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Brauer's permutation lemma

396 bytes removed, 23:59, 9 April 2010
Applications
==Applications==
Brauer's permutation lemma helps us exploit the [[conjugacy class-representation duality]] in an two interesting wayways. Let <math>C(G)</math> denote the set of conjugacy classes of a [[finite group]] <math>G</math> and <math>I(G)</math> denote the set of [[indecomposable linear representation]]s of <math>G</math>. Let <math>\chi(c,\rho)</math> denote the trace of <math>\rho(g)</math> where <math>g \in G</math> (i.e. the character value).
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>.
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 these.consider two different actions:
We thus get that for any * [[Application of Brauer's permutation lemma to Galois automorphism, the sizes of orbits of automorphisms on conjugacy classes under the automorphism, are the same as the sizes of orbits class-representation duality]]* [[Application of irreducible representations. In particular, the number of invariant Brauer's permutation lemma to group automorphisms on conjugacy classes equals the number of invariant irreducible representations.class-representation duality]]
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