Dimer representations of the Temperley-Lieb algebra

TL;DR

This paper introduces a novel dimer-based spin-chain representation of the Temperley-Lieb algebra at $eta=0$, which differs from traditional models by having half the dimension and reveals indecomposable zigzag modules.

Contribution

It presents a new dimer representation of the Temperley-Lieb algebra, expanding understanding of its structure and module types.

Findings

Dimer representation has dimension 2^{n-1}

Reveals indecomposable zigzag modules

Connects to the dimer model in statistical mechanics

Abstract

A new spin-chain representation of the Temperley-Lieb algebra $TL_n(\beta=0)$ is introduced and related to the dimer model. Unlike the usual XXZ spin-chain representations of dimension $2^n$, this dimer representation is of dimension $2^{n-1}$. A detailed analysis of its structure is presented and found to yield indecomposable zigzag modules.