Tensor space representations of Temperley-Lieb algebra and generalized permutation matrices

TL;DR

This paper classifies certain tensor space representations of the Temperley-Lieb algebra using orthogonal projections in complex tensor spaces, providing explicit solutions for specific ranks and parameter ranges.

Contribution

It offers a complete classification for rank one projections and constructs new solutions for rank two projections with varying parameters.

Findings

Complete classification of rank one solutions.

Construction of rank two solutions for specific parameter ranges.

Solutions applicable to tensor spaces of dimensions multiple of 3 and 4.

Abstract

Orthogonal projections in ${\mathbb C}^n \otimes {\mathbb C}^n$ of rank one and rank two that give rise to unitary tensor space representations of the Temperley-Lieb algebra $TL_N(Q)$ are considered. In the rank one case, a complete classification of solutions is given. In the rank two case, solutions with $Q$ varying in the ranges $[2n/3,\infty)$ and $[n/\sqrt{2},\infty)$ are constructed for $n=3k$ and $n=4k$, $k \in {\mathbb N}$, respectively.