The Rotor Model and Combinatorics

TL;DR

This paper investigates the rotor model's groundstate wavefunction under various boundary conditions, revealing connections to enumerations of alternating sign matrices, including new results on vertically symmetric matrices.

Contribution

It introduces three conjectures linking the rotor model's groundstate to enumerations of alternating sign matrices, including a new enumeration for vertically symmetric matrices.

Findings

Identifies conjectural links between the rotor model and alternating sign matrices.

Discovers the number A_V(2m+1;3) for vertically symmetric matrices.

Proposes three conjectures on these combinatorial connections.

Abstract

We examine the groundstate wavefunction of the rotor model for different boundary conditions. Three conjectures are made on the appearance of numbers enumerating alternating sign matrices. In addition to those occurring in the O($n=1$) model we find the number $A_{\rm V}(2m+1;3)$, which 3-enumerates vertically symmetric alternating sign matrices.