### 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 language | English (US) |
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Pages (from-to) | 702-715 |

Number of pages | 14 |

Journal | Journal of Algebra |

Volume | 319 |

Issue number | 2 |

DOIs | |

State | Published - Jan 15 2008 |

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### Keywords

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

### Cite this

*Journal of Algebra*,

*319*(2), 702-715. https://doi.org/10.1016/j.jalgebra.2006.01.059