Box splines, tensor product multiplicities and the volume function

TL;DR

This paper explores the connection between tensor product multiplicities in Lie algebras and a special volume function, introducing new computational techniques and explicit formulas for Littlewood-Richardson coefficients.

Contribution

It develops novel methods to compute multiplicities from the volume function using box spline deconvolution, answering a key open question and deriving explicit algebraic formulas.

Findings

Derived explicit formulas for Littlewood-Richardson coefficients in terms of the volume function.

Established new identities linking tensor product multiplicities, the volume function, and box splines.

Provided new proofs of existing theorems using the developed techniques.

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.