Irreducible character of degree greater than one takes value zero on some conjugacy class
From Groupprops
Statement
Suppose is a finite group and
is the character of an irreducible linear representation of
over
, such that the degree of the representation (and hence, of
) is greater than one. Then, there exists an element
(and hence, a Conjugacy class (?)) such that
.
Related facts
- Zero-or-scalar lemma
- Conjugacy class of more than average size has character value zero for some irreducible character
Examples
Representation | Degree | Conjugacy class(es) where the character is zero | Group | Linear representation theory information |
---|---|---|---|---|
standard representation of symmetric group:S3 | 2 | ![]() |
symmetric group:S3 | linear representation theory of symmetric group:S3 |
faithful irreducible representation of dihedral group:D8 | 2 | all the conjugacy classes of elements outside the center. | dihedral group:D8 | linear representation theory of dihedral group:D8 |
faithful irreducible representation of quaternion group | 2 | all the conjugacy classes of elements outside the center. | quaternion group | linear representation theory of quaternion group |
standard representation of symmetric group:S4 | 3 | ![]() |
symmetric group:S4 | linear representation theory of symmetric group:S4 |
Facts used
- Character orthogonality theorem
- Sufficiently large implies splitting: In particular, this exhibits a splitting field of characteristic zero that is a finite cyclotomic extension of the rationals.
- Cauchy-Schwartz inequality
- Characters are algebraic integers
Proof
Given: A finite group ,
the character of an irreducible representation of degree greater than
of the group
.
is the identity element,
is the degree, so
.
To prove: There exists such that
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() |
Fact (1) | Use Fact (1), and move the ![]() | ||
2 | ![]() |
Step (1) | Pull out the term for the identity element. | ||
3 | ![]() |
![]() |
Step (2) | Rearrange Step (2), use ![]() | |
4 | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Fact (2) | Fact (2) guarantees the existence of such a ![]() | ||
5 | ![]() |
Fact (3) | Steps (3), (4) | ||
6 | ![]() |
Step (5) | [SHOW MORE] | ||
7 | ![]() ![]() |
Steps (2), (4) | [SHOW MORE] | ||
8 | ![]() |
Step (7) | [SHOW MORE] | ||
9 | ![]() |
Steps (6), (8) | Step-combination direct | ||
10 | Each ![]() ![]() |
Fact (4) | Step (4) (definition of ![]() |
[SHOW MORE] | |
11 | For some ![]() ![]() |
Steps (9), (10) | [SHOW MORE] | ||
12 | For some ![]() ![]() |
Step (11) | Follows from Step (11) and the observation that ![]() |