Dual bases in Temperley-Lieb algebras

Abstract

This note is an announcement of the paper [BC16]. We derive a Laurent series expansion in $d$ for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra $\mathrm{TL}_k(d)$, converging for all complex loop parameters $d$ with $|d| \gt 2\cos\big(\frac{\pi}{k+1}\big)$. The coefficients appearing in each Laurent expansion are shown to have a natural combinatorial interpretation and their sign is explicitly understood. As an application, we solve a series of questions raised by Jones and improve substantially our understanding of the Jones Wenzl projection.