Symmetric powers and modular invariants of elementary abelian p-groups

Elmer, Jonathan ORCID logoORCID: (2017) Symmetric powers and modular invariants of elementary abelian p-groups. Journal of Algebra, 492 . pp. 157-184. ISSN 0021-8693 [Article] (doi:10.1016/j.jalgebra.2017.07.020)

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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.

Item Type: Article
Research Areas: A. > School of Science and Technology > Design Engineering and Mathematics
Item ID: 20138
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Depositing User: Jonathan Elmer
Date Deposited: 12 Jul 2017 14:47
Last Modified: 12 Jun 2022 05:28

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