An orthogonal realization of representations of the Temperley-Lieb algebra

TL;DR

This paper constructs an orthogonal basis for representations of the Temperley-Lieb algebra using maximal vectors in tensor powers of a quantum group module, providing new insights into its structure.

Contribution

It introduces a novel orthogonal basis for Temperley-Lieb algebra modules, connecting it with the standard cellular basis and expanding understanding of their representations.

Findings

Constructed orthogonal maximal vectors in tensor powers of quantum group modules.

Established an orthogonal basis for simple modules of the Temperley-Lieb algebra.

Linked the new basis to the existing cellular basis, enhancing structural understanding.

Abstract

Under a suitable hypothesis, we construct a full set of pairwise orthogonal maximal vectors in $V^{\otimes n}$, where $V=V(1)$ is the simple module of highest weight $1$ for the quantized enveloping algebra $\mathbf{U}(\mathfrak{sl}_2)$. We give a number of applications, one of which is an orthogonal basis of the simple modules for the Temperley-Lieb algebra $\text{TL}_n$. We relate this new orthogonal basis to the standard cellular basis.