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.

Received:
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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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