A refined Razumov-Stroganov conjecture

TL;DR

This paper extends the Razumov-Stroganov conjecture by relating the groundstate of the O(1) spin chain to refined combinatorial objects, including refined alternating sign matrices, loop configurations, and plane partitions.

Contribution

It introduces a refined conjecture connecting the groundstate of the monodromy matrix to various detailed combinatorial structures, enhancing the original Razumov-Stroganov conjecture.

Findings

Proposes a relation between the groundstate and refined alternating sign matrices.

Suggests a connection to refined fully-packed loop configurations.

Conjectures a link to refined totally symmetric self-complementary plane partitions.

Abstract

We extend the Razumov-Stroganov conjecture relating the groundstate of the O(1) spin chain to alternating sign matrices, by relating the groundstate of the monodromy matrix of the O(1) model to the so-called refined alternating sign matrices, i.e. with prescribed configuration of their first row, as well as to refined fully-packed loop configurations on a square grid, keeping track both of the loop connectivity and of the configuration of their top row. We also conjecture a direct relation between this groundstate and refined totally symmetric self-complementary plane partitions, namely, in their formulation as sets of non-intersecting lattice paths, with prescribed last steps of all paths.