两个链的乘积上Stanley-Thomas词的双有理提升
A birational lifting of the Stanley-Thomas word on products of two chains
AI总结：
本研究将Stanley-Thomas词从组合层面提升到分段线性、双有理和非交换层面，虽非双射，但足以证明同质性，并给出非交换反链行运动公式，为证明周期性奠定基础。
摘要：
近年来，某些组合作用及其在分段线性和双有理层面上的提升的动力系统已被研究，着眼于周期性、轨道结构和不变量等问题。行运动算子在某些有限偏序集上具有的一个关键性质是同质性，即一个统计量在所有轨道上的平均值相同。为了证明两个链偏序集的乘积中同质性的精细版本，J. Propp和第二作者使用了由R. Stanley和H. Thomas（非正式地）发现的等变双射。我们探索了这个“Stanley-Thomas词”在分段线性、双有理和非交换领域中的提升。尽管该映射不再是双射，因此不能直接用于证明周期性，但它仍然提供了足够的信息来证明分段线性和双有理层面上的同质性（这一结果先前由D. Grinberg、S. Hopkins和S. Okada证明）。即使在非交换层面上，偏序集标记的Stanley-Thomas词也随着反链行运动的提升而循环旋转。在此过程中，我们给出了一些非交换反链行运动的公式，我们希望这些公式将成为证明该层面预期周期性的第一步。
英文摘要：
The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key property enjoyed by the rowmotion operator on certain finite partially-ordered sets is homomesy, where the average value of a statistic is the same for all orbits. To prove refined versions of homomesy in the product of two chain posets, J. Propp and the second author used an equivariant bijection discovered (less formally) by R. Stanley and H. Thomas. We explore the lifting of this "Stanley--Thomas word" to the piecewise-linear, birational, and noncommutative realms. Although the map is no longer a bijection, so cannot be used to prove periodicity directly, it still gives enough information to prove the homomesy at the piecewise-linear and birational levels (a result previously shown by D. Grinberg, S. Hopkins, and S. Okada). Even at the noncommutative level, the Stanley--Thomas word of a poset labeling rotates cyclically with the lifting of antichain rowmotion. Along the way we give some formulas for noncommutative antichain rowmotion that we hope will be first steps towards proving the conjectured periodicity at this level.
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