Conjugation properties of tensor product multiplicities

TL;DR

This paper investigates the conjugation properties of tensor product multiplicities in Lie algebra representations, showing specific invariance and permutation properties for SU(3) and discussing potential extensions to affine algebras.

Contribution

It demonstrates that for SU(3), the multiplicity lists in tensor products are identical up to permutation when conjugating one factor, a property not generally true for other Lie algebras.

Findings

Multiplicity lists are identical up to permutation for SU(3) tensor products.

Conjugation invariance of multiplicities holds for SU(3) but not universally.

Conjecture on similar properties for affine SU(3) fusion products at finite levels.

Abstract

It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product lambda x mu of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products lambda x mu and lambda x bar{mu}, are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for the fusion product of the affine algebra of su(3) at finite levels, but this is not investigated in the present paper.