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I am following the text by Isaacs on character theory and I have a few questions.

From p. 10, it seems like an reducible representation is one whose matrix at each group element can be written in a block diagonal form. However, the proposition on p. 20 states that the matrix of a representation at every group element can be written as a diagonal matrix. This would imply that every representation is reducible, which is clearly nonsense. What am I missing here?

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    (Three questions)2011-04-13

2 Answers 2

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On your first question: It sounds like you've mixed up the quantifiers. For a representation to be reducible, there has to be a single similarity transform that brings all matrices for all group elements into block diagonal form simultaneously. The proposition on p. 20 (I don't have the book in front of me) probably merely states that each matrix individually can be diagonalized, each by a different similarity transform.

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For "also", this is corollary 2.9, page 17: two representations are isomorphic iff their characters are equal.

For "yet another", use lemma 5.2 (Frobenius reciprocity), lemma 5.14 (character of transitive perm rep), corollary 5.15 (orbit counting using inner product), problem 5.2 (easy Mackey) or problem 5.6 (full Mackey).