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Hook modules for general linear groups

Authors
  • Doty, Stephen1
  • Martin, Stuart2
  • 1 Loyola University Chicago, Mathematics and Statistics, Chicago, IL, 60626, USA , Chicago (United States)
  • 2 Magdalene College, Cambridge, CB3 0AG, United Kingdom , Cambridge (United Kingdom)
Type
Published Article
Journal
Archiv der Mathematik
Publisher
Birkhäuser-Verlag
Publication Date
Mar 27, 2009
Volume
92
Issue
3
Pages
206–214
Identifiers
DOI: 10.1007/s00013-009-2789-y
Source
Springer Nature
Keywords
License
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Abstract

For an arbitrary infinite field k of characteristic p > 0, we describe the structure of a block of the algebraic monoid Mn(k) (all n × n matrices over k), or, equivalently, a block of the Schur algebra S(n, p), whose simple modules are indexed by p-hook partitions. The result is known; we give an elementary and self-contained proof, based only on a result of Peel and Donkin’s description of the blocks of Schur algebras. The result leads to a character formula for certain simple GLn(k)-modules, valid for all n and all p. This character formula is a special case of one found by Brundan, Kleshchev, and Suprunenko and, independently, by Mathieu and Papadopoulo.

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