[Paper Review] On the inductive blockwise Alperin weight condition for type $\mathsf A$

Zhicheng Feng, Conghui Li+1 more|arXiv (Cornell University)|Aug 14, 2020

Finite Group Theory Research31 references4 citations

TL;DR

This paper proves the blockwise Alperin weight conjecture for finite special linear and unitary groups over finite fields, establishes the inductive blockwise Alperin weight condition for unipotent and maximal defect blocks, and classifies 2-blocks of these groups using semisimple 2'-elements and orbit representatives. It extends prior results by verifying the condition for groups of type A with square-free gcd(n, q−η)ℓ′ and for finite groups with abelian Sylow 3-subgroups.

ABSTRACT

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 $\mathsf A$. We also give a classification for the 2-blocks of special linear and unitary groups.

Motivation & Objective

• To verify the inductive blockwise Alperin weight (iBAW) condition for finite groups of type A, including special linear and unitary groups.
• To establish the blockwise Alperin weight conjecture for special linear and unitary groups at all primes.
• To classify the 2-blocks of finite special linear and unitary groups when the defining characteristic is odd.
• To extend previous results on unipotent and maximal defect blocks to broader classes of blocks and primes.
• To prove the blockwise Alperin weight conjecture for finite groups with abelian Sylow 3-subgroups.

Proposed method

• Utilizes the reduction results of Späth (2013) and Navarro–Tiep (2011) to reduce the blockwise Alperin weight conjecture to verifying the iBAW condition on finite simple groups of type A.
• Applies the classification of Brauer pairs for GLn(ηq) by Broué and the radical subgroup structure of SLn(ηq) from prior work to analyze 2-blocks.
• Employs the notion of d-Jordan-cuspidal pairs and semisimple 2'-elements to parametrize 2-blocks of SLn(ηq), generalizing earlier work for odd primes.
• Introduces (n,2)-admissible block symbols as a labeling set for 2-blocks, defined via orbits of roots of unity under Frobenius and degree conditions.
• Defines κ(𝔟) as the number of O₂′(ℤ)-fixed points on the set of blocks, which counts the number of 2-blocks of SLn(ηq) covered by a given 2-block of GLn(ηq).
• Uses group action and orbit counting to show that the set of (B𝔟)j, indexed over O₂′(ℤ)-orbits of admissible symbols and j=1,…,κ(𝔟), forms a complete set of 2-blocks.

Experimental results

Research questions

• RQ1 Does the inductive blockwise Alperin weight condition hold for unipotent and maximal defect blocks of SLn(ηq) at all primes?
• RQ2 Can the iBAW condition be verified for simple groups PSLn(ηq) when gcd(n, q−η)ℓ′ is square-free?
• RQ3 What is the complete parametrization of 2-blocks of SLn(ηq) for odd q, and how many 2-blocks are covered by a single 2-block of GLn(ηq)?
• RQ4 Does the blockwise Alperin weight conjecture hold for finite groups with abelian Sylow 3-subgroups?
• RQ5 Is the assumption of odd primes necessary in the d-Jordan-cuspidal pair classification for blocks of finite groups of Lie type?

Key findings

• The blockwise Alperin weight conjecture holds for all finite special linear and unitary groups and all primes.
• The inductive blockwise Alperin weight condition is verified for unipotent and maximal defect blocks of SLn(ηq) at any prime ℓ not dividing q.
• For simple groups PSLn(ηq) with gcd(n, q−η)ℓ′ square-free, the iBAW condition holds, and this implies the conjecture for n ≤ 7.
• The number of 2-blocks of SLn(ηq) covered by a given 2-block of GLn(ηq) is given by κ(𝔟), the number of O₂′(ℤ)-fixed points on the block symbol 𝔟.
• The set of (B𝔟)j, where 𝔟 runs over O₂′(ℤ)-orbit representatives of (n,2)-admissible block symbols and j=1,…,κ(𝔟), forms a complete and disjoint parametrization of all 2-blocks of SLn(ηq).
• The result shows that the assumption of odd primes in [26, Thm. A(e)] is necessary, as the upper bound for covered 2-blocks may exceed κ(𝔟) when 4∣q+η.