The boundary algebra of a GL$_m$-dimer

TL;DR

This paper proves that the boundary algebra associated with GL_m-dimers of triangulated polygons is invariant under different triangulations, revealing a fundamental algebraic property of these geometric structures.

Contribution

It establishes that the boundary algebra of GL_m-dimers remains isomorphic regardless of the triangulation of the polygon, a new invariance result in the study of dimer models.

Findings

Boundary algebras are isomorphic for different triangulations.

The boundary algebra is an invariant of the polygon triangulation.

Provides algebraic insight into the structure of GL_m-dimers.

Abstract

We consider GL$_m$-dimers of triangulations of regular convex $n$-gons, which give rise to a dimer model with boundary $Q$ and a dimer algebra $\Lambda_Q$. Let $e_b$ be the sum of the idempotents of all the boundary vertices, and $\mathcal{B}_Q:= e_b \Lambda_Q e_b$ the associated boundary algebra. In this article we show that given two different triangulations $T_1$ and $T_2$ of the $n$-gon, the boundary algebras are isomorphic, i.e. $e_b \Lambda_{Q_{T_1}} e_b \cong e_b \Lambda_{Q_{T_2}} e_b$.