CHAPTER 1

Introduction

1.1. How do you do ?

These lectures are for graduate students who know the basics of the represen-

tation theory of finite groups and artinian rings. Through these lectures, you will

be exposed to some recent research in mathematics. Since we have to skip the

proofs of several theorems at several points to keep these lectures elementary, you

are encouraged to read the original papers for these parts in the second reading.

In the first reading, I recommend trusting the results so as to make your life easy.

Although I skip proofs at several points, these notes are basically written in the

“theorem and proof ”style.

The main example we use is the quantum algebra of type Ar−1.

(1)

The purpose of

the first half of this book is to explain the general theory of crystal bases using this

example. In the second half, we explain several interesting results using Young dia-

grams. I hope that the reader finds it interesting to do research in “Combinatorial

Representation Theory”after reading these notes.

1.2. What are we interested in ?

When you start your professional education in mathematics, you soon encounter

the notion of a group. It is a mathematical device used to describe symmetries in

nature. It is an idea that developed concurrently in several areas, such as geometry,

number theory and the algebra of polynomial equations, with an axiomatic defini-

tion coming in the middle of the 19th century. As you already know, the Galois

theory is the most famous application of the group theory. In modern times groups

are widely used in many areas and they have little to do with equations. They are

also used in essential ways in physics and chemistry. For example, the groups used

in gauge theories and classification of elementary particles are called Lie groups.

A typical example of a Lie group is the matrix group

GL(n, C) = X ∈ M(n, n, C) | det(X) = 0 ,

but there are other examples as well. Let us consider a group G. Often G will

be described as a group of matrices; there are many different ways of doing this,

usually using matrices of different sizes, although the group behind them all is the

same. Namely, there are many ways to associate a matrix ρ(X) with X ∈ G in such

a way that the product of two elements corresponds to the product of the associated

matrices (i.e. ρ(X) ρ(Y ) = ρ(XY )). These ρ are called representations of G. Lie

himself worked with “infinitesimal groups”, understanding that for many things it

was usually enough to consider the Lie algebras. In today’s language, we can say

that an essential feature of Lie’s work is his discovery that by looking at the second

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http://dx.doi.org/10.1090/ulect/026/01