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 1
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