Graded representation theory of the cyclotomic quiver Hecke algebras of type
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 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 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.