On sums of tensor and fusion multiplicities

Abstract

The total multiplicity in the decomposition into irreducibles of the tensor product λ ⊗ µ of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them ν N ν λµ = ν N ν λµ . This also applies to the fusion multiplicities of affine algebras in conformal WZW theories. In that context, the statement is equivalent to a property of the modular S matrix, viz Σ(κ) := λ S λκ = 0 if κ is a complex representation. Curiously, this vanishing of Σ(κ) also holds when κ is a quaternionic representation. We provide proofs of all these statements. These proofs rely on a case-by-case analysis, maybe overlooking some hidden symmetry principle. We also give various illustrations of these properties in the contexts of boundary conformal field theories, integrable quantum field theories and topological field theories.

On some properties of SU(3) fusion coefficients

“…The one that we shall give below relies on a crucial property (a theorem that we recall below in sect. 5.1.1) that was actually obtained much later [13].…”

“…Because of theorem 3 of ref. [13], the sum x S ix vanishes if x is not of real type, hence the result. In the present case of SU(3), real irreps have highest weight {µ, µ}, with µ ∈ {0, · · · , k/2 }.…”

“…These properties implies that the total multiplicity in the decomposition into irreducibles of the product λ ⊗ µ of two irreducible representations is trivially invariant if we conjugate both of them. The following theorem was recently proven [13] Theorem 1. The total multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a simple 8 Lie algebra stays constant if we conjugate only one of them.…”

“…We have mult(λ ⊗ µ ⊗ ν) = mult(λ ⊗ µ ⊗ ν ) when the level is infinite (classical case), but their thresholds k min 0 are distinct: it is S + x − c for the first but S − c for the next since x vanishes in the latter case. 13 Indeed, from eq. (26) and footnote 12, for an arbitrary integer x, we have k min 0 (λ − (x, 0), µ − (x, 0); ν − (0, x)) − k min 0 (λ, µ; ν) = −x; moreover, if S1 ≥ S2, from eq.…”

Conjugation properties of tensor product and fusion coefficients

“…Counterexamples are provided by some finite subgroups of SU(3), see below in sect. 1.5, and also [1], and the detailed discussion in [2].…”