Paths to Understanding Birational Rowmotion on Products of Two Chains
Paths to Understanding Birational Rowmotion on Products of Two Chains
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理解两条链产品上的双理性行运动的路径
DOI：
10.5802/alco.43
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发表时间：
2018
影响因子：
--
通讯作者：
Tom Roby
中科院分区：
文献类型：
--
作者：
Gregg Musiker;Tom Roby
出生行进是有理函数对有限偏序集(偏序集)元素赋值空间上的一种作用。它是从偏序集$P$[AST11，BW74，CF95，Pan09，PR13，RuSh12，RuWa15+，SW12，ThWi17，Yil17]的序理想(等价于反链)上得到的，当在特殊的偏序集上迭代时，在周期性、循环筛选和同伦(每个轨道上的平均值是恒定的统计量)方面具有出人意料的良好性质[PR13]。在这一背景下，行动似乎与某些箭图上的Auslander-Reiten平移[Yil17]有关，并与Zamolodchikov周期中描述的$A_m\x A_n$类型的$Y$的混生行进运动有关。
我们用不相交的格路族的形式给出了两个链的乘积上二元行进映射的迭代作用的一个公式。这使得我们可以更简单地直接证明这样一个关键事实，即这个映射在长为$r$和$S$的链的乘积上的周期是$r+S+2$(最先由D.Grinberg和第二作者[GrRo15]证明)，以及关于这样的偏序集的沿档案的同伦的二元类比的证明。
Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset $P$ [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17], which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [PR13]. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers [Yil17], and birational rowmotion to $Y$-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity.
We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D. Grinberg and the second author [GrRo15]), as well as a proof of the birational analogue of homomesy along files for such posets.