  We prove a conjecture of Rouquier relating the decomposition numbers in
category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's
canonical basis of a higher level Fock space. Independent proofs of this
conjecture have also recently been given by Rouquier, Shan, Varagnolo and
Vasserot and by Losev, using different methods.
  Our approach is to develop two diagrammatic models for this category
$\mathcal{O}$; while inspired by geometry, these are purely diagrammatic
algebras, which we believe are of some intrinsic interest. In particular, we
can quite explicitly describe the representations of the Hecke algebra that are
hit by projectives under the $\mathsf{KZ}$-functor from the Cherednik category
$\mathcal{O}$ in this case, with an explicit basis.
  This algebra has a number of beautiful structures including categorifications
of many aspects of Fock space. It can be understood quite explicitly using a
homogeneous cellular basis which generalizes such a basis given by Hu and
Mathas for cyclotomic KLR algebras. Thus, we can transfer results proven in
this diagrammatic formalism to category $\mathcal{O}$ for a cyclotomic rational
Cherednik algebra, including the connection of decomposition numbers to
canonical bases mentioned above, and an action of the affine braid group by
derived equivalences between different blocks.
