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 x 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) >= 4r. For the latter class of representations, new examples are found. This includes explicit examples for r = 2,3,4 and any n >= r (with one exception) and a solution for n = r+1 with arbitrary r.