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

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.