On Modular Invariants of A Vector and A Covector

TL;DR

This paper investigates the invariant ring of a vector and covector under the action of SL2 over a finite field, constructing a basis, calculating the Hilbert series, and confirming a conjecture in invariant theory.

Contribution

It constructs a free module basis over a homogeneous system of parameters and proves the Gorenstein property of the invariant ring for SL2 over finite fields.

Findings

Constructed a free module basis for the invariant ring.

Calculated the Hilbert series of the invariant ring.

Proved the invariant ring is Gorenstein.

Abstract

Let $SL_{2}(F_{q})$ be the special linear group over a finite field $F_{q}$, $V$ be the 2-dimensional natural representation of $SL_{2}(F_{q})$ and $V^{\ast}$ be the dual representation. We denote by $F_{q}[V\oplus V^{\ast}]^{SL_{2}(F_{q})}$ the corresponding invariant ring of a vector and a covector for $SL_{2}(F_{q})$. In this paper, we construct a free module basis over some homogeneous system of parameters of $F_{q}[V\oplus V^{\ast}]^{SL_{2}(F_{q})}$. We calculate the Hilbert series of $F_{q}[V\oplus V^{\ast}]^{SL_{2}(F_{q})}$, and prove that it is a Gorenstein algebra. As an application, we confirm a special case of the recent conjecture of Bonnafe and Kemper in 2011.