Modules of the Temperley-Lieb algebra at zero
Eddy Li, Kenta Suzuki
TL;DR
This work provides a complete, explicit description of the module category of the Temperley-Lieb algebra $\mathsf{TL}_n(0)$ for even $n$ by realizing it as representations of a finite quiver algebra $\mathbb{C}\mathcal{Q}_{n/2}/J$ and establishing an equivalence of highest weight categories. It constructs an explicit exact sequence of standard modules $0\to W_n^n\to W_{n-2}^n\to\cdots\to W_0^n\to 0$, whose irreducibles are the images of the maps, and gives diagrammatic descriptions of the connecting morphisms $\phi_\,\ell^n$. The results yield a categorification of the Jones polynomial specialization at $t=-1$ and, in characteristic two, produce a corresponding exact sequence of Specht modules for the symmetric group, connecting TL(0) representation theory to both knot invariants and modular representation theory. The work also provides explicit HW-structure data and a concrete functor $\Phi$ realizing the category equivalence, with broader implications for diagrammatic and Hecke-algebraRelated representations.
Abstract
We explicitly describe the category of modules of the Temperley-Lieb algebra $\mathrm{TL}_n(β)$ under specialization $β=0$ 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.