Number of irreducible representations over reals equals number of equivalence classes under real conjugacy: Difference between revisions

From Groupprops
No edit summary
No edit summary
Line 22: Line 22:
==Facts used==
==Facts used==


# [[uses::Brauer's permutation lemma]]
# [[uses::Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations]]
# More?
==Proof==
==Proof==


Line 32: Line 31:
''This proof needs to be elaborated, but the essential idea is correct!''
''This proof needs to be elaborated, but the essential idea is correct!''


'''Proof''': Let <math>C(G)</math> be the set of conjugacy classes of <math>G</math> and <math>I(G)</math> be the set of characters of irreducible representations of <math>G</math> over <math>\mathbb{C}</math>. Consider the map <math>\tau:C(G) \to C(G)</math> that sends a conjugacy class to its inverse and the map <math>\sigma:I(G) \to I(G)</math> that sends an irreducible character to its complex conjugate character.
'''Proof''': We apply Fact (1) to the setup <math>d = 4</matH>, <math>r = -1</math>. {{fillin}}
 
Then, we see that <math>\tau</math> and <math>\sigma</math> both have order two, and that <math>\sigma(\chi)(g) = \chi(\tau(g))</math>. If we consider the character table matrix, then the action of one permutation on the rows equals the action of the other on the columns. Hence, by Fact (1), the permutations <math>\sigma</math> and <math>\tau</math> have the same cycle type. In particular, they have the same number of cycles.
 
For each cycle under <math>\sigma</math>, the character obtained by adding all the characters in that cycle gives a character over <math>\mathbb{C}</math> with values in <math>\R</math> and such that no proper subrepresentation has values in <math>\R</math>. This corresponds to (2).
 
Each cycle under <math>\tau</math> corresponds to an equivalence class under real conjugacy, which is (3).

Revision as of 17:15, 9 May 2011

Statement

The following are equal for a finite group G:

  1. The number of characters of G taking values in R arising from irreducible representations of G over R.
  2. The number of characters of G taking values in R arising from representations of G over C such that no proper nonzero subrepresentation takes values entirely in R.
  3. The number of equivalence classes of G under real conjugacy. Each such class arises as the union of a conjugacy class and the conjugacy class of inverse elements.
  4. The number of homomorphisms from Z to G, up to equivalence of automorphisms of Z and inner automorphisms of G.

Caveats and corollaries

The number of irreducible representations over reals is not the same as the number of irreducible representations over the complex numbers that can be realized over the reals. The latter number is either smaller or equal, and it is equal when the group is an ambivalent group, which means that every element is conjugate to its inverse.

Also, although the counts in (1) and (2) are equal, it is possible for a real character to arise from an irreducible representation over the complex numbers that is not realized over the reals. However, some multiple of that representation can be realized over the reals. This explains the equality of counts in (1) and (2). The smallest multiple used is termed the Schur index.

Related facts

Facts used

  1. Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations

Proof

Given: A finite group G

To prove: The number of irreducible representations of G over the real numbers equals the number of equivalence classes of elements of G under real conjugacy.

This proof needs to be elaborated, but the essential idea is correct!

Proof: We apply Fact (1) to the setup

d=4

,

r=−1

. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Retrieved from "https://groupprops.subwiki.org/w/index.php?title=Number_of_irreducible_representations_over_reals_equals_number_of_equivalence_classes_under_real_conjugacy&oldid=29635"