# Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements

## Contents

## Statement

Suppose is a finite group. Then, the following numbers are equal:

- The number of irreducible representations of over the complex numbers whose characters are real-valued. Note that this includes both real representations (representations realized over ), and quaternionic representations, which are not realized over but whose double is realized over (so they have Schur index 2).
- The number of conjugacy classes in of real elements, i.e., elements that are conjugate to their inverses.

## Related facts

## Facts used

- Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations (follows in turn from Brauer's permutation lemma): Suppose is a finite group and is an integer relatively prime to the order of . Suppose is a field and is a splitting field of of the form where is a primitive root of unity, with also relatively prime to (in fact, we can arrange to divide the order of because sufficiently large implies splitting). Suppose there is a Galois automorphism of that sends to . Consider the following two permutations:
- The permutation on the set of conjugacy classes of , denoted , induced by the mapping .
- The permutation on the set of irreducible representations of over , denoted , induced by the Galois automorphism of that sends to .

Then, these two permutations have the same cycle type. In particular, they have the same number of cycles, and the same number of fixed points, as each other.

## Proof

**Given**: A finite group

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

**Proof**: Let be the set of conjugacy classes of and be the set of irreducible representations of over .

Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | The cycle type of the permutation of induced by is the same as the cycle type of the permutation of induced by post-composing with complex conjugation. | Fact (1) | [SHOW MORE] | ||

2 | The number of fixed points for the permutation of induced by is the number of conjugacy classes of real elements in . | By definition | |||

3 | The number of fixed points for the permutation of induced by complex conjugation is the number of irreducible representations over the complex numbers with real character values. | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
| |||

4 | The result follows | Steps (1), (2), (3) | By Step (1), the permutation of and of have the same cycle type, hence the same number of fixed points. Steps (2) and (3) now complete the proof. |