Box splines, tensor product multiplicities and the volume function

McSwiggen, Colin

doi:10.5802/alco.164

McSwiggen, Colin ¹

¹ Brown University Division of Applied Mathematics 182 George St. Providence, RI 02906, USA

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

Abstract

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.

Received: 2020-05-01
Revised: 2020-11-28
Accepted: 2020-12-04
Published online: 2021-06-22

DOI: 10.5802/alco.164

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

Author's affiliations:

McSwiggen, Colin ¹

¹ Brown University Division of Applied Mathematics 182 George St. Providence, RI 02906, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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McSwiggen, Colin. Box splines, tensor product multiplicities and the volume function. Algebraic Combinatorics, Volume 4 (2021) no. 3, pp. 435-464. doi : 10.5802/alco.164. https://alco.centre-mersenne.org/articles/10.5802/alco.164/

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