促进增加画面：框架和同形
Promotion of increasing tableaux: frames and homomesies
摘要：
关于M的一个关键事实- P. Schützenberger（1972）在矩形标准Young tableaux上的提升算子是，每个条目迭代提升一次，恢复原始tableau。对于行和列严格递增的表格，H。Thomas和A. Yong（2009）介绍了一个$K$ -jeu de taquin理论，并将其应用于$K$ -理论舒伯特演算。作者（2014）研究了从该理论导出的$K$ -提升算子$\mathcal{P}$，但表明关键事实通常不会扩展到$K$ -提升这种增加的tableaux。在这里，我们表明，关键的事实持有标签的边界上的矩形。也就是说，对于$T$（一个矩形递增表，其条目以$q$为界），我们有$\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$，其中$\mathsf{Frame}(U)$表示$U$对其第一行和最后一行和列的限制。利用这一事实，我们得到了$2$ -提升轨道上某些统计量的平均值的一族同态结果，推广了J. Bloom，D. Saracino和作者（2016）任意矩形形状。
英文摘要：
A key fact about M.-P. Schützenberger's (1972) promotion operator on rectangular standard Young tableaux is that iterating promotion once per entry recovers the original tableau. For tableaux with strictly increasing rows and columns, H. Thomas and A. Yong (2009) introduced a theory of $K$-jeu de taquin with applications to $K$-theoretic Schubert calculus. The author (2014) studied a $K$-promotion operator $\mathcal{P}$ derived from this theory, but showed that the key fact does not generally extend to $K$-promotion of such increasing tableaux. Here we show that the key fact holds for labels on the boundary of the rectangle. That is, for $T$ a rectanglar increasing tableau with entries bounded by $q$, we have $\mathsf{Frame}(\mathcal{P}^q(T)) = \mathsf{Frame}(T)$, where $\mathsf{Frame}(U)$ denotes the restriction of $U$ to its first and last row and column. Using this fact, we obtain a family of homomesy results on the average value of certain statistics over $K$-promotion orbits, extending a $2$-row theorem of J. Bloom, D. Saracino, and the author (2016) to arbitrary rectangular shapes.
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