Cameron和Fon-Der-Flaass平面划分周期性猜想的极小模拟
Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
摘要：
设$P$是秩$r$的分次偏序集，$\mathbf{c}$是$c$ -元链.对于$P \times \mathbf{c}$的一个阶理想$I$，它的行运动$ψ(I)$是包含$I$的互补滤子的最小元素的最小理想。映射PDMATHAMT定义了理想集上的可逆动力学。我们说$P$有NRP（“非互质“）行运动，如果没有$ψ$ -轨道具有与$r+c+1$互质的基数。在与R。Patrias（2020），证明了P. Cameron和D. Fon-Der-Flaass通过建立两条链的乘积$P = \mathbf{a} \times \mathbf{b}$的NRP行运动，偏序集的序理想对应于包含下的格拉斯曼簇$\mathrm{Gr}_a(\mathbb{C}^{a+b})$的舒伯特簇。在这里，我们开始了一般的研究偏序集与NRP行运动。我们的第一个主要结果建立NRP行运动的所有极小偏序集$P$，偏序集的顺序理想反映了舒伯特分层的极小旗品种。我们的第二个主要结果是，NRP提升只依赖于$P$的可比图的同构类。
英文摘要：
Let $P$ be a graded poset of rank $r$ and let $\mathbf{c}$ be a $c$-element chain. For an order ideal $I$ of $P \times \mathbf{c}$, its rowmotion $ψ(I)$ is the smallest ideal containing the minimal elements of the complementary filter of $I$. The map $ψ$ defines invertible dynamics on the set of ideals. We say that $P$ has NRP ("not relatively prime") rowmotion if no $ψ$-orbit has cardinality relatively prime to $r+c+1$. In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product $P = \mathbf{a} \times \mathbf{b}$ of two chains, the poset whose order ideals correspond to the Schubert varieties of a Grassmann variety $\mathrm{Gr}_a(\mathbb{C}^{a+b})$ under containment. Here, we initiate the general study of posets with NRP rowmotion. Our first main result establishes NRP rowmotion for all minuscule posets $P$, posets whose order ideals reflect the Schubert stratification of minuscule flag varieties. Our second main result is that NRP promotion depends only on the isomorphism class of the comparability graph of $P$.
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