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 JWnp, an element of the Temperley-Lieb algebra TLn(2) with coefficients in Fp. We prove that these projectors give the indecomposable objects in the A˜1-Hecke category over Fp, or equivalently, they give the projector in EndSL2(Fp‾)((Fp2)⊗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 A˜1.