Degrees of irreducible representations: Difference between revisions
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* The degree of each irreducible representation of a group divides the [[order of a group|order]] of the group. {{proofat|[[degree of irreducible representation divides group order]]}} | * The degree of each irreducible representation of a group divides the [[order of a group|order]] of the group. {{proofat|[[degree of irreducible representation divides group order]]}} | ||
* The degree of each irreducible representation of a group divides the [[order of a group|order]] of the [[inner automorphism group]], or equivalently, the [[index of a subgroup|index]] of the [[center]]. | * The degree of each irreducible representation of a group divides the [[order of a group|order]] of the [[inner automorphism group]], or equivalently, the [[index of a subgroup|index]] of the [[center]]. {{proofat|[[degree of irreducible representation divides index of center]]}} | ||
* The degree of each irreducible representation of a group divides the [[index of a subgroup|index]] of any [[Abelian group|Abelian]] [[normal subgroup]]. {{proofat|[[degree of irreducible representation divides index of Abelian normal subgroup]]}} | * The degree of each irreducible representation of a group divides the [[index of a subgroup|index]] of any [[Abelian group|Abelian]] [[normal subgroup]]. {{proofat|[[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 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 a group|order]] of the [[inner automorphism group]]. {{proofat|[[Order of inner automorphism group bounds square of degree of irreducible representation]]}} | * The square of the degree of any irreducible representation is bounded from above by the [[order of a group|order]] of the [[inner automorphism group]]. {{proofat|[[Order of inner automorphism group bounds square of degree of irreducible representation]]}} | ||
Revision as of 20:51, 10 December 2007
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