Sam Hopkins

The CDE property for minuscule lattices

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.

Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on P that avoid the pattern p is denoted AvP(p). We extend a proof of Simion and Schmidt to show that AvP(132)=AvP(123) for any poset P, and we exactly classify the posets for which equality holds.

Interlacing networks

Motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence, we define interlacing networks, which are certain planar directed networks with a rigid structure of sources and sinks. We describe an involution that swaps paths in these networks and leads to Plucker-like three-term relations among path weights. We show that indeed these relations follow from the Plucker relations in the Grassmannian together with some simple rank properties of the matrices corresponding to our interlacing networks. The space of matrices obeying these rank properties forms the closure of a cell in the matroid stratification of the totally nonnegative Grassmannian. Not only does the octahedron recurrence for RSK follow immediately from the three-term relations for interlacing networks, but also these relations imply some interesting identities of Schur functions reminiscent of those obtained by Fulmek and Kleber. These Schur function identities lead to some results on Schur positivity for expressions of the form s ? s ? - s λ s µ .