# ALGEBRAIC COMBINATORICS

Box splines, tensor product multiplicities and the volume function
Algebraic Combinatorics, Volume 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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/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.
@article{ALCO_2021__4_3_435_0,
author = {McSwiggen, Colin},
title = {Box splines, tensor product multiplicities and the volume function},
journal = {Algebraic Combinatorics},
pages = {435--464},
publisher = {MathOA foundation},
volume = {4},
number = {3},
year = {2021},
doi = {10.5802/alco.164},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.164/}
}
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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