The toggle group, homomesy, and the Razumov-Stroganov correspondence

TL;DR

This paper explores the Razumov-Stroganov correspondence linking statistical physics and combinatorics, reformulating key proof components using posets, toggle groups, and homomesy, and establishing new homomesy results with potential broader impact.

Contribution

It introduces a poset-based reformulation of the Razumov-Stroganov proof and proves new homomesy results on general posets, expanding the theoretical framework.

Findings

Reformulation of the Razumov-Stroganov correspondence using posets

Two new homomesy results on general posets

Potential broader implications for combinatorics and statistical physics

Abstract

The Razumov-Stroganov correspondence, an important link between statistical physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello, relates the ground state eigenvector of the O(1) dense loop model on a semi-infinite cylinder to a refined enumeration of fully-packed loops, which are in bijection with alternating sign matrices. This paper reformulates a key component of this proof in terms of posets, the toggle group, and homomesy, and proves two new homomesy results on general posets which we hope will have broader implications.