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

TL;DR

This paper verifies the inductive blockwise Alperin weight condition for unipotent blocks of certain finite groups of Lie type E_6 and ^2E_6, under specific conditions on the parameters, advancing the understanding of modular representation theory.

Contribution

It proves the inductive blockwise Alperin weight condition for unipotent l-blocks of E_6 and ^2E_6 groups when 2,3 do not divide q and l ≥ 5, filling a gap in the theory.

Findings

The inductive blockwise Alperin weight condition holds for these groups under specified conditions.

The result applies to unipotent l-blocks with l ≥ 5.

Conditions on q exclude divisibility by 2 and 3.

Abstract

In this article, we consider the finite exceptional groups of Lie type $E_6$ and $^2E_6$. We prove the inductive blockwise Alperin weight condition holds for unipotent $l$-blocks of $E_6^{\varepsilon}(q)$ if $2,3\nmid q$, $l\geq 5$