Separating invariants for certain representations of the elementary Abelian $ p $-groups of rank two

Panpan Jia; Jizhu Nan; Yongsheng Ma; Panpan Jia; Jizhu Nan; Yongsheng Ma

doi:10.3934/math.20241250

AIMS Mathematics

2024, Volume 9, Issue 9: 25603-25618. doi: 10.3934/math.20241250

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Research article

Separating invariants for certain representations of the elementary Abelian $ p $-groups of rank two

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received: 11 July 2024 Revised: 24 August 2024 Accepted: 28 August 2024 Published: 03 September 2024
MSC : 13A50

For a finite group acting linearly on a vector space, a separating set is a subset of the invariant ring that separates the orbits. In this paper, we determined explicit separating sets in the corresponding rings of invariants for four families of finite dimensional representations of the elementary abelian $ p $-groups $ (\mathbb{Z}/p)^2 $ of rank two over an algebraically closed field of characteristic $ p $, where $ p $ is an odd prime. Our construction was recursive. The separating sets consisted only of transfers and norms, and the size of every separating set depended only on the dimension of the representation.
- elementary abelian $ p $-group,
- invariant theory,
- separating invariants
Citation: Panpan Jia, Jizhu Nan, Yongsheng Ma. Separating invariants for certain representations of the elementary Abelian $ p $-groups of rank two[J]. AIMS Mathematics, 2024, 9(9): 25603-25618. doi: 10.3934/math.20241250

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Abstract

For a finite group acting linearly on a vector space, a separating set is a subset of the invariant ring that separates the orbits. In this paper, we determined explicit separating sets in the corresponding rings of invariants for four families of finite dimensional representations of the elementary abelian $ p $-groups $ (\mathbb{Z}/p)^2 $ of rank two over an algebraically closed field of characteristic $ p $, where $ p $ is an odd prime. Our construction was recursive. The separating sets consisted only of transfers and norms, and the size of every separating set depended only on the dimension of the representation.

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