Modular Invariants of a Vector and a Covector: a proof of a conjecture of Bonnaf\'e and Kemper

TL;DR

This paper provides a minimal generating set for the invariant ring of a vector and covector under the general linear group over a finite field, confirming a conjecture and revealing its Gorenstein but non-complete intersection structure.

Contribution

It explicitly determines the generators of the invariant ring and proves it is Gorenstein but not a complete intersection, confirming Bonnafé and Kemper's conjecture.

Findings

The invariant ring has a minimal generating set.

The ring is Gorenstein.

The ring is not a complete intersection.

Abstract

Consider a finite dimensional vector space $V$ over a finite field $\mathbb{F}_q$. We give a minimal generating set for the ring of invariants $\mathbb{F}_q[V \oplus V^*]^{\text{GL}(V)}$, and show that this ring is a Gorenstein ring but is not a complete intersection. These results confirm a conjecture of Bonnaf\'e and Kemper.