# Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatorics

## TL;DR

This paper explores algebraic and combinatorial properties of quotients of type  C Temperley-Lieb algebras, introducing bases parameterized by fully commutative elements and enumerating them by affine length.

## Contribution

It provides a monomial basis for the two-boundary Temperley-Lieb and symplectic blob algebras, linking algebraic structures to combinatorial enumeration.

## Key findings

- Established monomial bases for both algebras
- Connected basis elements to subsets of fully commutative elements
- Enumerated these elements by affine length

## Abstract

We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.