Quantum groups are in fact not groups. They are also called quantized en-
veloping algebras, or quantum algebras for short. They were born in mathematical
physics and have evolved into a vast area of research. In particular, they have found
applications in algebraic groups and given rise to big progress in the Lusztig conjec-
ture for algebraic groups. But I have another story to tell: these quantum algebras
also gave rise to new combinatorial objects and have influenced the combinatorics
related to representation theories. This research area is called “combinatorial rep-
resentation theory”.
These lecture notes are based on my lectures delivered at Sophia university in
1997, and are intended for graduate students who have interests in this area.
In the preparation of the lectures, I benefitted from two important papers,
[Kashiwara] and [Lusztig]. In fact, my primary intention was to introduce the
reader to the theory of crystal bases and canonical bases by working out special
examples, quantum algebras of type
In the lectures, I have named fundamental theorems about crystal bases and
canonical bases of quantum algebras as the first, the second and the third main
theorems of Kashiwara and Lusztig. I hope that the naming is accepted by the
society of mathematicians.
The plan of the book is as follows. The first three chapters are a preparation to
start running. In the 4th to the 6th chapters, we establish basic notions of crystal
bases. We then introduce canonical bases in Chapter 7 and prove fundamental
theorems in the subsequent two chapters. These chapters have flavors of the gen-
eral theory, although we are content with our examples, the quantum algebras of
type Ar−1.
In the next two chapters, we turn to combinatorics. We prove the
combinatorial construction of crystal bases of Fock spaces due to Misra and Miwa.
In the 12th chapter, we summarize its applications to the representation theory of
cyclotomic Hecke algebras. The 13th and 14th chapters are devoted to the proof
of my main theorem stated in Chapter 12. The final chapter is a guide for further
reading. The list is not intended to be complete of course, and reflects my personal
research interests.
If the reader has some familiarity with representation theory, I recommend skip-
ping the first three chapters. If he/she has some specialty in this field, I recommend
starting with the 7th chapter.
I would like to thank Professor Bhama Srinivasan and Andrew Mathas for read-
ing the manuscript, and my wife Tomoko for many things. During the preparation
of these notes, I was partially supported by the JSPS-DFG Japanese-German coop-
erative science program “Representation Theory of Finite and Algebraic Groups”.
Susumu Ariki
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