Poincaré 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 Poincaré series for the induced modules and 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.

24 pages. Alg. Represent. Theory (2020)

References in corpus (2)

• Generalized McKay quivers of rank three
• Poincaré series, exponents of affine Lie algebras, and McKay-Slodowy correspondence

Cited by in corpus (1)

• Kostant's generating functions and McKay-Slodowy correspondence