论文标题
史丹利 - 托马斯（Stanley-Thomas）词关于两个连锁产品的产物
A birational lifting of the Stanley-Thomas word on products of two chains
论文作者
论文摘要
最近对某些组合作用的动态及其对在分段线性和异性层面上的动作提升的动力进行了研究，并着眼于周期性，轨道结构和不变的问题。 RowMotion操作员在某些有限的部分订购集中享有的一个关键属性是同质的，其中统计的平均值对于所有轨道都是相同的。为了证明在两个链Posets的乘积中的同性素的精致版本，J. Propp和第二作者使用了R. Stanley和H. Thomas发现的（较少形式）（较少形式的）。 我们探索了这个“史丹利 - 托马斯词”的提升，向分段线性，异性和非共同领域。尽管该地图不再是两者的，因此不能直接证明周期性，但它仍然提供了足够的信息来证明分段线性和异性级别的同源性（这是D. Grinberg，S。S. Hopkins和S. Okada先前显示的结果）。即使在非共同级别，史丹利 - 托马斯词的托马斯单词也随着抗抗小组的提升而周期性地旋转。在此过程中，我们为非共同抗小型行李箱提供了一些公式，我们希望我们将成为证明在此级别上猜想的周期性的第一步。
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.
