6 research outputs found

    Identities from representation theory

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    We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type CnC_n using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type BnB_n and the exterior representation in type CnC_n. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type CnC_n. By taking certain specializations, we obtain identities for qq-Catalan triangle numbers, the q,tq,t-Catalan number of Stump, qq-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.Comment: 68 pages, 8 figure

    International Congress of Mathematicians: 2022 July 6ā€“14: Proceedings of the ICM 2022

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    Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library
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