Box splines, tensor product multiplicities and the volume function

Abstract

We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra $\mathfrak{g}$ and a special function $\mathcal{J}$ associated to $\mathfrak{g}$, 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 $\mathcal{J}$, 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 $\mathcal{J}$. 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.

37 pages. v2: Significant improvements to the text. Main results unchanged