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 $T^2=QT$, then either $n^2=Q^{2}r$ and $Q^2=1,2,3$ or $n^2\ge 4r$. For the latter class of representations, new examples are found. This includes explicit examples for $r=2,3,4$ and any $n\ge r$ (with one exception) and a solution for $n=r+1$ with arbitrary $r$.