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arXiv · 1811.11901

Poincar\'e series, exponents of affine Lie algebras, and McKay-Slodowy correspondence

Naihuan Jing·Danxia Wang·Honglian Zhang
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Abstract

Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincar\'{e} series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)=\oplus_{k \geq 0}V^{\otimes k}$ are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincar\'e series associated with the distinguished pairs of subgroups of $\mathrm{SU}_2$ give rise to the exponents of all untwisted and twisted affine Lie algebras except ${\rm A}_{2n}^{(1)}$.

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Naihuan Jing, Danxia Wang, Honglian Zhang. 2018-11-29. Poincar\'e series, exponents of affine Lie algebras, and McKay-Slodowy correspondence. https://doi.org/10.1016/j.jalgebra.2019.10.042

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