Minuscule doppelgängers, the coincidental down-degree expectations property, and rowmotion

Abstract: We relate Reiner, Tenner, and Yong's coincidental down-degree expectations (CDE) property of posets to the minuscule doppelg\"{a}nger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelg\"{a}nger pairs behave "as if" they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelg\"{a}nger pairs pretending to have isomorphic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelg\"{a}nger 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).