Linear representation theory of dihedral group:D8

We shall use the dihedral group with the following presentation (here, $$e$$ is used to denote the identity element):

$$\langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

Summary information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.

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:

Sign representations with kernels as the maximal normal subgroups
The dihedral group has three normal subgroups of index two: the subgroup $$\langle a \rangle$$, the subgroup $$\langle a^2, x \rangle$$, and the subgroup $$\langle a^2, ax \rangle$$. For each such subgroup, there is an irreducible one-dimensional representation sending elements in that subgroup to $$1$$ and elements outside that subgroup to $$-1$$.

These representations make sense over all fields, but in characteristic two, they become the same as the trivial representation.

Here is the representation with kernel $$\langle a \rangle$$:

Here is the representation with kernel $$\langle a^2, x \rangle$$:

Here is the representation with kernel $$\langle a^2, ax \rangle$$:

Character table
 This character table works over characteristic zero:

The same character table works over any characteristic not equal to 2 where the elements 1,-1,0,2,-2 are interpreted over the field.



Here is the size-degree-weighted character table, i.e., each cell entry is obtained by multiplying the character value by the size of the conjugacy class and then dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers.

Table of matrix entries
This table satisfies the grand orthogonality theorem. Note that unlike the character table, this table is not canonical but rather, for the degree two irreducible representation, depends on the choice of basis.

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 $$\langle a \rangle$$. The two sign representations with kernels $$\langle a^2, x \rangle$$ and $$\langle a^2,ax\rangle$$ are exchanged by an outer automorphism.

Isoclinism and projective representations
Please compare this with projective representation theory of Klein four-group.

Direct sum decomposition
If $$K$$ is any field whose characteristic is not 2, then the group ring $$K[D_8]$$ splits as a direct sum of two-sided ideals corresponding to the irreducible representations:

$$K[D_8] \cong M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_2(K) = K \oplus K \oplus K \oplus K \oplus M_2(K)$$

More generally, if $$R$$ is any commutative unital ring that is uniquely 2-divisible, then we can write:

$$R[D_8] \cong M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_2(R) = R \oplus R \oplus R \oplus R \oplus M_2(R)$$

Note that the ring of integers $$\mathbb{Z}$$ does not satisfy the condition for this direct sum decomposition to hold. Instead we need to use the ring $$\mathbb{Z}[1/2]$$ (In general, we need to use a ring that is uniquely divisible by all primes dividing the order of the group).

Explicit decomposition and idempotents
We can write:

$$R[D_8] = M_1(R)e_1 \oplus M_1(R)e_2 \oplus M_1(R)e_3 \oplus M_1(R)e_4 \oplus M_2(R)e_5$$

where $$e_1,e_2,e_3,e_4,e_5$$ are idempotents. These are called primitive central idempotents.

Note that $$e$$ here denotes the identity of the group, and can also be written as $$1$$ since it gives the identity of the group ring.

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).


 * 1) The trivial representation on the center induces a representation obtained as a sum of the four one-dimensional representations.
 * 2) The sign representation on the center (which comprises $$\pm 1$$) induces the double of the two-dimensional irreducible representation of the dihedral group.
 * 3) The trivial representation on the cyclic subgroup generated by $$a$$ induces a representation on the whole group that is the sum of a trivial representation and the representation with the $$a$$-kernel.
 * 4) A representation on $$\langle a \rangle$$ that sends $$a$$ to $$-1$$ induces a representation of the whole group that is the sum of the sign representations for the other two kernels.
 * 5) A representation on $$\langle a \rangle$$ that sends $$a$$ to $$i$$ (now viewed as a complex number) induces the two-dimensional irreducible representation.

Verification of Artin's induction theorem
Artin's induction theorem states that the characters induced from characters on cyclic subgroups span the space of class functions. Points (2) and (5) cover the case of the two-dimensional irreducible representation.

Degrees of irreducible representations
These can be computed using the CharacterDegrees function:

gap> CharacterDegrees(DihedralGroup(8)); [ [ 1, 4 ], [ 2, 1 ] ]

Character table
The characters of the irreducible representations can be computed using Irr and CharacterTable:

gap> Irr(CharacterTable(DihedralGroup(8))); [ Character( CharacterTable(  ),   [ 1, 1, 1, 1, 1 ] ), Character( CharacterTable(  ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable(  ),   [ 1, 1, -1, 1, -1 ] ), Character( CharacterTable(  ), [ 1, -1, -1, 1, 1 ] ), Character( CharacterTable(  ),   [ 2, 0, 0, -2, 0 ] ) ]

The character table can be displayed more nicely using the Display function:

gap> Display(CharacterTable(DihedralGroup(8))); CT1

2 3  2  2  3  2

1a 2a 4a 2b 2c

X.1    1  1  1  1  1 X.2    1 -1  1  1 -1 X.3    1  1 -1  1 -1 X.4    1 -1 -1  1  1 X.5    2. . -2.

Irreducible representations
The irreducible representations can be accessed using GAP's IrreducibleRepresentations function:

gap> IrreducibleRepresentations(DihedralGroup(8)); [ Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ] ]