Poincaré Series for Tensor Invariants and the McKay Correspondence

2014/07/15 by Georgia Benkart, Benkart, Georgia
Mathematics · #05E10 #14E16 #20C05 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:05E10 #msc:14E16 #msc:20C05

paper · pdf · doi:10.48550/arxiv.1407.3997

Minor changes from the first edition. The paper has been accepted for publication in Advances in Mathematics

arxiv created 2015/12/16 · arxiv updated 2015/12/17

Abstract

For a finite group G and a finite-dimensional G-module V, we prove a general result on the Poincaré series for the G-invariants in the tensor algebra T(V). We apply this result to the finite subgroups G of the 2-by-2 special unitary matrices and their natural module V of 2-by-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.

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