Box splines, tensor product multiplicities and the volume function

McSwiggen, Colin

Algebraic Combinatorics, Tome 4 (2021) no. 3, pp. 435-464.

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

Classification : 05E10, 14C17, 53D20
Mots clés : Box splines, tensor product multiplicities, Littlewood–Richardson coefficients, Horn’s problem, volume function, Berenstein–Zelevinsky polytopes.

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