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For a finite group G and a finite-dimensional G-module V, we prove a generalresult on the Poincar\'e series for the G-invariants in the tensor algebraT(V). We apply this result to the finite subgroups G of the 2-by-2 specialunitary matrices and their natural module V of 2-by-1 column vectors. Becausethese subgroups are in one-to-one correspondence with the simply laced affineDynkin diagrams by the McKay correspondence, the Poincar\'e series obtained arethe generating functions for the number of walks on the simply laced affineDynkin diagrams.

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