Set-valued tableaux and cells of Gelfand-Zetlin polytopes

Abstract

Two combinatorial rules are known for the Grassmannian Grothendieck polynomial $G_{\lambda }^{(\beta )}$: 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 $\lambda =(2,1,0)$.

arXiv comment17 pages