Poincar\'{e} series of relative symmetric invariants for SL$_n(\mathbb{C})$

Abstract

Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincar\'{e} series for the induced modules and restriction modules. In particular, this provides a uniform formula of the Poincar\'{e} series for the symmetric invariants in terms of the McKay-Slodowy correspondence. Moreover, we also derive a global version of the Poincar\'e series in terms of Tchebychev polynomials in the sense that one needs only the dimensions of the subgroups and their group-types to completely determine the Poincar\'e series.