Poincar\'e Series for Tensor Invariants and the McKay Correspondence

TL;DR

This paper establishes a general formula for the Poincaré series of G-invariants in tensor algebras for finite groups and applies it to subgroups of SU(2), linking the results to affine Dynkin diagrams via the McKay correspondence.

Contribution

It provides a new general result on Poincaré series for tensor invariants and connects these to affine Dynkin diagrams through the McKay correspondence.

Findings

Derived explicit Poincaré series for finite subgroups of SU(2).

Linked Poincaré series to generating functions for walks on affine Dynkin diagrams.

Established a connection between tensor invariants and the McKay correspondence.

Abstract

For a finite group G and a finite-dimensional G-module V, we prove a general result on the Poincar\'e 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 Poincar\'e series obtained are the generating functions for the number of walks on the simply laced affine Dynkin diagrams.