Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector

TL;DR

This paper explicitly determines the invariant rings of certain unipotent and triangular matrix groups acting on a vector space and its dual over finite fields, showing these rings are complete intersections with explicit generators and relations.

Contribution

It provides a complete description of invariant rings for $U_n$ and $B_n$ acting on a vector and its dual over finite fields, including explicit generators and relations, establishing these as complete intersections.

Findings

Invariant rings are complete intersections for all $n$ and $q$.

Explicit generators and relations for the invariant rings are provided.

The results add to the limited known series with uniform invariant ring descriptions.

Abstract

Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where $V =\gfq^n$ and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in $\GL_n(\gfq)$. In fact, we determine $\gfq[V \oplus V^*]^G$ for $G = U_n$ and $G =B_n$. The result is a complete intersection for all values of $n$ and $q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of "doubly parametrized" series of group actions whose invariant rings are known to have a uniform description.