Poincaré series for tensor invariants and the McKay correspondence

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• 作者：Georgia Benkart ^{benkart@math.wisc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14E16 ; 05E10 ; 20C05
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：26 February 2016
• 年：2016
• 卷：290
• 期：Complete
• 页码：236-259
• 全文大小：2142 K

文摘

For a finite group G$\mathsf{G}$ and a finite-dimensional G$\mathsf{G}$-module V$\mathsf{V}$, we prove a general result on the Poincaré series for the G$\mathsf{G}$-invariants in the tensor algebra T(V)=⨁_k≥0V^⊗k$\mathsf{T}(\mathsf{V})={⨁}_{k\ge 0}{\mathsf{V}}^{\otimes k}$. We apply this result to the finite subgroups G$\mathsf{G}$ of the 2×2$2\times 2$ special unitary matrices and their natural module V$\mathsf{V}$ of 2×1$2\times 1$ column vectors. Because these subgroups are in one-to-one correspondence with the simply laced affine Dynkin diagrams by the McKay correspondence, the Poincaré series obtained are the generating functions for the number of walks on the simply laced affine Dynkin diagrams.