Degrees of irreducible representations

This term is related to: linear representation theory
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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 $\varphi:G \to GL(n,K)$ 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.

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.

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:

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

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

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:

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.