# Symmetric powers and modular invariants of elementary abelian p-groups

Article

Elmer, J. 2017. Symmetric powers and modular invariants of elementary abelian p-groups.

*Journal of Algebra.*492, pp. 157-184. https://doi.org/10.1016/j.jalgebra.2017.07.020

Type | Article |
---|---|

Title | Symmetric powers and modular invariants of elementary abelian p-groups |

Authors | Elmer, J. |

Abstract | Let E be a elementary abelian p-group of order q = p^n. Let W be a faithful indecomposable representation of E with dimension 2 over a field k of characteristic p, and let V = S^m(W ) with m < q. We prove that the rings of invariants k[V ]^E are generated by elements of degree ≤ q and relative transfers. This extends recent work of Wehlau on modular invariants of cyclic groups of order p. If m < p we prove that k[V ]^E is generated by invariants of degree ≤ 2q −3, extending a result of Fleischmann, Sezer, Shank and Woodcock for cyclic groups of order p . Our methods are primarily representation-theoretic, and along the way we prove that for any d < q with d + m ≥ q, S^d (V^∗) is projective relative to the set of subgroups of E with order ≤ m, and that the sequence S^d (V^∗) is periodic with period q, modulo summands which are projective relative to the same set of subgroups. These results extend results of Almkvist and Fossum on cyclic groups of prime order. |

Publisher | Elsevier |

Journal | Journal of Algebra |

ISSN | 0021-8693 |

Publication dates | |

Online | 04 Aug 2017 |

Print | 15 Dec 2017 |

Publication process dates | |

Deposited | 12 Jul 2017 |

Accepted | 03 Jul 2017 |

Output status | Published |

Accepted author manuscript | License File Access Level Restricted |

Digital Object Identifier (DOI) | https://doi.org/10.1016/j.jalgebra.2017.07.020 |

Language | English |

First submitted version | File Access Level Restricted |

https://repository.mdx.ac.uk/item/867qy

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