Irreducible character of degree greater than one takes value zero on some conjugacy class
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 .
|Representation||Degree||Conjugacy class(es) where the character is zero||Group||Linear representation theory information|
|standard representation of symmetric group:S3||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||-- class of 3-cycles||symmetric group:S4||linear representation theory of symmetric group:S4|
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 .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||Fact (1)||Use Fact (1), and move the to the other side.|
|2||Step (1)||Pull out the term for the identity element.|
|3||, i.e., the representation has degree more than one||Step (2)||Rearrange Step (2), use .|
|4|| Suppose is a finite degree cyclotomic extension of that is a splitting field for . Let be the Galois group of the field extension . Then, acts on the set of irreducible representations, with an automorphism acting by:
||Fact (2)||Fact (2) guarantees the existence of such a .|
|5||Fact (3)||Steps (3), (4)|
|6||Step (5)||[SHOW MORE]|
|7||for every||Steps (2), (4)||[SHOW MORE]|
|8||Step (7)||[SHOW MORE]|
|9||Steps (6), (8)||Step-combination direct|
|10||Each is an integer, so each is a nonnegative integer.||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 .|