Modules of the Temperley-Lieb algebra at zero

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.

💡 Research Summary

The paper studies the Temperley‑Lieb algebra TLₙ(β) specialized at β = 0, focusing on even n. The authors first recall that for generic β the standard modules Wₙ^ℓ are irreducible, but at β = 0 they acquire length two and fit into short exact sequences 0 → Lₙ^{ℓ+2} → Wₙ^ℓ → Lₙ^ℓ → 0, where Lₙ^ℓ are the simple quotients. They then construct projective covers Pₙ^ℓ by inducing from TL_{n‑1}(0) and compute Hom‑spaces between these projectives and the standards, showing that non‑zero maps occur only when the indices differ by at most two.

A key insight is the organization of the projectives {Pₙ^{2}, Pₙ^{4}, …, Pₙ^{n}} into a diagram that mirrors a straight‑line quiver Q_{n/2}. By defining morphisms ω_ℓ: Pₙ^ℓ → Pₙ^{ℓ+2} and γ_ℓ: Pₙ^{ℓ+2} → Pₙ^ℓ and checking that they satisfy the relations a_{i+1}a_i = 0, b_i b_{i+1} = 0, and a_i b_i = b_{i+1} a_{i+1}, the authors identify End(P_·) with the path algebra C Q_{n/2} modulo the ideal J generated by those relations. Consequently, the functor Hom(P_·, –) yields an equivalence of highest‑weight categories \