For a prime number p and any natural number n we introduce, by giving an explicit recursive formula, the p-Jones-Wenzl projector p JWn, an element of the Temperley-Lieb algebra T Ln(2) with coefficients in Fp. We prove that these projectors give the indecomposable objects in theà 1 -Hecke category over Fp, or equivalently, they give the projector in End SL 2 (Fp) ((F 2 p ) ⊗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 forà 1 .

The combinatorial invariance conjecture (due independently to Lusztig and Dyer) predicts that if $[x,y]$ and $[x^{\prime},y^{\prime}]$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan–Lusztig polynomials are equal, that is, $P_{x,y}(q)=P_{x^{\prime},y^{\prime}}(q)$. We prove this conjecture for the affine Weyl group of type $\widetilde {A}_2$. This is the first infinite group with non-trivial Kazhdan–Lusztig polynomials where the conjecture is proved.

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