Irreducible character of degree greater than one takes value zero on some conjugacy class

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Statement

Suppose G is a finite group and \chi is the character of an irreducible linear representation of G over \mathbb{C}, such that the degree of the representation (and hence, of \chi) is greater than one. Then, there exists an element g \in G (and hence, a Conjugacy class (?)) such that \chi(g) = 0.

Related facts

Examples

Representation Degree Conjugacy class(es) where the character is zero Group Linear representation theory information
standard representation of symmetric group:S3 2 (1,2) -- class of 2-transpositions 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 (1,2,3) -- class of 3-cycles symmetric group:S4 linear representation theory of symmetric group:S4

Facts used

  1. Character orthogonality theorem
  2. Sufficiently large implies splitting: In particular, this exhibits a splitting field of characteristic zero that is a finite cyclotomic extension of the rationals.
  3. Cauchy-Schwartz inequality
  4. Characters are algebraic integers

Proof

Given: A finite group G, \chi the character of an irreducible representation of degree greater than 1 of the group G. e is the identity element, d = \chi(e) is the degree, so d > 1.

To prove: There exists g \in G \setminus \{ e \} such that \chi(g) = 0.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 \sum_{g \in G} |\chi(g)|^2 = |G| Fact (1) Use Fact (1), and move the 1/|G| to the other side.
2 d^2 + \sum_{g \in G \setminus \{ e \}} |\chi(g)|^2 = |G| Step (1) Pull out the term for the identity element.
3 \sum_{g \in G \setminus \{ e \}} |\chi(g)|^2 = |G| - d^2 < |G| - 1 d > 1, i.e., the representation has degree more than one Step (2) Rearrange Step (2), use d > 1.
4 Suppose K is a finite degree cyclotomic extension of \mathbb{Q} that is a splitting field for G. Let H be the Galois group of the field extension K/\mathbb{Q}. Then, H acts on the set of irreducible representations, with an automorphism \sigma acting by:
\varphi \mapsto (g \mapsto \sigma(\varphi(g))
Fact (2) Fact (2) guarantees the existence of such a K.
5 \sum_{g \in G \setminus \{ e \}} \left[\prod_{\sigma \in H} |\sigma(\chi(g))| \right]^2 \le \prod_{\sigma \in H} \left[ \sum_{g \in G \setminus \{ e \}} |\sigma(\chi(g))|^2 \right]^{1/|H|} Fact (3) Steps (3), (4)
6 \sum_{g \in G \setminus \{ e \}} |N_K(\chi(g))|^2 \le \prod_{\sigma \in H} \left[ \sum_{g \in G \setminus \{ e \}} |\sigma(\chi(g))|^2 \right]^{1/|H|} Step (5) [SHOW MORE]
7 \sum_{g \in G \setminus \{ e \}} |\sigma(\chi(g))|^2 = |G| - d^2 < |G| - 1 for every \sigma \in H Steps (2), (4) [SHOW MORE]
8  \prod_{\sigma \in H} \left[ \sum_{g \in G \setminus \{ e \}} |\sigma(\chi(g))|^2 \right]^{1/|H|} < |G| - 1 Step (7) [SHOW MORE]
9 \sum_{g \in G \setminus \{ e \}} |N_K(\chi(g))|^2 < |G| - 1 Steps (6), (8) Step-combination direct
10 Each N_K(\chi(g)) is an integer, so each |N_K(\chi(g))|^2 is a nonnegative integer. Fact (4) Step (4) (definition of K) [SHOW MORE]
11 For some g \in G \setminus \{ e \}, N_K(\chi(g)) = 0 Steps (9), (10) [SHOW MORE]
12 For some g \in G \setminus \{ e \}, \chi(g) = 0 Step (11) Follows from Step (11) and the observation that N_K(x) = 0 \iff x = 0.