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.
Highlights
The Razumov-Stroganov conjecture [15], proved in 2011 by L
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
Summary
The Razumov-Stroganov conjecture [15], proved in 2011 by L. Define a partial ordering on n × n ASMs by componentwise comparison of the corresponding height function matrices This poset is a self-dual distributive lattice which is the MacNeille completion of the Bruhat order on the symmetric group; this was proved in [11]. We denote the poset of join irreducibles of this distributive lattice as An, so that J(An) is in bijection with the set of n × n ASMs. In Definition 3.3 we give an explicit construction of An, and in Proposition 3.5 we give an explicit bijection, which will be of use later, from the order ideals J(An) to height function matrices of order n. Before we discuss the proof, we need to explain the statement of the RazumovStroganov correspondence
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