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

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13

Characters

Suppose that ρ: G → GL(n, image) is a representation of the finite group G. With each n × n matrix gρ (gG) we associate the complex number given by adding all the diagonal entries of the matrix, and call this number χ(g). The function χ: Gimage is called the character of the representation ρ. Characters of representations have many remarkable properties, and they are the fundamental tools for performing calculations in representation theory. For example, we shall show later that two representations have the same character if and only if ...

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