## $$p$$-Jones-Wenzl idempotents

### Gaston Burrull, Nicolas Libedinsky and Paolo Sentinelli

#### 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 $${\mathbf F}_p$$. We prove that these projectors give the indecomposable objects in the $$\tilde{A}_1$$-Hecke category over $${\mathbf F}_p$$, or equivalently, they give the projector in $$\mathrm{End}_{\mathrm{SL}_2(\overline{{\mathbf F}_p})}(({\mathbf F}_p^2)^{\otimes n})$$ 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 $$\tilde{A}_1$$.

: Primary 20G05; secondary 05E10.

This paper is available as a pdf (424kB) file. It is also on the arXiv: arxiv.org/abs/1902.00305.

 Monday, February 4, 2019