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

TL;DR

This paper develops a unified framework for the representation theory of cyclotomic KLR algebras of affine types A and C, introducing deformations, content systems, and cellular bases, leading to new results especially in type C.

Contribution

It introduces a graded deformation and content systems for affine types A and C, providing new tools and results for their representation theory, including cellular bases and categorification.

Findings

Constructed all irreducible representations of deformed cyclotomic KLR algebras.

Established dual cellular bases for non-semisimple KLR algebras of types A and C.

Proved categorification of highest weight modules and classified simple modules.

Abstract

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.