University of Sydney Algebra Seminar

Robert Coquereaux (Centre de Physique Th�orique, CNRS)

Friday 8th November, 12:05-12:55pm, Carslaw 373

Sum rules for tensor and fusion multiplicities

We prove that the total multiplicity in the decomposition into irreducibles representations (irreps) of the tensor products of two finite-dimensional irreps of a simple Lie algebra is invariant under conjugation of one of them. This sum rule also applies to fusion multiplicities of integrable irreps of affine algebras at a given level, in conformal WZW theories, or to the multiplicities of irreps (with non-zero dimension) of quantum groups at roots of unity. In the latter cases it is related to a property of the modular $S$-matrix. The same sum rule does not hold, in general, for fusion categories built from representations of finite groups or from their (Drinfeld) quantum doubles. The results, obtained with J.-B. Zuber in 2011 and 2012, will be illustrated with examples taken from representation theory of compact Lie groups, Lie groups at level $k$ (CFT), finite subgroups of $\mathrm{SU}2$ or $\mathrm{SU}3$, and their quantum doubles.