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 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).

Keywords:

• AND PHRASES. Coincidental down-degree expectations (CDE)
• doppelgängers
• P-partitions
• rowmotion
• homomesy
• cyclic sieving phenomenon
• minuscule posets
• root posets

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 06A07
• 06A11
• 05E18

Acknowledgements

I thank Vic Reiner for many useful discussions throughout, and in particular for helpful references. I thank Nathan Williams for useful comments about doppelgängers and bijections, and for making me aware of the paper [24]. I thank Joel Kamnitzer, Brendon Rhoades, and Hugh Thomas for useful comments about Conjecture 4.23. Hugh Thomas informed me that he and his collaborators were also thinking about cyclic sieving for minuscule P-partitions. I thank Jim Propp for informing me that he had independently and earlier (around 2016) conjectured the Type A case of Conjecture 4.28. I thank David Einstein for explaining to me the bounded nature of piecewise-linear rowmotion (see Remark 4.33). I thank Bruce Westbury for explaining some possible connections to crystals and cactus group actions (see Remark 4.26). Finally, I thank the anonymous referee for careful attention to the manuscript and useful comments. I was supported by NSF grant $\#1802920.$ Sage mathematics software [25, 26] was indispensable for testing conjectures. The Sage code used for these tests is available upon request and is included with the arXiv submission of this paper.

Notes

1 Very recently, the part of our conjecture concerning the orbit structure of rowmotion for minuscule doppelgänger pairs was resolved in the affirmative; see Remark 4.15.

2 Traditionally (as in [[97], Chapter 3]) a P-partition is defined to be order-reversing rather than order-preserving; but we follow [38] here in defining it to be order-preserving.

3 The conventional indexing (as in [[97], Chapter 3]) would define the order polynomial to be what is in our notation ${\Omega }_P(\ell -1),$ but this is immaterial; again, we follow [38].

4 This transfer map is slightly different than the one in [95]: the difference is essentially given by replacing P by $P^{*}.$ The differences are immaterial and this definition ofϕ is cleaner for our later applications.

5 The correspondence between rectangle poset order ideals under rowmotion and binary words under rotation was also independently discovered by Thomas; see [[77], §3.3.2], where this correspondence is termed the “Stanley-Thomas word.”