Linear representation theory of dihedral group:D8
This article gives specific information, namely, linear representation theory, about a particular group, namely: dihedral group:D8.
View linear representation theory of particular groups | View other specific information about dihedral group:D8
We shall use the dihedral group with the following presentation:
.
Summary
Item | Value |
---|---|
Degrees of irreducible representations over a splitting field | 1,1,1,1,2 maximum: 2, lcm: 2, number: 5, sum of squares: 8 |
Schur index values of irreducible representations | 1,1,1,1,1 |
Smallest ring of realization for all irreducible representations (characteristic zero) | |
Smallest field of realization for all irreducible representations (characteristic zero) | (hence, it is a rational representation group) |
Condition for being a splitting field for this group | Any field of characteristic not two is a splitting field. |
Smallest size splitting field | field:F3, i.e., the field with three elements. |
Orbits over a splitting field under action of automorphism group | orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), and 1 (degree 2 representation) number: 4 |
Other groups with the same character table | quaternion group (see linear representation theory of quaternion group) |
Family contexts
Family name | Parameter values | General discussion of linear representation theory of family |
---|---|---|
dihedral group | degree , order | linear representation theory of dihedral groups |
prime-cube order group:U(3,p) | Case , somewhat different from the odd primes | linear representation theory of prime-cube order group:U(3,p) |
COMPARE AND CONTRAST: View linear representation theory of groups of order 8 to compare and contrast the linear representation theory with other groups of order 8.
Representations
Summary information
Below is summary information on irreducible representations. Note that a particular representation may make sense, and be irreducible, only for certain kinds of fields -- see the "Values not allowed for field characteristic" and "Criterion for field" columns to see the condition the field must satisfy for the representation to be irreducible there.
Name of representation type | Number of representations of this type | Values not allowed for field characteristic | Criterion for field | What happens over a splitting field? | Kernel | Degree | Schur index | What happens by reducing the -representation over bad characteristics? |
---|---|---|---|---|---|---|---|---|
trivial | 1 | -- | any | remains the same | whole group | 1 | 1 | -- |
sign representation with kernel cyclic of order four | 1 | -- | any | remains the same | cyclic maximal subgroup of dihedral group:D8 | 1 | 1 | There are no bad characteristics, but it is noteworthy that in characteristic two, this representation is the same as the trivial representation. |
sign representation with kernel a Klein four-subgroup | 2 | -- | any | remains the same | Klein four-subgroups of dihedral group:D8 | 1 | 1 | There are no bad characteristics, but it is noteworthy that in characteristic two, this representation is the same as the trivial representation. |
two-dimensional irreducible | 1 | 2 | any | remains the same | trivial subgroup, i.e., it is a faithful linear representation | 2 | 1 | The exact form of the new representation depends on the choice of matrices before we go mod 2, but the kernel becomes one of the Klein four-subgroups of dihedral group:D8, and we thus get a representation of cyclic group:Z2 in characteristic two that sends the non-identity element to . This has an invariant one-dimensional subspace and is not irreducible. |
Trivial representation
The trivial or principal representation is a one-dimensional representation sending every element of the group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements:
Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
1 | ||||
1 | ||||
1 | ||||
1 | ||||
1 | ||||
1 | ||||
1 | ||||
1 |
Sign representations with kernels as the maximal normal subgroups
The dihedral group has three normal subgroups of index two: the subgroup , the subgroup , and the subgroup . For each such subgroup, there is an irreducible one-dimensional representation sending elements in that subgroup to and elements outside that subgroup to .
Here is the representation with kernel :
Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
1 | ||||
1 | ||||
1 | ||||
1 | ||||
-1 | ||||
-1 | ||||
-1 | ||||
-1 |
Here is the representation with kernel :
Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
1 | ||||
-1 | ||||
1 | ||||
-1 | ||||
1 | ||||
-1 | ||||
1 | ||||
-1 |
Here is the representation with kernel :
Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
1 | ||||
-1 | ||||
1 | ||||
-1 | ||||
-1 | ||||
1 | ||||
-1 | ||||
1 |
Two-dimensional irreducible representation
The dihedral group of order eight has a two-dimensional irreducible representation, where the element acts as a rotation (by an angle of , and the element acts as a reflection about the first axis. The matrices are:
There are many other choices of two-dimensional representation, but these are all equivalent as linear representations.
Below is a description of the matrices based on the above choice:
Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|
2 | ||||
0 | ||||
-2 | ||||
0 | ||||
0 | ||||
0 | ||||
0 | ||||
0 |
Over other fields
These representations generalize to any field of characteristic not equal to .
Character table
This character table works over characteristic zero:
Rep/Conj class | |||||
---|---|---|---|---|---|
Trivial representation | 1 | 1 | 1 | 1 | 1 |
-kernel | 1 | 1 | 1 | -1 | -1 |
-kernel | 1 | 1 | -1 | 1 | -1 |
-kernel | 1 | 1 | -1 | -1 | 1 |
2-dimensional | 2 | -2 | 0 | 0 | 0 |
Realizability information
Smallest ring of realization
Representation | Smallest ring over which it is realized | Smallest set of elements in matrix entries |
---|---|---|
trivial representation | -- ring of integers | |
-kernel | -- ring of integers | |
-kernel | -- ring of integers | |
-kernel | -- ring of integers | |
two-dimensional irreducible | -- ring of integers |
Smallest ring of realization as orthogonal matrices
Representation | Smallest ring over which it is realized | Smallest set of elements in matrix entries |
---|---|---|
trivial representation | -- ring of integers | |
-kernel | -- ring of integers | |
-kernel | -- ring of integers | |
-kernel | -- ring of integers | |
two-dimensional irreducible | -- ring of integers |
Orthogonality relations and numerical checks
- Any field of characteristic not equal to two is a splitting field for the dihedral group.
- The number of irreducible representations over such a field is , which is equal to the number of conjugacy classes.
- The number of one-dimensional representations equals , which is the order of the abelianization.
- The sum of squares of degrees is , equal to the order of the group. This confirms the fact that sum of squares of degrees of irreducible representations equals order of group.
- The degrees of irreducible representations all divide , which is the smallest possible index of an abelian normal subgroup. This confirms the fact that degree of irreducible representation divides index of abelian normal subgroup.
- The order of the inner automorphism group (which is ) bounds the square of the degree of every irreducible representation. This confirms the fact that order of inner automorphism group bounds square of degree of irreducible representation.
- The character table satisfies the orthogonality relations: in particular, the row orthogonality theorem and the column orthogonality theorem.
Action of automorphism group
The automorphism group of the dihedral group preserves the trivial representation, the two-dimensional representation, and the sign representation whose kernel is the cyclic group . The two sign representations with kernels and are exchanged by an outer automorphism.
Relation with representations of subgroups
Induced representations from subgroups
Since the dihedral group is a finite nilpotent group, it is in particular a finite supersolvable group, and hence, it is a monomial-representation group: every irreducible representation can be realized as a monomial representation, i.e., every irreducible representation is induced from a degree one representation of a subgroup. (Point (5) below explains how the two-dimensional irreducible representation is induced).
- The trivial representation on the center induces a representation obtained as a sum of the four one-dimensional representations.
- The sign representation on the center (which comprises ) induces the double of the two-dimensional irreducible representation of the dihedral group.
- The trivial representation on the cyclic subgroup generated by induces a representation on the whole group that is the sum of a trivial representation and the representation with the -kernel.
- A representation on that sends to induces a representation of the whole group that is the sum of the sign representations for the other two kernels.
- A representation on that sends to (now viewed as a complex number) induces the two-dimensional irreducible representation.