Brauer's permutation lemma
Brauer's permutation lemma has the following equivalent forms:
- If a row permutation and a column permutation have the same effect on a nonsingular matrix, then they must have the same number of cycles of a given length
- The symmetric group is a conjugacy-closed subgroup in the general linear group over any field of characteristic zero
- If two permutation matrices are conjugate in the general linear group over a field of characteristic zero, then they have the same number of cycles of each length, viz, are conjugate in the symmetric group itself
- If two permutation representations of a cyclic group are conjugate in the general linear group over a field of characteristic zero, they are also conjugate in the symmetric group
Notice that in the last formulation we cannot replace cyclic group by an arbitrary finite group.
Brauer's permutation lemma helps us exploit the conjugacy class-representation duality in an interesting way. Let denote the set of conjugacy classes of a finite group and denote the set of indecomposable linear representations of . Let denote the trace of where (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 and column is .
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.
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.