  Let $W$ be an affine Weyl group, and let $\Bbbk$ be a field of characteristic
$p>0$. The diagrammatic Hecke category $\mathcal{D}$ for $W$ over $\Bbbk$ is a
categorification of the Hecke algebra for $W$ with rich connections to modular
representation theory. We explicitly construct a functor from $\mathcal{D}$ to
a matrix category which categorifies a recursive representation $\xi:
\mathbb{Z}W \rightarrow M_{p^r}(\mathbb{Z}W)$, 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.
