On extensions of simple modules over symmetric and algebraic groups
type: Item
instance of: scholarly article
title: On extensions of simple modules over symmetric and algebraic groups
zbMATH Open document ID: 0981.20006
published in: Journal of Algebra
publication date: 2001-11-20T00:00:00Z
review text: Let \(G\) be a connected reductive algebraic group \(G\) over an algebraically closed field \(F\) of characteristic \(p>0\). Further, let \(L(\lambda)\) and \(L(\mu)\) be simple \(G\)-modules corresponding to dominant weights \(\lambda\) and \(\mu\). Under the standard partial ordering on weights, if \(\mu\) does not strictly dominate \(\lambda\), the group of extensions of \(L(\lambda)\) by \(L(\mu)\) is isomorphic to the group of homomorphisms from the radical of \(\Delta(\lambda)\) to \(L(\mu)\). (Here \(\Delta(\lambda)\) is the Weyl module with highest weight \(\lambda\).) That is, \(\text{Ext}^1(L(\lambda),L(\mu))\cong\text{Hom}_G(\text{rad }\Delta(\lambda),L(\mu))\). The main result of this paper is an analogous isomorphism for extensions of \(F\Sigma_n\)-modules. For the symmetric group, \(\lambda\) and \(\mu\) are partitions of \(n\) and the Weyl module is replaced by the corresponding Specht module. The analogous isomorphism is shown to hold when \(p>2\), \(\lambda\) and \(\mu\) are partitions of \(n\) with at most \(p-1\) parts, and \(\lambda\) does not strictly dominate \(\mu\) (relative to the standard partial ordering of partitions). Note here the reversal of the dominance condition. The authors give examples to show that the conditions on \(p\) and the partitions are in fact necessary. The isomorphism is then used to make computations of extensions between simple modules whose corresponding partitions are two row partitions of \(n\). In the process, some computations of extensions are also made for algebraic groups.
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 1400117
zbMATH Keywords: symmetric groups
zbMATH Keywords: extensions
zbMATH Keywords: connected reductive algebraic groups
zbMATH Keywords: partitions
zbMATH Keywords: Specht modules
zbMATH Keywords: simple modules
zbMATH Keywords: weights
zbMATH Keywords: Weyl modules
author: Alexander S. Kleshchev
author: Jagat Sheth
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1400117