Degrees of irreducible representations

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This term is related to: linear representation theory
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Definition

The degrees of irreducible representations for a group associate to it the multiset giving, for each irreducible representation of the group, the degree of that representation. (This term is typically used for the degrees of the irreducible representations over a splitting field, usually, the field $\mathbb{C}$ of complex numbers).

Facts

For finite groups (links to proofs will be given soon):

The degree of each irreducible representation of a group divides the order of the group. For full proof, refer: degree of irreducible representation divides group order
The degree of each irreducible representation of a group divides the order of the inner automorphism group, or equivalently, the index of the center. For full proof, refer: degree of irreducible representation divides index of center
The degree of each irreducible representation of a group divides the index of any Abelian normal subgroup. For full proof, refer: degree of irreducible representation divides index of Abelian normal subgroup
The sum of the squares of degrees of irreducible representations is the order of the group. For full proof, refer: Sum of squares of degrees of irreducible representations equals order of group
The square of the degree of any irreducible representation is bounded from above by the order of the inner automorphism group. For full proof, refer: Order of inner automorphism group bounds square of degree of irreducible representation

Particular cases

Group	Order	Degrees of irreducible representations	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	$4$	$1,1,1,1$
Klein four-group	$4$	$1,1,1,1$
cyclic group:Z5	$5$	$1,1,1,1,1$
symmetric group:S3	$6$	$1,1,2$
cyclic group:Z6	$6$	$1,1,1,1,1,1$
cyclic group:Z7	$7$	$1,1,1,1,1,1,1$
cyclic group:Z8	$8$	$1,1,1,1,1,1,1,1$
direct product of Z4 and Z2	$8$	$1,1,1,1,1,1,1,1$
dihedral group:D8	$8$	$1,1,1,1,2$	linear representation theory of dihedral group:D8
quaternion group	$8$	$1,1,1,1,2$	linear representation theory of quaternion group
elementary abelian group:E8	$8$	$1,1,1,1,1,1,1,1$
cyclic group:Z9	$9$	$1,1,1,1,1,1,1,1,1$
elementary abelian group:E9	$9$	$1,1,1,1,1,1,1,1,1,1$
dihedral group:D10	$10$	$1,1,2,2$	linear representation theory of dihedral groups
dihedral group:D12	$12$	$1,1,1,1,2,2$
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)	$24$	$1,1,1,2,2,2,3$