Chapter 1
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
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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