Ext Groups between Irreducible $\text{GL}_n(q)$-modules in Cross Characteristic

TL;DR

This paper extends the translation of Ext group calculations from the general linear group over finite fields to $q$-Schur algebras, providing new formulas and vanishing results for irreducible modules in cross characteristic.

Contribution

It generalizes previous work by relating Ext groups for $ ext{GL}_n(q)$ to those over $q$-Schur algebras for all degrees, including new formulas and vanishing criteria.

Findings

Established formulas relating Ext groups of $ ext{GL}_n(q)$ to $q$-Schur algebra Ext groups.

Proved the absence of non-split self-extensions for certain irreducible modules.

Developed a method for vanishing results of higher Ext groups, demonstrated through examples.

Abstract

Let $G=\text{GL}_n(q)$ be the general linear group over the finite field $\mathbb{F}_q$ of $q$ elements, and let $k$ be an algebraically closed field of characteristic $r >0$ such that $r$ does not divide $q(q-1)$. In 1999, Cline, Parshall, and Scott showed that under these assumptions, cohomology calculations for $G$ may be translated to Ext$^i$ calculations over a $q$-Schur algebra. The aim of this paper is to extend the results of Cline, Parshall, and Scott and show that Ext$^i$ calculations for $\text{GL}_n(q)$ may also be translated to Ext$^i$ calculations over an appropriate $q$-Schur algebra (both for $i=1$ and $i>1$). To that end, we establish formulas relating certain Ext groups for $\text{GL}_n(q)$ to Ext groups for the $q$-Schur algebra $S_q(n,n)_k$. As a consequence, we show that there are no non-split self-extensions of irreducible $kG$-modules belonging to the unipotent principal Harish-Chandra series. As an application in higher degree, we describe a method which yields vanishing results for higher Ext groups between irreducible $kG$-modules and demonstrate this method in a series of examples.