Unique factorization in invariant power series rings

David Benson, Peter Webb

Research output: Contribution to journalArticle

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Abstract

Let G be a finite group, k a perfect field, and V a finite-dimensional kG-module. We let G act on the power series k {left open bracket} V {right open bracket} by linear substitutions and address the question of when the invariant power series k {left open bracket} V {right open bracket}G form a unique factorization domain. We prove that for a permutation module for a p-group in characteristic p, the answer is always positive. On the other hand, if G is a cyclic group of order p, k has characteristic p, and V is an indecomposable kG-module of dimension r with 1 ≤ r ≤ p, we show that the invariant power series form a unique factorization domain if and only if r is equal to 1, 2, p - 1 or p. This contradicts a conjecture of Peskin.

Original languageEnglish (US)
Pages (from-to)702-715
Number of pages14
JournalJournal of Algebra
Volume319
Issue number2
DOIs
StatePublished - Jan 15 2008

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Keywords

  • Invariant theory
  • Modular representation
  • Symmetric powers
  • Unique factorization

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