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 then constructing all of the irreducible representations of the deformed cyclotomic KLR algebras using content systems, which are recursively defined using Rouquier's Q-polynomials.This leads to a generalisation of the Young's seminormal forms for the symmetric groups in the KLR setting.Quite amazingly, the same theory captures the representation theory of the cyclotomic KLR algebras of affine types A and C, with the main difference being that the definition of the residue sequence of a tableau depends on the Cartan type.We use our semisimple deformations to construct two "dual" cellular bases for the non-semisimple KLR algebras of affine types A and C. As applications 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, give 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.We record with deep sadness the passing of Anton Evseev on February 21, 2017.