Groupprops, The Group Properties Wiki (pre-alpha)
Take a short survey about Math Resources on the Internet.
Degrees of irreducible representations
From Groupprops
This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory
Contents |
Definition
The degrees of irreducible representations for a group over a field 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 
of complex numbers).
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 or 
.
Counting and arithmetic results
- Number of irreducible representations equals number of conjugacy classes
- Sum of squares of degrees of irreducible representations equals group order
Divisibility results
All results here are for degrees of irreducible representations over 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
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 exponent | 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 need not be less than exponent | degree of irreducible representation | exponent of the group |
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
