Graded representation theory of the cyclotomic quiver Hecke algebras of type \(A\)

Andrew Mathas

Abstract

This is a DRAFT chapter based on a series of lectures that I gave at the National University of Singapore in April 2013. They 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 thory and try to explain what the grading gives us which we didn't have before. Combinatorics and cellular algebra techniques are used extensively, with a few results from geometry and 2-representation theory being quoted. 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 using the graded branching rules, a cellular algebra approach to adjustment matrices and an optimistic conjecture for the graded dimensions of the simple modules. Comments, corrections and suggestions are very welcome!

AMS Subject Classification: Primary 20G43; secondary 20C08, 20C30.