Abstract

We relate Reiner, Tenner, and Yong’s coincidental down-degree expectations (CDE) property of posets to the minuscule doppelgänger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelgänger pairs behave “as if” they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelgänger pairs pretending to have isomorphic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelgänger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type).

摘要

我们将 Reiner、Tenner 和 Yong 的巧合下度期望 (CDE)属性与Hamaker、Patrias、Pechenik 和 Williams 研究的微小分身对联系起来。通过这种关系，我们提出了一系列猜想，这些猜想表明微小的分身对表现得“好像”它们具有同构的可比图，即使它们没有。我们通过考虑行运动进一步探索了微小的分身对假装具有同构可比图的想法订单理想的运算符。我们推测极小的分身对的成员在行运动下的行为方式相同，就像它们具有同构可比图时一样。此外，我们推测这些对在爱因斯坦和普罗普引入的行运动的分段线性和双理性提升下继续以相同的方式表现。这个猜想促使我们研究双理性行运动表现出的同质性（在 Propp 和 Roby 的意义上）。我们为已知或推测具有有限双有理行运动顺序的偏序集的主要示例（即：巧合类型的小偏序集和根偏序集）建立了反链基数同调的双有理类似物。