Preface

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

Ar−1.(1)

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.

(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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