Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$

This paper initiates a systematic study of the cyclotomic KLR algebras of affine types $A$ and $C$. We start by introducing a graded deformation of these algebras and the constructing all of the irreducible representations of the deformed cyclotomic KLR algebras using content systems and a generalisation of the Young's seminormal forms for the symmetric groups. Quite amazingly, this theory simultaneously captures the representation theory of the cyclotomic KLR algebras of types $A$ and $C$, with the main difference being the definition of residue sequences of tableaux. We then use our semisimple deformations to construct two "dual" cellular bases for the non-semisimple KLR algebras of affine types $A$ and $C$. As applications of this theory we recover many of the main features from the representation theory in type $A$, simultaneously proving them for the cyclotomic KLR algebras of types $A$ and $C$. These results are completely new in type $C$ and we, usually, more direct proofs in type $A$. In particular, we show that these algebras categorify the irreducible integrable highest weight modules of the corresponding Kac-Moody algebras, we construct and classify their simple modules, we investigate links with canonical bases and we generalise Kleshchev's modular branching rules to these algebras.