$p$-Jones-Wenzl idempotents

Abstract

For a prime number $p$ and any natural number $n$ we introduce, by giving an explicit recursive formula, the $p$-Jones-Wenzl projector ${}^p{\mathrm{JW}}_n$, an element of the Temperley-Lieb algebra $T{L}_n(2)$ with coefficients in ${\mathbb{F}}_p$. We prove that these projectors give the indecomposable objects in the ${\tilde{A}}_1$-Hecke category over ${\mathbb{F}}_p$, or equivalently, they give the projector in ${\mathrm{End}}_{{\mathrm{SL}}_2(\overline{{\mathbb{F}}_p})}(({\mathbb{F}}_p^2{)}^{\otimes n})$ to the top tilting module. The way in which we find these projectors is by categorifying the fractal appearing in the expression of the $p$-canonical basis in terms of the Kazhdan-Lusztig basis for ${\tilde{A}}_1$.

15 pages, 21 figures. Many minor changes. Major change of notation. Final version