Representations of the Temperley-Lieb Algebra via a New Inner Product on Half-Diagrams

TL;DR

This paper introduces a new inner product on diagrams representing the Temperley-Lieb algebra, providing a natural combinatorial basis and orthogonalization method that generalizes existing constructions related to Catalan numbers.

Contribution

It presents a novel inner product and orthogonal basis for the Temperley-Lieb algebra using combinatorial bijections and generalizations of Catalan-numbered structures.

Findings

Constructed a new inner product on diagram representations.

Developed a natural combinatorial basis for the algebra.

Provided an orthogonalization method similar to existing approaches.

Abstract

We describe an inner product on the diagrams on which the Temperley-Lieb algebra can be represented. We exhibit several constructions which are in natural combinatorial bijection with these diagrams, which are generalizations of various constructions counted by the Catalan numbers. We use a method similar to the existing ones for orthogonalizing the Temperley-Lieb algebra to construct an orthogonal basis for the vector space over these diagrams.