# $p$-Jones-Wenzl idempotents

## TL;DR

This paper introduces explicit recursive formulas for $p$-Jones-Wenzl projectors in the Temperley-Lieb algebra over ${
m f F}_p$, linking them to indecomposable objects in the $ ilde{A}_1$-Hecke category and categorifying a fractal related to the $p$-canonical basis.

## Contribution

It provides the first explicit recursive construction of $p$-Jones-Wenzl projectors over ${
m f F}_p$, connecting algebraic, categorical, and combinatorial structures.

## Key findings

- Explicit recursive formulas for $p$-Jones-Wenzl projectors.
- Identification of these projectors as indecomposable objects in the $ ilde{A}_1$-Hecke category.
- Categorification of a fractal related to the $p$-canonical basis.

## Abstract

For a prime number $p$ and any natural number $n$ we introduce, by giving an explicit recursive formula, the $p$-Jones-Wenzl projector ${}^p\operatorname{JW}_n$, an element of the Temperley-Lieb algebra $TL_n(2)$ 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 $\tilde{A}_1$.