Iterative Properties of Birational Rowmotion II: Rectangles and Triangles
Iterative Properties of Birational Rowmotion II: Rectangles and Triangles
复制标题
双有理行运动的迭代性质 II：矩形和三角形
DOI：
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发表时间：
2014
影响因子：
0.7
通讯作者：
Tom Roby
中科院分区：
文献类型：
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作者：
Darij Grinberg;Tom Roby
双有理行运动（英语：Birational rowmotion）-一个与任何有限偏序集P$相关联的双有理映射-已经由爱因斯坦和普罗普引入，作为P$的序理想集上的经典行运动映射的一个意义深远的推广。继续我们对这个双有理行运动的探索，我们证明了它在$\left（p，q\right）$-矩形偏序集上有阶$p+q$（即，在$p$-元链与$q$-元链的乘积上）;我们还计算了它在一些三角形偏序集上的阶。在所有提到的情况下，它都有有限的（和显式可计算的）阶，这是它在一般有限偏序集上没有表现出的性质（不像经典的行运动，它是有限集的置换）。我们的证明在矩形偏序集的情况下，使用了Volkov（arXiv：hep-th/0606094）引入的一个想法来证明Zamolodchikov周期性猜想的$AA$情况;事实上，许多偏序集上的双有理行运动的有限阶可以被认为是Zamolodchikov周期性的类似物。我们评论怀疑，但到目前为止，神秘的，连接到根偏序集理论。
Birational rowmotion — a birational map associated to any finite poset $P$ — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of $P$. Continuing our exploration of this birational rowmotion, we prove that it has order $p+q$ on the $\left( p, q\right) $-rectangle poset (i.e., on the product of a $p$-element chain with a $q$-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the $AA$ case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets.