An orthogonal realization of representations of the Temperley-Lieb algebra

Under a suitable hypothesis, we construct a full set of pairwise orthogonal maximal vectors in $V^{\otimes n}$, where $V=V(1)$ is the simple module of highest weight $1$ for the quantized enveloping algebra $\mathbf{U}({\mathfrak{s}\mathfrak{l}}_2)$. We give a number of applications, one of which is an orthogonal basis of the simple modules for the Temperley-Lieb algebra ${\text{TL}}_n$. We relate this new orthogonal basis to the standard cellular basis.