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Register a new userCyclic sieving for longest reduced words in the hyperoctahedral group
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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 the end, and we show that the orbit structure of this action is encoded by the generating function for the major index on R(w_0).
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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.
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We obtain an explicit combinatorial formula for certain parabolic Kostka-Shoji polynomials associated with the cyclic quiver, generalizing results of Shoji and of Liu and Shoji.
Let $G$ be a finite cyclic group, written additively, and let $A, B$ be nonempty subsets of $G$. We will say that $G= A+B$ is a textit{factorization} if for each $g$ in $G$ there are unique elements $a, b$ of $G$ such that $g=a+b, ain A, bin B$. In particular, if $A$ is a complete set of residues $modulo$ $|A|$, then we call the factorization a textit{coset factorization} of $G$. In this paper, we mainly study a factorization $G= A+B$, where $G$ is a finite cyclic group and $A=[0,n-k-1]cup{i_0,i_1,ldots i_{k-1}}$ with $|A|=n$ and $ngeq 2k+1$. We obtain the following conclusion: If $(i)$ $kleq 2$ or $(ii)$ The number of distinct prime divisors of $gcd(|A|,|B|)$ is at most $1$ or $(iii)$ $gcd(|A|,|B|)=pq$ with $gcd(pq,frac{|B|}{gcd(|A|,|B|)})=1$, then $A$ is a complete set of residues $modulo$ $n$.
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