Brauer's permutation lemma: Difference between revisions

Revision as of 23:59, 9 April 2010

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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Statement

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.

Applications

Brauer's permutation lemma helps us exploit the conjugacy class-representation duality in two interesting ways. Let $C (G)$ denote the set of conjugacy classes of a finite group $G$ and $I (G)$ denote the set of indecomposable linear representations of $G$ . Let $χ (c, ρ)$ denote the trace of $ρ (g)$ where $g \in G$ (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 $ρ$ and column $c$ is $χ (c, ρ)$ .

We can now consider two different actions:

@@ Line 13: / Line 13: @@
 ==Applications==
-Brauer's permutation lemma helps us exploit the [[conjugacy class-representation duality]] in an interesting way. 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).
+Brauer's permutation lemma helps us exploit the [[conjugacy class-representation duality]] in two interesting ways. 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.
@@ Line 19: / Line 19: @@
 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.
+We can now consider two different actions:
-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.
+* [[Application of Brauer's permutation lemma to Galois automorphisms on conjugacy class-representation duality]]
+* [[Application of Brauer's permutation lemma to group automorphisms on conjugacy class-representation duality]]