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In this paper, we study a new cyclic sieving phenomenon on the set $mathsf{SST}_n(lambda)$ of semistandard Young tableaux with the cyclic action $mathsf{c}$ arising from its $U_q(mathfrak{sl}_n)$-crystal structure. We prove that if $lambda$ is a Young diagram with $ell(lambda) < n$ and $gcd( n, |lambda| )=1$, then the triple $left( mathsf{SST}_n(lambda), mathsf{C}, q^{- kappa(lambda)} s_lambda(1,q, ldots, q^{n-1}) right) $ exhibits the cyclic sieving phenomenon, where $mathsf{C}$ is the cyclic group generated by $mathsf{c}$. We further investigate a connection between $mathsf{c}$ and the promotion $mathsf{pr}$ and show the bicyclic sieving phenomenon given by $mathsf{c}$ and $mathsf{pr}^n$ for hook shape.
In this paper, we compute explicitly the $q$-dimensions of highest weight crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving phenomenon.
In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an op
The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridges $q=-1$ phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization
We show that the set R(w_0) of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically, R(w_0) possesses a natural cyclic action given by moving the first letter of a word to
By considering the specialisation $s_{lambda}(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $lambda$ in terms of two properties of the boxes in the diagram f