Poincaré Series of Relative Symmetric Invariants for $\text {SL}_{n}(\mathbb {C})$

Let (N,G), where $N\unlhd G\leq \text {SL}_{n}(\mathbb {C})$, 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 ) = ⊕k≥ 0Sk(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.