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.
Categories: math.AC math.AT