### Abstract

*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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### Cite this

*p*-groups.

*Journal of Algebra*,

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

**Symmetric powers and modular invariants of elementary abelian p-groups.** / Elmer, Jonathan.

Research output: Contribution to journal › Article

*p*-groups',

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

*p*-groups. Journal of Algebra. 2017 Dec;492:157-184. https://doi.org/10.1016/j.jalgebra.2017.07.020

}

TY - JOUR

T1 - Symmetric powers and modular invariants of elementary abelian p-groups

AU - Elmer, Jonathan

N1 - Special thanks go to Professor David Benson a number of invaluable conversations at the genesis of this work. Thanks also to Dr. Müfit Sezer for his assistance with the proof of Proposition 4.3, and an anonymous referee for some helpful remarks.

PY - 2017/12

Y1 - 2017/12

N2 - 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 mq and relative transfers. This extends recent work of Wehlau [18] on modular invariants of cyclic groups of order p. If mp. Our methods are primarily representation-theoretic, and along the way we prove that for any dE 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.

AB - 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 mq and relative transfers. This extends recent work of Wehlau [18] on modular invariants of cyclic groups of order p. If mp. Our methods are primarily representation-theoretic, and along the way we prove that for any dE 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.

KW - modular representation theory

KW - invariant theory

KW - elementary abelian p-groups

KW - symmetric powers

KW - relative stable module category

UR - http://eprints.mdx.ac.uk/20138/

U2 - 10.1016/j.jalgebra.2017.07.020

DO - 10.1016/j.jalgebra.2017.07.020

M3 - Article

VL - 492

SP - 157

EP - 184

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -