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

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.