Set-valued tableaux and cells of Gelfand-Zetlin polytopes
Abstract
Two combinatorial rules are known for the Grassmannian Grothendieck polynomial G(β)λ: a sum over set-valued tableaux of shape λ, due to Buch, and a sum over the efficient cells of a cellular decomposition of the Gelfand-Zetlin polytope GZ(λ), 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 λ=(2,1,0).
Related
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BibTeX
@misc{smirnov2026setvalued,
title = {{Set-valued tableaux and cells of Gelfand-Zetlin polytopes}},
author = {Evgeny Smirnov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08760},
eprint = {2609.08760},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08760},
}
RIS
TY - UNPB
TI - Set-valued tableaux and cells of Gelfand-Zetlin polytopes
AU - Evgeny Smirnov
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.08760
UR - https://arxiv.org/abs/2609.08760
AB - Two combinatorial rules are known for the Grassmannian Grothendieck polynomial G^(β)_λ: a sum over set-valued tableaux of shape λ, due to Buch, and a sum over the efficient cells of a cellular decomposition of the Gelfand-Zetlin polytope GZ(λ), 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 λ=(2,1,0).
ER -