The blocks and weights of finite special linear and unitary groups

TL;DR

This paper classifies the $ ext{l}$-blocks of finite special linear and unitary groups, describes their $ ext{l}$-weights, and verifies the Alperin weight conjecture under certain conditions, advancing understanding of modular representation theory.

Contribution

It provides a classification of $ ext{l}$-blocks, describes how to derive $ ext{l}$-weights from general linear groups, and verifies the Alperin weight conjecture for these groups under specific conditions.

Findings

Classification of $ ext{l}$-blocks for $SL_n( ext{epsilon} q)$

Description of $ ext{l}$-weights from $GL_n( ext{epsilon} q)$

Verification of the Alperin weight conjecture under certain conditions

Abstract

This paper has two main parts. Firstly, we give a classification of the $\ell$-blocks of finite special linear and unitary groups $SL_n(\epsilon q)$ in the non-defining characteristic $\ell\ge 3$. Secondly, we describe how the $\ell$-weights of $SL_n(\epsilon q)$ can be obtained from the $\ell$-weights of $GL_n(\epsilon q)$ when $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$, and verify the Alperin weight conjecture for $SL_n(\epsilon q)$ under the condition $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent $\ell$-block of $SL_n(\epsilon q)$ if $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$.