Crystals, semistandard tableaux and cyclic sieving phenomenon

2019/06/18 by Oh, Young-Tak, Park, Euiyong
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1906.07420

Abstract

In this paper, we study a new cyclic sieving phenomenon on the set SST_n(λ) of semistandard Young tableaux with the cyclic action c arising from its U_q(\mathfraksl_n)-crystal structure. We prove that if λ is a Young diagram with ℓ(λ) < n and gcd( n, |λ| )=1, then the triple ( SST_n(λ), C, q^{- κ(λ)} s_λ(1,q, …, q^n-1) ) exhibits the cyclic sieving phenomenon, where C is the cyclic group generated by c. We further investigate a connection between c and the promotion pr and show the bicyclic sieving phenomenon given by c and prⁿ for hook shape.

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