Brauer's permutation lemma
This article gives the statement, and proof, of a particular subgroup in a group being conjugacy-closed: in other words, any two elements of the subgroup that are conjugate in the whole group, are also conjugate in the subgroup
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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
- Symmetric group is not subset-conjugacy-closed in general linear group: In particular, in formulation (4), we cannot replace cyclic group by an arbitrary finite group.
Brauer's permutation lemma helps us exploit the conjugacy class-representation duality in two interesting ways. 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 consider two different actions: