Unique factorization in invariant power series rings

David John Benson, P. J. Webb

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 [V] by linear substitutions and address the question of when the invariant power series k[V](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. (C) 2006 Elsevier Inc. All rights reserved.

Original language English 702-715 14 Journal of Algebra 319 2 https://doi.org/10.1016/J.JALGEBRA.2006.01.059 Published - Jan 2008

Keywords

• invariant theory
• symmetric powers
• unique factorization
• modular representation
• characteristic-P
• completions

Cite this

In: Journal of Algebra, Vol. 319, No. 2, 01.2008, p. 702-715.

Research output: Contribution to journalArticle

Benson, David John ; Webb, P. J. / Unique factorization in invariant power series rings. In: Journal of Algebra. 2008 ; Vol. 319, No. 2. pp. 702-715.
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