Heaps reduction, decorated diagrams, and the affine Temperley-Lieb algebra of type $C$

TL;DR

This paper introduces a combinatorial framework with decoration and reduction algorithms to analyze the affine Temperley-Lieb algebra of type C, providing explicit algorithms for its representation and inverse.

Contribution

It offers a novel combinatorial approach with algorithms for the affine Temperley-Lieb algebra of type C, including a faithful representation and its inverse.

Findings

Explicit algorithmic description of Ernst's representation map

Construction of the inverse map for the representation

Demonstration of faithfulness of the representation

Abstract

In this paper we propose a combinatorial framework to study a diagrammatic representation of the affine Temperley-Lieb algebra of type C introduced by Ernst. In doing this, we define two procedures, a decoration algorithm on diagrams and a reduction algorithm on heaps of independent interest. Using this approach, an explicit algorithmic description of Ernst representation map is provided from which its faithfulness can be deduced. We also give a construction of the inverse map.