The boundary algebra of a GL${}_m$-dimer

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. $e_b{\Lambda }_{Q_{T_1}}{e}_b\cong e_b{\Lambda }_{Q_{T_2}}{e}_b$.

22 pages, 17 figures