Promotion and Cyclic Sieving on Rectangular $δ$-Semistandard Tableaux

Let $δ=(δ_1,\ldots,δ_n)$ be a string of letters $h$ and $v$. We define a Young tableau to be $δ$-semistandard if the entries are weakly increasing along rows and columns, and the entries $i$ form a horizontal strip if $δ_i=h$ and a vertical strip if $δ_i=v$. We define $δ$-promotion on such tableaux via a modified jeu-de-taquin. The first main result is that $δ$-promotion has period $n$ on rectangular $δ$-semistandard tableaux, generalizing the results of Haiman and Rhoades for standard and semistandard tableaux. The second main result states that the set of rectangular $δ$-semistandard tableaux for fixed $δ$ and content $γ$ exhibits the cyclic sieving phenomenon with the generalized Kostka polynomial. To do so we follow Fontaine-Kamnitzer and associate to $(δ,γ)$ an $SL_m$-invariant space Inv$(V_{λ^1}\otimes\cdots\otimes V_{λ^n})$ where each $V_{λ^i}$ is an alternating or symmetric representation. We show that the Satake basis of the corresponding invariant space is indexed by the set of tableaux corresponding to $(δ,γ)$ and is permuted by rotation of tensor factors. We then diagonalize the rotation action using the fusion product. This cyclic sieving generalizes the result of Rhoades, and of Fontaine-Kamnitzer (in type A), and is closely related to that of Westbury.

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