Veronica Shalotenko | Searcharxiv

arXiv2020

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

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.

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arXiv2019

Bounds on the Dimension of Ext for Finite Groups of Lie Type

Let $G$ be a finite group of Lie type defined in characteristic $p$, and let $k$ be an algebraically closed field of characteristic $r>0$. We will assume that $r \neq p$ (so, we are in the non-defining characteristic case). Let $V$ be a finite-dimensional irreducible left $kG$-module. In 2011, Guralnick and Tiep found bounds on the dimension of $H^1(G,V)$ in non-defining characteristic, which are independent of $V$. The aim of this paper is to generalize the work of Gurlanick and Tiep. We assume that $G$ is split and use methods of modular Harish-Chandra theory to find bounds on the dimension of $\mathrm{Ext^1}$ between certain irreducible $kG$-modules. We then use Dipper and Du's algorithms to illustrate our bounds in a series of examples.

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