Modules of the Temperley-Lieb algebra at zero

Abstract

We explicitly describe the category of modules of the Temperley-Lieb algebra ${\mathrm{TL}}_n({\hat{I}}^2)$ under specialization ${\hat{I}}^2=0$ for even $n$ in terms of a quiver algebra, analogous to a result of Berest-Etingof-Ginzburg. In particular, we explicitly construct an exact sequence of the standard modules of ${\mathrm{TL}}_n(0)$, which categorifies a numerical coincidence regarding the evaluation of the Jones polynomial at $t=-1$. We furthermore deduce a consequence in the representation theory of symmetric groups over characteristic two.

17 pages