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Universal Deformation Rings for the Symmetric Group S 4

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Abstract

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S 4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S 4,V) for every kS 4-module V which has stable endomorphism ring k and show that R(S 4,V) is isomorphic to either k, or W[t]/(t 2,2t), or the group ring W[ℤ/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.

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Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242-1419, USA

    Frauke M. Bleher

  2. Department of Mathematics, CSU San Bernardino, San Bernardino, CA, 92407-2397, USA

    Giovanna Llosent

Authors
  1. Frauke M. Bleher
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  2. Giovanna Llosent
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Corresponding author

Correspondence to Frauke M. Bleher.

Additional information

The first author was supported in part by NSF Grants DMS01-39737 and DMS06-51332 and NSA Grant H98230-06-1-0021.

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Bleher, F.M., Llosent, G. Universal Deformation Rings for the Symmetric Group S 4 . Algebr Represent Theor 13, 255–270 (2010). https://doi.org/10.1007/s10468-008-9120-7

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  • DOI: https://doi.org/10.1007/s10468-008-9120-7

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