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arXiv · 2609.08760

Set-valued tableaux and cells of Gelfand-Zetlin polytopes

Evgeny Smirnov
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Abstract

Two combinatorial rules are known for the Grassmannian Grothendieck polynomial $G^{(\beta)}_\lambda$: a sum over set-valued tableaux of shape $\lambda$, due to Buch, and a sum over the efficient cells of a cellular decomposition of the Gelfand-Zetlin polytope $GZ(\lambda)$, due to E.Presnova and the author. All coefficients in both sums equal $1$. We construct an explicit bijection between the two indexing sets which matches the summands term by term, carrying the number of excess entries of a tableau to the dimension of the corresponding cell; in particular the two rules are equivalent, either being deducible from the other. The efficiency condition on cells turns out to be the column-strictness of tableaux. We then transport Yu's square-root crystal operators to the cells and find that they respect dimension, along a double $i$-string the cells alternate between two consecutive dimensions, but not incidence: consecutive cells of such a string need not share a point, already for $\lambda=(2,1,0)$.

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Evgeny Smirnov. 2026-09-08. Set-valued tableaux and cells of Gelfand-Zetlin polytopes. https://arxiv.org/abs/2609.08760

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