Bounding Ext for modules for algebraic groups, finite groups, and quantum groups

2009/09/28 by Brian Parshall, Parshall, Brian, Leonard Scott +1 · 1 citation
Mathematics · #20G #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20G

paper · pdf · doi:10.48550/arxiv.0909.5207

This version significantly improves (and corrects) the earlier version. In many of the results, there is now no restriction on the size of the prime $p$. In addition, a number of new results (e.g., on PIMS and tilting modules) are proved

arxiv created 2010/02/23 · arxiv updated 2010/02/26

Abstract

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that dim\Ext_G¹(L,L')<c, for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L,L'. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for \Extⁿ, for any integer n≥ 0, using a constant depending only on n and the root system. Weaker versions of this are proved in the algebraic and finite group cases, sufficient to give similar results for algebraic and generic cohomology. The results both use, and have consequences for, Kazhdan-Lusztig polynomials. An appendix proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.

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• Variations on a theme of Cline and Donkin

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• Unbounding Ext
• On modules arising from quantum groups at p^r-th roots of unity
• Bounds on the Dimension of Ext for Finite Groups of Lie Type
• On the Ext groups between Weyl modules for GL_n
• Some remarks on Ext groups