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

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.

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
• 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