Cyclotomic quiver Hecke algebras of type A

TL;DR

This paper surveys the representation theory of cyclotomic Hecke algebras of type A, emphasizing the KLR grading and its connections to classical and graded theories, with detailed examples and new insights.

Contribution

It provides a comprehensive introduction to cyclotomic quiver Hecke algebras of type A, including a complete description of semisimple KLR algebras and proofs of key categorification theorems.

Findings

Complete description of semisimple KLR algebras of type A

Extensive discussion on graded Specht modules

Proof of the Ariki-Brundan-Kleshchev categorification theorem

Abstract

This chapter is based on a series of lectures that I gave at the National University of Singapore in April 2013. The notes survey the representation theory of the cyclotomic Hecke algebras of type A with an emphasis on understanding the KLR grading and the connections between the "classical" ungraded representation theory and the rapidly emerging graded theory. They are fairly self-contained and they try to give a leisurely introduction to these algebras, with many examples and calculations that don't appear elsewhere. We make extensive use of the interactions between the ungraded and graded representation theory and try to explain what the grading gives us that we didn't have before. Combinatorics and cellular algebra techniques are used throughout, with a few results from geometry and 2-representation theory being quoted from the literature. Highlights include a complete description of the semisimple KLR algebras of type A using just the KLR relations, extensive discussion about graded Specht modules, a proof of the Ariki-Brundan-Kleshchev graded categorification theorem based on the graded branching rules, a cellular algebra approach to adjustment matrices and a (possibly optimistic) conjecture for the graded dimensions of the simple modules.