On modular invariants of a vector and a covector

摘要

Let GL_2(F_q) be the general linear group over a finite field F_q, V be the 2-dimensional natural representation of GL_2(F_q) and V~? be the dual representation. We denote by F_q [V?V~?]~(GL2)(F_q) the corresponding invariant ring of a vector and a covector forGL_2(F_q). In this paper, we prove that F_q [V ? V~?]~(GL2)(F_q) is a Gorenstein algebra. This result confirms a special case (n = 2) of the recent conjecture of Bonnafé and Kemper (J Algebra 335:96-112, 2011).