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

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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 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.

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Acknowledgments

N. Jing would like to thank the partial support of Simons Foundation grant no. 523868 and NSFC grant no. 11531004. H. Zhang would like to thank the support of NSFC grant no 11871325.

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Correspondence to Danxia Wang.

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Presented by: Alistair Savage

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Jing, N., Wang, D. & Zhang, H. Poincaré Series of Relative Symmetric Invariants for \(\text {SL}_{n}(\mathbb {C})\). Algebr Represent Theor 24, 601–623 (2021). https://doi.org/10.1007/s10468-020-09962-0

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