Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements
From Groupprops
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
- Number of irreducible representations equals number of conjugacy classes
- Number of irreducible representations over reals equals number of equivalence classes under real conjugacy
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
.
- The permutation on the set of conjugacy classes of
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 ![]() ![]() ![]() |
Fact (1) | [SHOW MORE] | ||
2 | The number of fixed points for the permutation of ![]() ![]() ![]() |
By definition | |||
3 | The number of fixed points for the permutation of ![]() |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | |||
4 | The result follows | Steps (1), (2), (3) | By Step (1), the permutation of ![]() ![]() |