Sum of squares of degrees of irreducible representations equals order of group

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This fact is related to: linear representation theory
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Statement

Suppose G is a finite group, K is a splitting field for G, and \chi_1,\chi_2, \dots, \chi_r are the characters of the irreducible linear representations (up to equivalence) of G over K. Let d_i be the degree of \chi_i. In other words, d_i are the Degrees of irreducible representations (?) of G. Then:

\! d_1^2 + d_2^2 + \dots + d_r^2 = |G|

This fact is instrumental in defining the Plancherel measure on the set of irreducible representations of a finite group, which assigns a measure of d_i^2/|G| to the i^{th} irreducible representation.

Note also that the d_is (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

Similar facts

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 n n 1 (n times) 1 (n times)
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 n 2n 1 (4 times), 2 ((n-2)/2 times) algebraic simplification. See also linear representation theory of dihedral groups
dihedral group of odd degree n 2n 1 (2 times), 2 ((n-1)/2 times) algebraic simplification. See also linear representation theory of dihedral groups
symmetric group of degree n n! For each partition \lambda of n, an irreducible representation of degree d_\lambda, which is the number of Young tableaux of shape \lambda Robinson-Schensted correspondence. See also linear representation theory of symmetric groups
general affine group of degree one over a finite field of size q q(q - 1) 1 (q - 1 times), q - 1 (1 time) 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 q q(q+1)(q-1)^2 1 (q - 1 times), q (q - 1 times), q + 1 ((q - 1)/(q - 2)/2 times), q - 1 (q(q - 1)/2 times) 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 q q(q+1)(q-1) = q^3 - q Case q odd: 1 (2 times), q - 1 ((q - 1)/2 times), q (2 times), q + 1 ((q - 3)/2 times)
Case q even: 1 (1 time), q - 1 (q/2 times), q (1 time), q + 1 ((q - 2)/2 times)
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 q q(q+1)(q-1) = q^3 - q Case q odd: 1 (1 time), (q - 1)/2 (2 times), (q + 1)/2 (2 times), q - 1 ((q - 1)/2 times), q (1 time), q + 1 ((q - 3)/2 times)
Case q even: 1 (1 time), q - 1 (q/2 times), q (1 time), q + 1 ((q - 2)/2 times)
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 q q(q+1)(q-1)/2 = (q^3 - q)/2 if q odd; q^3 - q if q even Case q odd: 1 (1 time), (q + 1)/2 (2 times), q - 1 ((q - 1)/4 or (q - 3)/4 times depending on congruence class of q mod 4), q (1 time), q + 1 ((q - 5)/4 or (q - 3)/4 times depending on congruence class of q mod 4)
Case q even: 1 (1 time), q - 1 (q/2 times), q (1 time), q + 1 ((q - 2)/2 times)
Algebraic simplification. See linear representation theory of projective special linear group of degree two over a finite field

Facts used

  1. Maschke's averaging lemma, which we use to say that every representation is completely reducible.
  2. 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 G with irreducible representations having characters \chi_1, \chi_2,\dots, \chi_r and degrees d_1, d_2, \dots, d_r.

To prove: d_1^2 + d_2^2 + \dots + d_r^2 = |G|

Proof: We let \rho be the regular representation of G, i.e., the permutation representation obtained by using the regular group action. Let \alpha be the character of \rho.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 \alpha takes the value |G| at the identity element of G, and zero elsewhere. [SHOW MORE]
2 The inner product \langle \alpha, \chi_i \rangle equals d_i for all i. Step (1) [SHOW MORE]
3 \alpha is the sum \sum_{i=1}^r d_i\chi_i Facts (1),(2) \chi_i are characters of (all the) irreducible representations. Step (2) [SHOW MORE]
4 The value of \alpha at the identity element is \sum_{i=1}^r d_i^2. Step (3) [SHOW MORE]
5 \sum_{i=1}^r d_i^2 = |G| 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.