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

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 $r$.