Chapter 1

Introduction

Very roughly speaking, representation theory studies symmetry in

linear spaces. It is a beautiful mathematical subject which has many

applications, ranging from number theory and combinatorics to ge-

ometry, probability theory, quantum mechanics, and quantum field

theory.

Representation theory was born in 1896 in the work of the Ger-

man mathematician F. G. Frobenius. This work was triggered by a

letter to Frobenius by R. Dedekind. In this letter Dedekind made the

following observation: take the multiplication table of a finite group

G and turn it into a matrix XG by replacing every entry g of this

table by a variable xg. Then the determinant of XG factors into a

product of irreducible polynomials in {xg}, each of which occurs with

multiplicity equal to its degree. Dedekind checked this surprising fact

in a few special cases but could not prove it in general. So he gave

this problem to Frobenius. In order to find a solution of this problem

(which we will explain below), Frobenius created the representation

theory of finite groups.

The goal of this book is to give a “holistic” introduction to rep-

resentation theory, presenting it as a unified subject which studies

representations of associative algebras and treating the representa-

tion theories of groups, Lie algebras, and quivers as special cases. It

is designed as a textbook for advanced undergraduate and beginning

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http://dx.doi.org/10.1090/stml/059/01