发表机构
HSE University; Independent University of Moscow; Guangdong Technion – Israel Institute of Technology（高等经济大学; 莫斯科独立大学; 广东以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Grassmannian Grothendieck多项式的两种组合规则（基于set-valued tableaux和基于Gelfand-Zetlin多胞体有效胞腔）构造显式双射，逐项匹配求和项，证明两者等价，并研究晶体算子在胞腔上的性质。
AI 中文摘要
关于 Grassmannian Grothendieck 多项式 $G^{(\beta)}_\lambda$ 已知有两种组合规则：一种是对形状为 $\lambda$ 的 set-valued tableaux 求和，由 Buch 给出；另一种是对 Gelfand-Zetlin 多胞体 $GZ(\lambda)$ 的胞腔分解中的有效胞腔求和，由 this http URL 和作者给出。两种求和中的所有系数均等于 $1$。我们构造了两个指标集之间的显式双射，该双射逐项匹配求和项，将 tableaux 的多余条目数对应到相应胞腔的维数；特别地，这两种规则是等价的，任一规则可由另一规则推导得出。胞腔的有效性条件恰好对应于 tableaux 的列严格性。随后，我们将 Yu 的平方根晶体算子移植到胞腔上，并发现它们保持维数：沿着双重 $i$-串，胞腔在两个连续维数之间交替，但不保持关联性：这样的串中相邻胞腔不必共享一个点，即使对于 $\lambda=(2,1,0)$ 也是如此。
英文摘要
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 $λ=(2,1,0)$.
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