Noncrossing partitions, fully commutative elements and bases of the Temperley-Lieb algebra

TL;DR

This paper introduces a new basis for the Temperley-Lieb algebra using combinatorial bijections and Bruhat order, providing explicit formulas for basis change coefficients.

Contribution

It presents a novel basis for the Temperley-Lieb algebra derived from noncrossing partitions and fully commutative elements, connecting combinatorics with algebraic structures.

Findings

New basis defined via noncrossing partitions and fully commutative elements

Closed formulas for coefficients in basis change expansions

Properties of the basis related to Bruhat order and existing bases

Abstract

We introduce a new basis of the Temperley-Lieb algebra. It is defined using a bijection between noncrossing partitions and fully commutative elements together with a basis introduced by Zinno, which is obtained by mapping the simple elements of the Birman-Ko-Lee braid monoid to the Temperley-Lieb algebra. The combinatorics of the new basis involve the Bruhat order restricted to noncrossing partitions. As an application we can derive properties of the coefficients of the base change matrix between Zinno's basis and the well-known diagram or Kazhdan-Lusztig basis of the Temperley-Lieb algebra. In particular, we give closed formulas for some of the coefficients of the expansion of an element of the diagram basis in the Zinno basis.