Ext Groups between Irreducible GL_n(q)-modules in Cross Characteristic

2020/10/04 by Veronica Shalotenko, Shalotenko, Veronica
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2010.01629

Abstract

Let G=GL_n(q) be the general linear group over the finite field \mathbbF_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 Extⁱ calculations over a q-Schur algebra. The aim of this paper is to extend the results of Cline, Parshall, and Scott and show that Extⁱ calculations for GL_n(q) may also be translated to Extⁱ 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 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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