3 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
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