Modular invariants of a vector and a covector for some elementary abelian p-groups

Abstract

Let ${\mathbb{F}}_p$ be the prime field of order p > 0 and G be an elementary abelian p-group. For some n-dimensional cohyperplane G-representations V over ${\mathbb{F}}_p$, we show that ${\mathbb{F}}_p{[V\oplus V^{*}]}^G$, the invariant ring of a vector and a covector is a complete intersection by exhibiting an explicit generating set (in fact, a SAGBI basis) and exposing all relations among the generators.

Keywords:

• Complete intersections
• elementary abelian p-groups
• modular invariants

2020 Mathematics Subject Classification:

• 13A50

Acknowledgments

The author would like to thank her Ph.D advisor Professor Yin Chen for his supervision and help. She also wants to thank Professors H. Eddy A. Campbell, Xianhui Fu, and David L. Wehlau for their careful reading, comments, and encouragements. Many thanks go to the anonymous referee for his/her valuable comments and suggestions on the first version of this paper. The symbolic computation language MAGMA [1] (http://magma.maths.usyd.edu.au/) was very helpful.