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
TypeArticle
TitleSymmetric powers and modular invariants of elementary abelian p-groups
AuthorsElmer, 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.

LanguageEnglish
PublisherElsevier
JournalJournal of Algebra
ISSN0021-8693
Publication dates
Online04 Aug 2017
Print15 Dec 2017
Publication process dates
Deposited12 Jul 2017
Accepted03 Jul 2017
Output statusPublished
Accepted author manuscript
License
File Access Level
Restricted
First submitted version
File Access Level
Restricted
Digital Object Identifier (DOI)https://doi.org/10.1016/j.jalgebra.2017.07.020
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