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Representations and Characters of Groups, Second Edition by Martin Liebeck, Gordon James

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16

Character tables and orthogonality relations

The irreducible characters of a finite group G are class functions, and the number of them is equal to the number of conjugacy classes of G. It is therefore convenient to record all the values of all the irreducible characters of G in a square matrix. This matrix is called the character table of G. The entries in a character table are related to each other in subtle ways, many of which are encapsulated in the orthogonality relations (Theorem 16.4). Much of the later material in the book will be devoted to understanding character tables. The motivation for this is Theorem 14.21, which tells us that every G-module is determined by its character. Thus, many problems in representation theory can be ...

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