We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra and a special function associated to , 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 , 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 . 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.
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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 = {http://www.numdam.org/articles/10.5802/alco.164/} }
TY - JOUR AU - McSwiggen, Colin TI - Box splines, tensor product multiplicities and the volume function JO - Algebraic Combinatorics PY - 2021 SP - 435 EP - 464 VL - 4 IS - 3 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.164/ DO - 10.5802/alco.164 LA - en ID - ALCO_2021__4_3_435_0 ER -
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. http://www.numdam.org/articles/10.5802/alco.164/
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