Poincare Series of Relative Symmetric Invariants for SLn(C)

Let (N,G), where N G <= SLn(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 ) = circle plus S-k >= 0(k)(V ) by giving explicit formulas of the Poincare series for the induced modules and the restriction modules. In particular, this provides a uniform formula of the Poincare series for the symmetric invariants in terms of the McKay-Slodowy correspondence. Moreover, we also derive a global version of the Poincare 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 Poincare series.