  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.

Representation Theory;Combinatorics;Quantum Algebra;Rings and Algebras