Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups

Let W be an affine Weyl group, and let k be a field of characteristic p> 0. The diagrammatic Hecke category D for W over k is a categorification of the Hecke algebra for W with rich connections to modular representation theory. We explicitly construct a functor from D to a matrix category which categorifies a recursive representation ξ : ZW → Mpr (ZW), where r is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are “smaller” by a factor of p. It also explains the presence of self-similarity in the p-canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the p-canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent p-canonical tilting character formula due to Achar–Makisumi–Riche–Williamson.