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≥ 0S^k(V ) by giving explicit formulas of the Poincaré series for the induced modules and the restriction modules. In particular, this provides a uniform formula of the Poincaré series for the symmetric invariants in terms of the McKay-Slodowy correspondence. Moreover, we also derive a global version of the Poincaré 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 Poincaré series.

令（Ñ，G ^），其中\（N \ unlhdģ\当量\文本{SL} _ {N}（\ mathbb {C}）\） ，是一对有限群和V有限维基本ģ -模块。我们研究了ģ在对称代数-invariants小号（V）=⊕ _{ķ ≥0}小号^ķ（V）给出诱导模块和限制模块的庞加莱级数的明确公式。特别地，这根据McKay-Slodowy对应关系为对称不变量提供了Poincaré级数的统一公式。此外，从Tchebychev多项式的角度出发，我们还可以推导出Poincaré系列的全局版本，因为人们仅需要子组的维数及其组类型来完全确定Poincaré系列。