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

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