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

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