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Register a new userCrystal structure on King tableaux and semistandard oscillating tableaux
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Seung Jin Lee
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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 open problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(2m)$ in terms of semistandard oscillating tableaux.
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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.
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 for $lambda$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $lambda$-tableaux.
We prove that a family of average weights for oscillating tableaux are polynomials in two variables, namely, the length of the oscillating tableau and the size of the ending partition, which generalizes a result of Hopkins and Zhang. Several explicit and asymptotic formulas for the average weights are also derived.
This paper completely characterizes the standard Young tableaux that can be reconstructed from their sets or multisets of $1$-minors. In particular, any standard Young tableau with at least $5$ entries can be reconstructed from its set of $1$-minors.
Motivated by Stanleys results in cite{St02}, we generalize the rank of a partition $lambda$ to the rank of a shifted partition $S(lambda)$. We show that the number of bars required in a minimal bar tableau of $S(lambda)$ is max$(o, e + (ell(lambda) mathrm{mod} 2))$, where $o$ and $e$ are the number of odd and even rows of $lambda$. As a consequence we show that the irreducible projective characters of $S_n$ vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schurs $Q_{lambda}$ symmetric functions in terms of the power sum symmetric functions.
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