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.
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