有限偏序集的迭代性质
Iterative properties of birational rowmotion
AI总结：
本文研究了有限偏序集的birational rowmotion映射，证明其在(p,q)-矩形偏序集上有有限阶，并探讨了其在三角形和skeletal类偏序集上的性质，揭示了其与Zamolodchikov周期猜想的类比关系。
摘要：
我们研究了任意有限偏序集P相关的birational映射，这是Einstein和Propp发现的rowmotion的经典推广。经典rowmotion是P的order ideals集合的置换。本文证明birational rowmotion在(p,q)-矩形偏序集上具有阶p+q，并计算了某些三角形和skeletal类偏序集的阶。所有情况均显示birational rowmotion具有有限阶，这一性质在一般有限偏序集上不成立。证明使用了Volkov的方法，将birational rowmotion的有限阶视为Zamolodchikov周期猜想的类比。还讨论了其与根poset理论的潜在联系，并探讨了skeletal偏序集上的经典rowmotion。
英文摘要：
We study a birational map associated to any finite poset P. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of P. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we prove that birational rowmotion has order p+q on the (p, q)-rectangle poset (i.e., on the product of a p-element chain with a q-element chain); we furthermore compute its orders on some triangle-shaped posets and on a class of posets which we call "skeletal" (this class includes all graded forests). In all cases mentioned, birational rowmotion turns out to have a 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. We also make a digression to study classical rowmotion on skeletal posets, since this case has seemingly been overlooked so far.
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