A relation for the Jones-Wenzl projector and tensor space representations of the Temperley-Lieb algebra

TL;DR

This paper proves a key relation for the Jones-Wenzl projector and explores its implications for tensor space representations of the Temperley-Lieb algebra, providing new examples and conditions for such representations.

Contribution

It establishes a relation for the Jones-Wenzl projector and characterizes tensor space representations based on a specific matrix condition, introducing new explicit examples.

Findings

Derived conditions for tensor space representations based on matrix rank and eigenvalues.

Constructed explicit examples for ranks 2, 3, 4 with various dimensions.

Identified new classes of representations satisfying the established relations.

Abstract

A relation for the Jones-Wenzl projector is proven. It has the following consequence for representations of the Temperley-Lieb algebra on tensor product spaces: if such a representation is built from a Hermitian $n \times n$ matrix $T$ of rank $r$ such that $T^2=Q T$, then either $n^2 = Q^2 r$ and $Q^2 =1,2,3$ or $n^2 \geq 4 r$. For the latter class of representations, new examples are found. This includes explicit examples for $r=2,3,4$ and any $n \geq r$ (with one exception) and a solution for $n=r+1$ with arbitrary $r$.