\(p\)-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 \(\tilde{A}_1\).

AMS Subject Classification: Primary 20G05; secondary 05E10.