Modules of the Temperley-Lieb algebra at zero

TL;DR

This paper explicitly describes the module category of the Temperley-Lieb algebra at zero for even n, linking it to a quiver algebra and connecting to Jones polynomial evaluations and symmetric group representations.

Contribution

It provides an explicit quiver algebra description of modules of TL_n(0) for even n, and constructs an exact sequence of standard modules that categorifies a Jones polynomial evaluation.

Findings

Explicit quiver algebra description of TL_n(0) modules

Construction of an exact sequence categorifying Jones polynomial at t=-1

Implications for symmetric group representations in characteristic two

Abstract

We explicitly describe the category of modules of the Temperley-Lieb algebra $\mathrm{TL}_n(\beta)$ under specialization $\beta=0$ 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.