Sum of squares of degrees of irreducible representations equals order of group
This fact is related to: linear representation theory
View other facts related to linear representation theoryView terms related to linear representation theory |
Contents
Statement
Suppose is a finite group,
is a splitting field for
, and
are the characters of the irreducible linear representations (up to equivalence) of
over
. Let
be the degree of
. In other words,
are the Degrees of irreducible representations (?) of
. Then:
This fact is instrumental in defining the Plancherel measure on the set of irreducible representations of a finite group, which assigns a measure of to the
irreducible representation.
Note also that the s (up to rearrangement) are the same for all splitting fields -- see degrees of irreducible representations are the same for all splitting fields.
Related facts
Alternative formulations
- Regular representation over splitting field has multiplicity of each irreducible representation equal to its degree
- Group ring over splitting field is direct sum of matrix rings for each irreducible representation
- Peter-Weyl theorem: A generalization to compact groups.
Similar facts
- Number of irreducible representations equals number of conjugacy classes
- Character orthogonality theorem
- Column orthogonality theorem
- Grand orthogonality theorem
See also the many facts about the degrees of irreducible representations under degrees of irreducible representations.
Particular cases
Groups
Group/group type | Order | Degrees of irreducible representations | Squares of degrees of irreducible representations |
---|---|---|---|
trivial group | 1 | 1 | 1 |
abelian group of order ![]() |
![]() |
1 (![]() |
1 (![]() |
symmetric group:S3 | 6 | 1,1,2 | 1,1,4 |
dihedral group:D8 | 8 | 1,1,1,1,2 | 1,1,1,1,4 |
quaternion group | 8 | 1,1,1,1,2 | 1,1,1,1,4 |
dihedral group:D10 | 10 | 1,1,2,2 | 1,1,4,4 |
alternating group:A4 | 12 | 1,1,1,3 | 1,1,1,9 |
direct product of S3 and Z2 | 12 | 1,1,1,1,2,2 | 1,1,1,1,4,4 |
dicyclic group:Dic12 | 12 | 1,1,1,1,2,2 | 1,1,1,1,4,4 |
symmetric group:S4 | 24 | 1,1,2,3,3 | 1,1,4,9,9 |
special linear group:SL(2,3) | 24 | 1,1,1,2,2,2,3 | 1,1,1,4,4,4,9 |
Families
Family of groups | Order | Degrees of irreducible representations | Family-specific explanation for sum of squares |
---|---|---|---|
dihedral group of even degree ![]() |
![]() |
![]() ![]() ![]() ![]() |
algebraic simplification. See also linear representation theory of dihedral groups |
dihedral group of odd degree ![]() |
![]() |
![]() ![]() ![]() ![]() |
algebraic simplification. See also linear representation theory of dihedral groups |
symmetric group of degree ![]() |
![]() |
For each partition ![]() ![]() ![]() ![]() |
Robinson-Schensted correspondence. See also linear representation theory of symmetric groups |
general affine group of degree one over a finite field of size ![]() |
![]() |
![]() ![]() ![]() ![]() |
See linear representation theory of general affine group of degree one over a finite field |
general linear group of degree two over a finite field of size ![]() |
![]() |
1 (![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Algebraic simplification. See linear representation theory of general linear group of degree two over a finite field |
projective general linear group of degree two over a finite field of size ![]() |
![]() |
Case ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() |
Algebraic simplification. See linear representation theory of projective general linear group of degree two over a finite field |
special linear group of degree two over a finite field of size ![]() |
![]() |
Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() |
Algebraic simplification. See linear representation theory of special linear group of degree two over a finite field |
projective special linear group of degree two over a finite field of size ![]() |
![]() ![]() ![]() ![]() |
Case ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() |
Algebraic simplification. See linear representation theory of projective special linear group of degree two over a finite field |
Facts used
- Maschke's averaging lemma, which we use to say that every representation is completely reducible.
- Orthogonal projection formula, which in turn uses character orthogonality theorem. See inner product of functions for the notation.
Proof
Proof in characteristic zero
Note: We can in fact use this proof to also show that there are only finitely many equivalence classes of irreducible representations, though the formulation below does not quite do that.
Given: A finite group with irreducible representations having characters
and degrees
.
To prove:
Proof: We let be the regular representation of
, i.e., the permutation representation obtained by using the regular group action. Let
be the character of
.
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() ![]() |
[SHOW MORE] | |||
2 | The inner product ![]() ![]() ![]() |
Step (1) | [SHOW MORE] | ||
3 | ![]() ![]() |
Facts (1),(2) | ![]() |
Step (2) | [SHOW MORE] |
4 | The value of ![]() ![]() |
Step (3) | [SHOW MORE] | ||
5 | ![]() |
Steps (1), (4) | [SHOW MORE] |
Proof in other characteristics
This follows from the characteristic zero proof, and the fact that degrees of irreducible representations are the same for all splitting fields.