of 2-weights for Sn and the set of 2-regular partitions of n.

Formula omitted] is “Good” for the McKay, Alperin weight, and related local–global conjectures

The so-called “local–global” conjectures in the representation theory of finite groups relate the representation theory of G to that of certain proper subgroups, such as the normalizers of particular p-groups. Recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups. Here, we show that G=Sp6(2a) is “good” for these reductions for the McKay conjecture, the Alperin weight conjecture, and their blockwise versions.

On Alperin's weight conjecture for $p$-blocks of $p$-solvable groups

For $p$-solvable groups, a strong form of Alperin's weight conjecture has been proved by T. Okuyama (unpublished). L. Barker has refined this theorem by taking Green correspondence into account. We prove here a relative version of Barker's theorem.

On the inductive Alperin–McKay and Alperin weight conjecture for groups with abelian Sylow subgroups

We study the inductive Alperin–McKay conjecture, the inductive Isaacs–Navarro refinement and the inductive blockwise Alperin weight conjecture for groups of Lie type in the generic case of abelian Sylow ℓ-subgroups. We also show that the alternating groups, the Suzuki groups and the Ree groups satisfy the inductive condition necessary for Späthʼs reduction of the blockwise Alperin weight conjecture to the case of simple groups.

Two remarks on the reduction of Alperin's weight conjecture

The so-called inductive McKay condition on finite simple groups, due to Isaacs–Malle–Navarro, has recently been reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro–Tiep. This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.

The strengthened Alperin Weight Conjecture for p-solvable groups

The Alperin Weight Conjecture is a well-known conjecture, and it is central to the modern representation theory of finite groups. It is known to be true for many types of finite groups, but it remains open. In particular, the conjecture is known to be true for finite p-solvable groups. In this paper, we prove that a very strong form of the conjecture holds for all finite p-solvable groups. It follows that the strengthening of the Alperin Weight Conjecture that includes Galois automorphisms over the p-adic numbers holds for all finite p-solvable groups. This strengthening of the Alperin Weight Conjecture was suggested in an earlier work of Navarro on the McKay Conjecture. The present paper provides the first direct evidence (beyond the calculation of individual cases and the case of groups of odd order) for this strengthening.

A reduction theorem for the blockwise Alperin weight conjecture

Abstract.We show that the blockwise version of the Alperin weight conjecture is true if for every finite non-abelian simple group a set of conditions holds. Furthermore we prove that several series of simple groups satisfy these assumptions. This refines recent work of Navarro and Tiep, who proved an analogous reduction theorem for the non-blockwise version of the Alperin weight conjecture.

Equivariant Alperin–Robinsonʼs Conjecture reduces to almost-simple [formula omitted]-groups

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions. Our purpose here is to show that the equivariant refinement of the Alperinʼs Conjecture for blocks formulated by Geoffrey Robinson in the eighties can be reduced to checking the same statement on any central k⁎-extension of any finite almost-simple group, or of any finite simple group up to verifying an “almost necessary” condition. In Appendix A we develop some old arguments that we need in the proof.

The AWC-goodness and essential rank of sporadic simple groups

In a recent paper, Navarro and Tiep defined the property AWC-good for finite simple groups. They proved that the Alperin Weight Conjecture holds for every finite group if every finite simple group is AWC-good. We show that every sporadic simple group is AWC-good. Our computational proof requires to construct many radical subgroups of sporadic simple groups up to conjugacy; we provide these groups in an extensive appendix. As another application, for every sporadic simple group G and prime p, we determine the essential p-rank of G, that is, the number of G-conjugacy classes of essential subgroups of a Sylow p-subgroup D of G. The essential p-rank is closely related to the minimal cardinality of a conjugation family for the Frobenius category FG(D).

Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8

Using the classification of finite simple groups, we prove Alperin's weight conjecture and the character theoretic version of Broué's abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

A reduction theorem for the Alperin weight conjecture

The Alperin Weight Conjecture was stated by J. Alperin in 1986 and constitutes one of the main problems in the representation theory of finite groups. In this paper, we provide a reduction to simple groups and as an application we prove the conjecture in several cases.

The Alperin weight conjecture and Uno’s conjecture for the Monster 𝕄, p odd

AbstractSuppose thatpis 3,5,7,11 or 13. We classify the radicalp-chains of the Monster 𝕄 and verify the Alperin weight conjecture and Uno’s reductive conjecture for 𝕄, the latter being a refinement of Dade’s reductive conjecture and the Isaacs–Navarro conjecture.

On stable equivalences and blocks with one simple module

Using a stable equivalence due to Rouquier, we show that Alperin's weight conjecture holds for any p-block of a finite group with defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules.

Endomorphism rings of permutation modules

Let k be an algebraically closed field of characteristic p and G a finite group. Then the permutation module k P G of G on the cosets of a Sylow p -subgroup P is via Fitting correspondence strongly related to its endomorphism ring End k G ( k P G ) . On the other hand, each Green correspondent in G of a weight module of G occurs as a direct summand of k P G . This fact suggests to analyze both structures, the permutation module and the associated endomorphism ring towards hints at a proof for Alperin's weight conjecture. We present a selection of such investigations for different groups and characteristics. In particular we focus on the socle and head constituents of the indecomposable direct summands of k P G and of the PIMs of E E .

A Bijection for the Alperin Weight Conjecture in S n

The Alperin weight conjecture states that if G is a finite group and p is a prime, then the number of irreducible Brauer characters of a group G should be equal to the number of conjugacy classes of p-weights of G. This conjecture is known to be true for the symmetric group S n , however there is no explicit bijection given between the two sets. In this paper we develop an explicit bijection between the p-weights of S n and a certain set of partitions that is known to have the same cardinality as the irreducible Brauer characters of S n . We also develop some properties of this bijection, especially in relation to a certain class of partitions whose corresponding Specht modules over fields of characteristic p are known to be irreducible.