The CDE property for minuscule lattices

TL;DR

This paper advances the understanding of the CDE property in finite posets, particularly minuscule lattices, and explores its applications in combinatorics and dynamical algebraic combinatorics.

Contribution

It resolves several conjectures about CDE posets and completes the proof that all minuscule lattices possess the CDE property.

Findings

All minuscule lattices are CDE.

CDE property leads to formulas for set-valued tableaux.

CDE property explains homomesy in rowmotion and gyration.

Abstract

Reiner, Tenner, and Yong recently introduced the coincidental down-degree expectations (CDE) property for finite posets and showed that many nice posets are CDE. In this paper we further explore the CDE property, resolving a number of conjectures about CDE posets put forth by Reiner-Tenner-Yong. A consequence of our work is the completion of a case-by-case proof that any minuscule lattice is CDE. We also explain two major applications of the study of CDE posets: formulas for certain classes of set-valued tableaux; and homomesy results for rowmotion and gyration acting on sets of order ideals.