Diagram calculus for a type affine $C$ Temperley--Lieb algebra, II

TL;DR

This paper proves the faithfulness of a diagrammatic representation for an affine type C Temperley--Lieb algebra and links basis diagrams to the algebra's monomial basis, advancing diagram calculus understanding.

Contribution

It establishes the faithfulness of the diagrammatic representation and connects basis diagrams to the algebra's monomial basis for the affine type C case.

Findings

Diagrammatic representation is faithful.

Basis diagrams correspond to monomial basis elements.

Advances diagram calculus for affine type C Temperley--Lieb algebra.

Abstract

In a previous paper, we presented an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley--Lieb algebra having a basis indexed by the fully commutative elements of the Coxeter group of type affine $C$. We also provided an explicit description of a basis for the diagram algebra. In this paper, we show that the diagrammatic representation is faithful and establish a correspondence between the basis diagrams and the so-called monomial basis of the Temperley--Lieb algebra of type affine $C$.