On the inductive blockwise Alperin weight condition for classical groups

Feng, Zhicheng; Li, Zhenye; Zhang, Jiping

doi:10.1016/j.jalgebra.2019.07.022

Cited by 18 publications

(24 citation statements)

References 61 publications

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“…• G = Spin 2n+1 (q) (with n ≥ 2), Spin − 2n (q) (with n ≥ 4) or Spin + 2n (q) (with n ≥ 5) and both q and ℓ are odd (cf. [28]),…”

Section: Then Any Block B In S Is Baw-goodmentioning

confidence: 99%

“…[23]). We mention that for the case G = Sp 2n (q) and odd ℓ, the statement in [28] assumes that q is odd. For even q, the proof for odd q in [28] also applies, using the weights given in [43, §6].…”

Section: Then Any Block B In S Is Baw-goodmentioning

confidence: 99%

“…In particular, the unipotent blocks are an important class of quasi-isolated blocks. We mention that the unipotent blocks had been checked to satisfy the inductive BAW condition for many cases, especially for most of the classical groups [27,28]; see Remark 5.9.…”

Section: Introductionmentioning

confidence: 99%

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Jordan decomposition for weights and the blockwise Alperin weight conjecture

Feng¹,

Li²,

Zhang³

2021

Preprint

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“…• G = Spin 2n+1 (q) (with n ≥ 2), Spin − 2n (q) (with n ≥ 4) or Spin + 2n (q) (with n ≥ 5) and both q and ℓ are odd (cf. [28]),…”

Section: Then Any Block B In S Is Baw-goodmentioning

confidence: 99%

Section: Then Any Block B In S Is Baw-goodmentioning

confidence: 99%

See 1 more Smart Citation

Jordan decomposition for weights and the blockwise Alperin weight conjecture

Feng¹,

Li²,

Zhang³

2021

Preprint

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“…Du), pcli17@pku.edu.cn (P. Li), zsy0509@pku.edu.cn (S. Zhao). groups, simple groups of type 2 F 4 , see [37]; simple groups of Lie type in the defining characteristic, simple groups of type 2 B 2 and 2 G 2 , see [41]; simple groups of type G 2 and 3 D 4 , see [40]; some cases of type A, see [16], [17] and [19]; the case of type C under the assumption that the decomposition matrix with respect to E(G, l ′ ) is unitriangular, see [20] and [29]; unipotent blocks of classical groups, see [18].…”

Section: W(b)| = |Ibr(b)|mentioning

confidence: 99%

On the inductive blockwise Alperin weight condition for the unipotent blocks of finite groups of Lie type E_6

Du,

Li,

Zhao

2021

Preprint

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“…In [14], the case for type C and linear primes are considered using a different method; they also consider some other cases for classical groups.…”

Section: Introductionmentioning

confidence: 99%

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

2019

Journal of Algebra

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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 Sp 2n (q) and odd non-defining characteristics. If the decomposition matrix of Sp 2n (q) is unitriangular with respect to an Aut(Sp 2n (q))-stable basic set (this assumption holds for linear primes), we can establish an equivariant bijection between the irreducible Brauer characters and weights. , then (R, ϕ) is called a B-weight. Denote the set of all G-conjugacy classes of B-weights by W(B). In the sequel, the term "weight" will mean a single weight or a conjugacy class of weights depending on the context. J. L. Alperin gives the following conjecture in [1] called blockwise Alperin weight (BAW) conjecture.Conjecture (Alperin). Let G be a finite group, ℓ a prime and B an ℓ-block of G,This conjecture has been reduced to the simple groups, which means that the BAW conjecture holds for a finite group G if all non-abelian simple groups involved in G satisfy the inductive blockwise Alperin weight (iBAW) condition. See [31] for a definition of the iBAW condition and see [19] for another version. Basically speaking, the iBAW condition consists of two parts: the first one requires to establish an equivariant bijection between irreducible Brauer characters and weights; the second one (normally embedded conditions) requires to consider the extension of irreducible Brauer characters and weight characters. This inductive condition has been verified for several cases, such as: simple alternate groups, many sporadic simple groups, simple groups of Lie type and the defining characteristic, some few cases of simple groups of Lie type and primes different from the defining characteristic; see [7], [10], [13], [19], [21], [24], [28], [29], [31], etc.The purpose of this paper is to start the consideration of the iBAW condition for PSp 2n (q) with q a power of an odd prime p and an odd prime ℓ p. To do this, we first need to establish a blockwise bijection between the irreducible Brauer characters and weights of the universal covering group Sp 2n (q) of PSp 2n (q).Theorem 1. Let p be an odd prime, q = p f and ℓ an odd prime different from p, then the blockwise Alperin weight conjecture holds for Sp 2n (q) for ℓ.In [15], the classification of blocks and the partition of irreducible ordinary characters into blocks for CSp 2n (q) are obtained. Using the results of Fong and Srinivasan in [15] and the results of Lusztig in [20], we can classify the blocks of Sp 2n (q) and partition irreducible ordinary characters into blocks. To give a parametrization of irreducible Brauer characters, we use the basic set E(Sp 2n (q), ℓ...

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On the inductive blockwise Alperin weight condition for classical groups

Abstract: 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.

Cited by 18 publications

References 61 publications

Jordan decomposition for weights and the blockwise Alperin weight conjecture

Jordan decomposition for weights and the blockwise Alperin weight conjecture

On the inductive blockwise Alperin weight condition for the unipotent blocks of finite groups of Lie type E_6

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

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