2019/09/26 by McSwiggen, Colin
#05E10 (Primary) 14C17 #53D20 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1909.12278
We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra \mathfrakg and a special function J associated to \mathfrakg, called the volume function. The volume function arises in connection with the randomized Horn's problem in random matrix theory and has a related significance in symplectic geometry. Building on box spline deconvolution formulae of Dahmen-Micchelli and De Concini-Procesi-Vergne, we develop new techniques for computing the multiplicities from J, answering a question posed by Coquereaux and Zuber. In particular, we derive an explicit algebraic formula for a large class of Littlewood-Richardson coefficients in terms of J. We also give analogous results for weight multiplicities, and we show a number of further identities relating the tensor product multiplicities, the volume function and the box spline. To illustrate these ideas, we give new proofs of some known theorems.