### Abstract

Let

*E*be an elementary abelian*p*-group of order q=pn. Let*W*be a faithful indecomposable representation of*E*with dimension 2 over a field k of characteristic*p*, and let V=Sm(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 [18] 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 [6] 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, ⁎Sd(V⁎) is projective relative to the set of subgroups of*E*with order ≤*m*, and that the sequence ⁎Sd(V⁎)d≥0 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 [1] on cyclic groups of prime order.Original language | English |
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Pages (from-to) | 157-184 |

Number of pages | 28 |

Journal | Journal of Algebra |

Volume | 492 |

Early online date | 4 Aug 2017 |

DOIs | |

Publication status | Published - Dec 2017 |

### Keywords

- modular representation theory
- invariant theory
- elementary abelian p-groups
- symmetric powers
- relative stable module category

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## Cite this

Elmer, J. (2017). Symmetric powers and modular invariants of elementary abelian

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