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Second cohomology for finite groups of Lie type | | Brian D. Boe
; Brian Bonsignore
; Theresa Brons
; Jon F. Carlson
; Leonard Chastkofsky
; Christopher M. Drupieski
; Niles Johnson
; Daniel K. Nakano
; Wenjing Li
; Phong Thanh Luu
; Tiago Macedo
; Nham Vo Ngo
; Brandon L. Samples
; Andrew J. Talian
; Lisa Townsley
; Benjamin J. Wyser
; | | Date: |
2 Oct 2011 | | Abstract: | Let $G$ be a simple, simply-connected algebraic group defined over
$mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(mathbb{F}_q) subset G$
be the subgroup of $mathbb{F}_q$-rational points. Let $L(lambda)$ be the
simple rational $G$-module of highest weight $lambda$. In this paper we
establish sufficient criteria for the restriction map in second cohomology
$H^2(G,L(lambda))
ightarrow H^2(G(mathbb{F}_q),L(lambda))$ to be an
isomorphism. In particular, the restriction map is an isomorphism under very
mild conditions on $p$ and $q$ provided $lambda$ is less than or equal to a
fundamental dominant weight. Even when the restriction map is not an
isomorphism, we are often able to describe $H^2(G(mathbb{F}_q),L(lambda))$ in
terms of rational cohomology for $G$. We apply our techniques to compute
$H^2(G(mathbb{F}_q),L(lambda))$ in a wide range of cases, and obtain new
examples of nonzero second cohomology for finite groups of Lie type. | | Source: | arXiv, 1110.0228 | | Services: | Forum | Review | PDF | Favorites | |
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