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Brauer's permutation lemma

127 bytes added, 23:21, 16 September 2007
Note that since the field has characteristic zero, the irreducible representations are the same as indecomposable representations.
Consider the matrix with rows indexed by indecomposable representations, columns indexed by conjugacy classes, and the entry in row <math>\rho</math> and column <math>c</math> is <math>\chi(c,\rho)</math>. In other words, the matrix is the [[character table]]. The automorphism group acts on the rows and on the columns in a way that the entries are not affected. {{further|[[Conjugacy class-representation duality]]}}
Applying We can now apply Galois automorphisms (viz, maps that raise to exponents which are in the Galois group of the sufficiently large field, over the given field) to both the conjugacy classes and the irreducible representations. The effect of a Galois automorphism on the rows is the same as its effect on the columns. Hence, we can apply Brauer's permutation lemma to this, we these. We thus get that for any Galois automorphism, the sizes of orbits of conjugacy classes under the automorphism, are the same as the sizes of orbits of irreducible representations. In particular, the number of invariant conjugacy classes equals the number of invariant irreducible representations.
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