Linear representation theory of dihedral group:D16

Summary
We shall use the dihedral group of order 16 with the following presentation:

$$\langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

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.

Below are representations that are irreducible over some non-splitting field but split further over a splitting field.

Trivial representation
The trivial representation or principal representation (whose character is called the trivial character or principal character) sends all elements of the group to the $$1 \times 1$$ matrix $$(1)$$:

Sign representation with kernel $$\langle a \rangle$$
This representation is a one-dimensional representation sending everything in the cyclic subgroup $$\langle a \rangle$$ (see Z8 in D16) to $$(1)$$ and everything outside it to $$(-1)$$.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Sign representations with kernels $$\langle a^2, x \rangle$$ and $$\langle a^2, ax \rangle$$
These are sign representations with kernels one of the D8 in D16 subgroups. There are two such representations, one for each subgroup.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Sign representation with kernel $$\langle a^2, x \rangle$$:

Sign representation with kernel $$\langle a^2, ax \rangle$$

Two-dimensional irreducible unfaithful representation
This representation has kernel equal to $$\langle a^4 \rangle$$ -- center of dihedral group:D16. It descends to a faithful irreducible two-dimensional representation of the quotient group, which is isomorphic to dihedral group:D8.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Character table
 

Here are the size-degree-weighted characters (obtained by multiplying the character value by the size of the conjugacy class and dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers.

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

gap> CharacterDegrees(DihedralGroup(16)); [ [ 1, 4 ], [ 2, 3 ] ]

Character table
The character table can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(DihedralGroup(16))); [ Character( CharacterTable(  ),   [ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable(  ), [ 1, -1, 1, 1, 1, -1, 1 ] ), Character( CharacterTable(  ),   [ 1, 1, -1, 1, 1, -1, -1 ] ), Character( CharacterTable(  ),   [ 1, -1, -1, 1, 1, 1, -1 ] ), Character( CharacterTable(  ),   [ 2, 0, 0, -2, 2, 0, 0 ] ), Character( CharacterTable(  ), [ 2, 0, E(8)-E(8)^3, 0, -2, 0, -E(8)+E(8)^3 ] ), Character( CharacterTable(  ),   [ 2, 0, -E(8)+E(8)^3, 0, -2, 0, E(8)-E(8)^3 ] ) ]

It can be displayed in nicer form using the Display function:

gap> Display(CharacterTable(DihedralGroup(16))); CT2

2 4  2  3  3  4  2  3

1a 2a 8a 4a 2b 2c 8b

X.1    1  1  1  1  1  1  1 X.2    1 -1  1  1  1 -1  1 X.3    1  1 -1  1  1 -1 -1 X.4    1 -1 -1  1  1  1 -1 X.5    2. . -2 2  .  . X.6     2. A. -2 . -A X.7    2. -A. -2 .  A

A = E(8)-E(8)^3 = ER(2) = r2

Irreducible representations
The irreducible representations can be computed using the IrreducibleRepresentations function:

gap> IrreducibleRepresentations(DihedralGroup(16)); [ Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(8), 0 ], [ 0, -E(8)^3 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -E(8), 0 ], [ 0, E(8)^3 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ] ]