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The multistep homology of the simplex and representations of symmetric groups

Part of: Homological algebra Representation theory of groups

Published online by Cambridge University Press:  20 June 2019

MARK WILDON
MARK WILDON*
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Eghan Hill, Eghan TW20 OEX, United Kingdom. e-mail: mark.wildon@rhul.qc.uk
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Abstract

The symmetric group on a set acts transitively on the set of its subsets of a fixed size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

MSC classification

Primary: 20C30: Representations of finite symmetric groups
Secondary: 18G35: Chain complexes 20C20: Modular representations and characters
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 231 - 253
Copyright
© Cambridge Philosophical Society 2019

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