On Modular Invariants of A Vector and A Covector

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.