A simple counting argument of the irreducible representations of SU(N) on mixed product spaces

TL;DR

This paper provides an alternative proof for counting irreducible representations of SU(N) on tensor products, extending the result to mixed spaces involving both vector spaces and their duals, with implications for symmetry and duality.

Contribution

It introduces a new proof method using projection operators and generalizes the counting result to mixed tensor product spaces involving duals.

Findings

Number of irreducible representations remains unchanged when exchanging V with its dual in mixed tensor spaces.

The proof method simplifies understanding of invariant algebra structures.

The result applies to both pure and mixed tensor product spaces.

Abstract

It is a well known result that the number of irreducible representations of SU(N) on a tensor product containing k factors of a vector space V is given by the number of involutions in the symmetric group on k letters. In this paper, we present an alternative proof for this fact using a basis of projection and transition operators of the algebra of invariants of SU(N). This proof easily generalizes to the irreducible representations of SU(N) on mixed tensor product spaces (consisting of factors of V as well as its dual space). This implies that the number of irreducible representation of SU(N) on such a space remains unchanged if one exchanges factors V for its dual and vice versa, as long as the total number of factors remains unchanged.