On Modular Invariants of A Vector and A Covector

Abstract

Let $S{L}_2(F_q)$ be the special linear group over a finite field $F_q$, $V$ be the 2-dimensional natural representation of $S{L}_2(F_q)$ and $V^{\ast }$ be the dual representation. We denote by $F_q[V\oplus V^{\ast }{]}^{S{L}_2(F_q)}$ the corresponding invariant ring of a vector and a covector for $S{L}_2(F_q)$. In this paper, we construct a free module basis over some homogeneous system of parameters of $F_q[V\oplus V^{\ast }{]}^{S{L}_2(F_q)}$. We calculate the Hilbert series of $F_q[V\oplus V^{\ast }{]}^{S{L}_2(F_q)}$, 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.

11 pages

References in corpus (2)

• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• On the quotient ring by diagonal harmonics