Degrees of irreducible representations

This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

Definition

Over a field

The degrees of irreducible representations for a group over a field associate to it the multiset giving, for each irreducible linear representation (considered up to equivalence of linear representations, so only one representation is considered per equivalence class) of the group, the degree of that representation, i.e., the dimension of the vector space on which the action is happening. For an irreducible representation $\phi :G\to GL(n,K)$ over a field $K$, the degree is $n$.

Typical context: finite group and splitting field

The term degrees of the irreducible representations is typically used for a finite group over a splitting field for the group.

For a finite group, any two splitting fields of the same characteristic give rise to the same bunch of degrees of irreducible representations. Thus, for a finite group, we can talk of the degrees of irreducible representations in a particular characteristic (as long as the characteristic is either zero or a prime not dividing the order of the group) and this is understood to mean the degrees of irreducible representations in a splitting field of that characteristic, such as an algebraically closed field.

If the group is a finite group and $p$ is a prime number not dividing the order of the group, then the degrees of irreducible representations in characteristic $p$ are the same as the degrees of irreducible representations in characteristic zero. Further information: degrees of irreducible representations are the same for all splitting fields

Default context: finite group and characteristic zero

For a finite group, if no other information is specified, we interpret the degrees of irreducible representations as all being in characteristic zero, e.g., over the field of complex numbers. As mentioned above, these are the same as the degrees of irreducible representations in any characteristic not dividing the order of the group.

Related notions

• lcm of degrees of irreducible representations is the least common multiple of the degrees of irreducible representations.
• Maximum degree of irreducible representation is the maximum of the degrees of irreducible representations.

Facts

Unless otherwise stated, all results here are over splitting fields. In particular, they hold for algebraically closed fields whose characteristic does not divide the order of the group, such as $\mathbb{C}$ or $\overline{\mathbb{Q}}$. Note that degrees of irreducible representations are the same for all splitting fields.

Counting and arithmetic results

• Number of irreducible representations equals number of conjugacy classes
• Number of one-dimensional representations equals order of abelianization
• Sum of squares of degrees of irreducible representations equals order of group
• Sum of squares of degrees of irreducible representations whose restriction to the center is a given character equals order of inner automorphism group

Relationship with conjugacy class size statistics

For most small orders of groups, knowing the degrees of irreducible representations allows us to compute the conjugacy class size statistics and vice versa, simply on the strength of the counting and arithmetic results on the degrees and the conjugacy class sizes. However, this is not universally the case:

• Degrees of irreducible representations need not determine conjugacy class size statistics
• Conjugacy class size statistics need not determine degrees of irreducible representations
• Maximum degree of irreducible representation does not give bound on maximum conjugacy class size
• Maximum conjugacy class size does not give bound on maximum degree of irreducible representation

Relation with other other aspects of the group structure

• Nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order (and vice versa too)
• Degrees of irreducible representations need not determine nilpotency class (counterexamples of order $p^5$, see linear representation theory of groups of order 32 and linear representation theory of groups of prime-fifth order)

Divisibility results

All results here are for degrees of irreducible representations over splitting fields. The proofs given on the pages may work only for splitting fields of characteristic zero, though modified versions can be used for other splitting fields (or alternatively, we can combine with the fact that degrees of irreducible representations are the same for all splitting fields):

Statement	What divides ...	divides what
degree of irreducible representation divides group order	degree of irreducible representation	order of the group
degree of irreducible representation divides order of inner automorphism group	degree of irreducible representation	index of center, or equivalently, order of inner automorphism group
degree of irreducible representation divides index of abelian normal subgroup	degree of irreducible representation	index of an abelian normal subgroup; in particular, of a subgroup maximal among abelian normal subgroups
Schur index divides degree of irreducible representation	Schur index of irreducible representation	degree of irreducible representation

Numerical bounds

• Order of inner automorphism group bounds square of degree of irreducible representation
• Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
• Degree of irreducible representation is bounded by index of abelian subgroup

Divisibility non-results

Statement	What need not divide ...	need not divide what
degree of irreducible representation need not divide exponent	degree of irreducible representation	exponent of the group
degree of irreducible representation need not divide order of derived subgroup	degree of irreducible representation	order of derived subgroup
square of degree of irreducible representation need not divide order	square of degree of irreducible representation	order of the group

Numerical non-bounds

Statement	What is not bounded ...	not bounded by what
degree of irreducible representation may be greater than exponent	degree of irreducible representation	exponent of the group
degree of irreducible representation may be greater than order of derived subgroup	degree of irreducible representation	order of derived subgroup

Coverage of prime factors

• Degrees of irreducible representations need not cover all prime factors
• Ito-Michler theorem: A partial opposite result which states that if a particular prime does not divide any of the degrees of irreducible representations, then the corresponding Sylow subgroup must be both abelian and normal. The converse is also true.

Some results that hold for fields that are not splitting fields

• Degree of irreducible representation of nontrivial finite group is strictly less than order of group
• Maximum degree of irreducible real representation is at most twice maximum degree of irreducible complex representation
• Degree of irreducible representation over field of characteristic coprime to order divides product of order and Euler phi-function of exponent

Particular cases

Particular groups

Group	Order	Degrees of irreducible representations over $\mathbb{C}$ or $\overline{\mathbb{Q}}$	More information
trivial group	1	1
cyclic group:Z2	2	1,1	linear representation theory of cyclic group:Z2, linear representation theory of cyclic groups
cyclic group:Z3	3	1,1,1	linear representation theory of cyclic group:Z3, linear representation theory of cyclic groups
cyclic group:Z4	4	1,1,1,1	linear representation theory of cyclic group:Z4, linear representation theory of cyclic groups
Klein four-group	4	1,1,1,1	linear representation theory of Klein four-group, linear representation theory of elementary abelian groups
cyclic group:Z5	5	1,1,1,1,1	linear representation theory of cyclic group:Z5, linear representation theory of cyclic groups
symmetric group:S3	6	1,1,2	linear representation theory of symmetric group:S3, linear representation theory of symmetric groups
cyclic group:Z6	6	1,1,1,1,1,1	linear representation theory of cyclic group:Z6, linear representation theory of cyclic groups
cyclic group:Z7	7	1,1,1,1,1,1,1	linear representation theory of cyclic group:Z7, linear representation theory of cyclic groups
cyclic group:Z8	8	1,1,1,1,1,1,1,1	linear representation theory of cyclic group:Z8, linear representation theory of cyclic groups
direct product of Z4 and Z2	8	1,1,1,1,1,1,1,1	linear representation theory of direct product of Z4 and Z2
dihedral group:D8	8	1,1,1,1,2	linear representation theory of dihedral group:D8, linear representation theory of dihedral groups
quaternion group	8	1,1,1,1,2	linear representation theory of quaternion group, linear representation theory of dicyclic groups
elementary abelian group:E8	8	1,1,1,1,1,1,1,1	linear representation theory of elementary abelian group:E8, linear representation theory of elementary abelian groups
cyclic group:Z9	9	1,1,1,1,1,1,1,1,1	linear representation theory of cyclic groups
elementary abelian group:E9	9	1,1,1,1,1,1,1,1,1,1	linear representation theory of elementary abelian groups
dihedral group:D10	10	1,1,2,2	linear representation theory of dihedral groups
dihedral group:D12 (also direct product of S3 and Z2)	12	1,1,1,1,2,2	linear representation theory of dihedral groups
dicyclic group:Dic12	12	1,1,1,1,2,2	linear representation theory of dicyclic groups
alternating group:A4	12	1,1,1,3	linear representation theory of alternating group:A4
dihedral group:D14	14	1,1,2,2,2	linear representation theory of dihedral groups
symmetric group:S4	24	1,1,2,3,3	linear representation theory of symmetric group:S4, linear representation theory of symmetric groups
special linear group:SL(2,3)	24	1,1,1,2,2,2,3	linear representation theory of special linear group:SL(2,3), linear representation theory of special linear groups of degree two

Group families

For various group families, the degrees of irreducible representations can be described in terms of parameters for members of the families. The descriptions are sometimes quite complicated, so we simply provide links:

Family	Information on linear representation theory, section on degrees of irreducible representations
finite abelian group	In this case, all degrees are 1, and there are as many as the order of the group. Linear representation theory of finite abelian groups
symmetric group	Linear representation theory of symmetric groups#Degrees of irreducible representations
alternating group	Linear representation theory of alternating groups#Degrees of irreducible representations
general linear group of degree two over a finite field	Linear representation theory of general linear group of degree two over a finite field#Degrees of irreducible representations
special linear group of degree two over a finite field	Linear representation theory of special linear group of degree two over a finite field#Degrees of irreducible representations
projective general linear group of degree two over a finite field	Linear representation theory of projective general linear group of degree two over a finite field#Degrees of irreducible representations
projective special linear group of degree two over a finite field	Linear representation theory of projective special linear group of degree two over a finite field#Degrees of irreducible representations
general affine group of degree one over a finite field	Linear representation theory of general affine group of degree one over a finite field#Degrees of irreducible representations

For a complete list, see Category:Linear representation theory of group families.

Grouping by order

Given the order of a group, there is a finite, usually small, collection of possibilities for the list of degrees of irreducible representations. Below are links to some small orders and the information on degrees. We omit very small orders, some of which are already covered in the table above.

Order	Number of possible lists of degrees of irreducible representations	Information on degrees of irreducible representations
8	2	Linear representation theory of groups of order 8#Degrees of irreducible representations
12	3	Linear representation theory of groups of order 12#Degrees of irreducible representations
16	3	Linear representation theory of groups of order 16#Degrees of irreducible representations
18	?	Linear representation theory of groups of order 18#Degrees of irreducible representations
20	?	Linear representation theory of groups of order 20#Degrees of irreducible representations
24	5	Linear representation theory of groups of order 24#Degrees of irreducible representations
48	13	Linear representation theory of groups of order 48#Degrees of irreducible representations

GAP implementation

To find the degrees of irreducible representations for a finite group over the complex numbers (and hence over any splitting field of characteristic zero), GAP has the CharacterDegrees function. This returns a list of pairs, where the first member of each pair is a degree of irreducible representation and the second member is the number of equivalence classes of irreducible representations with that degree. Here's an example:

gap> CharacterDegrees(SL(2,5));
[ [ 1, 1 ], [ 2, 2 ], [ 3, 2 ], [ 4, 2 ], [ 5, 1 ], [ 6, 1 ] ]

In this example, the input group is special linear group:SL(2,5), constructed using GAP's SL function, and the output indicates that there is 1 irreducible representation of degree 1, 2 of degree 2, 2 of degree 3, 2 of degree 4, 1 of degree 5, and 1 of degree 6. The list of degrees is thus 1,2,2,3,3,4,4,5,6.