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

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 $T{L}_N(Q)$ 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 $[2n\mathrm{/}3,\mathrm{\infty })$ and $[n\mathrm{/}\sqrt{2},\mathrm{\infty })$ are constructed for $n=3k$ and $n=4k$, $k\in \mathbb{N}$, respectively.