Combinatorial properties of Temperley Lieb algebras

TL;DR

This paper explores the combinatorial properties of two polynomial families within Temperley-Lieb algebras, providing recursions, formulas, and symmetry insights that enhance understanding of their structure.

Contribution

It introduces and analyzes two polynomial families in Temperley-Lieb algebras, revealing their combinatorial properties and establishing new recursive and explicit formulas.

Findings

Derived recursion relations for the polynomials

Established symmetry properties of the polynomials

Provided explicit formulas for constant terms

Abstract

We consider two families of polynomials that play the same role in the Temperley Lieb algebra of a Coxeter group as the Kazhdan Lusztig and R polynomials play in the Hecke algebra of the group. We study these polynomials from a combinatorial point of view. More precisely we obtain recursions, non recursive formulas, symmetry properties, and expressions for the constant terms, of these polynomials.