Linear representation theory of dihedral group:D16

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Summary

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

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

Item Value
degrees of irreducible representations over a splitting field (such as $\mathbb{C}$ or $\overline{\mathbb{Q}}$) 1,1,1,1,2,2,2
maximum: 2, lcm: 2, number: 7, sum of squares: 16
Schur index values of irreducible representations over a splitting field 1,1,1,1,1,1,1
smallest ring of realization (characteristic zero) $\mathbb{Z}[\sqrt{2}] = \mathbb{Z}[t]/(t^2 - 2)$
smallest splitting field, i.e., smallest field of realization (characteristic zero) $\mathbb{Q}(\sqrt{2}) = \mathbb{Q}[t]/(t^2 - 2)$
condition for a field to be a splitting field The characteristic should not be equal to 2, and the polynomial $t^2 - 2$ must split.
For a finite field of size $q$, this is equivalent to saying that $q \equiv \pm 1 \pmod 8$.
smallest splitting field in characteristic $p \ne 2$ Case $p \equiv \pm 1 \pmod 8$: prime field $\mathbb{F}_p$
Case $p \equiv \pm 3 \pmod 8$: $\mathbb{F}_{p^2}$, a quadratic extension of $\mathbb{F}_p$.
smallest size splitting field Field:F7.
degrees of irreducible representations over the rational numbers 1,1,1,1,2,4

Family contexts

Family name Parameter values General discussion of linear representation theory of family
dihedral group degree $n = 8$, order $2n = 16$ linear representation theory of dihedral groups
COMPARE AND CONTRAST: View linear representation theory of groups of order 16 to compare and contrast the linear representation theory with other groups of order 16.

Representations

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.

Name of representation type Number of representations of this type Degree Schur index Criterion for field Kernel Quotient by kernel (on which it descends to a faithful representation) Characteristic 2
trivial 1 1 1 any whole group trivial group works
sign representation with kernel $\langle a \rangle$ 1 1 1 any Z8 in D16: $\langle a \rangle$ cyclic group:Z2 works, same as trivial
sign representation with kernel a maximal dihedral subgroup 2 1 1 any D8 in D16: $\langle a^2, x \rangle$ or $\langle a^2, ax \rangle$ cyclic group:Z2 works, same as trivial
two-dimensional irreducible, not faithful 1 2 1 any center of dihedral group:D16: $\langle a^4 \rangle$ dihedral group:D8 indecomposable but not irreducible
two-dimensional faithful irreducible 2 2 1 The polynomial $t^2 - 2$ must split, i.e., $2$ must have a square root trivial subgroup, i.e., it is a faithful linear representation dihedral group:D16 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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

Name of representation type Number of representations of this type Degree Criterion for field What happens over a splitting field? Kernel Quotient by kernel (on which it descends to a faithful representation)
four-dimensional faithful irreducible 1 4 The polynomial $t^2 - 2$ must not split, i.e., $2$ must not have a square root splits into the two two-dimensional faithful irreducibles. trivial subgroup, i.e., it is a faithful linear representation dihedral group:D16

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 $(1)$:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
$e$ $(1)$ $t - 1$ $t - 1$ 1
$a$ $(1)$ $t - 1$ $t - 1$ 1
$a^2$ $(1)$ $t - 1$ $t - 1$ 1
$a^3$ $(1)$ $t - 1$ $t - 1$ 1
$a^4$ $(1)$ $t - 1$ $t - 1$ 1
$a^5$ $(1)$ $t - 1$ $t - 1$ 1
$a^6$ $(1)$ $t - 1$ $t - 1$ 1
$a^7$ $(1)$ $t - 1$ $t - 1$ 1
$x$ $(1)$ $t - 1$ $t - 1$ 1
$ax$ $(1)$ $t - 1$ $t - 1$ 1
$a^2x$ $(1)$ $t - 1$ $t - 1$ 1
$a^3x$ $(1)$ $t - 1$ $t - 1$ 1
$a^4x$ $(1)$ $t - 1$ $t - 1$ 1
$a^5x$ $(1)$ $t - 1$ $t - 1$ 1
$a^6x$ $(1)$ $t - 1$ $t - 1$ 1
$a^7x$ $(1)$ $t - 1$ $t - 1$ 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:

Elements Matrix Characteristic polynomial Minimal polynomial Trace, character value
$\{ e,a,a^2,a^3,a^4,a^5,a^6,a^7 \}$ $(1)$ $t - 1$ $t - 1$ 1
$\{ x,ax,a^2x,a^3x,a^4x,a^5x,a^6x,a^7x \}$ $(-1)$ $t + 1$ $t + 1$ -1

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 $\langle a^2, x \rangle$:

Elements Matrix Characteristic polynomial Minimal polynomial Trace, character value
$\{ e,a^2,a^4,a^6,x,a^2x,a^4x,a^6x \}$ $(1)$ $t - 1$ $t - 1$ 1
$\{ a,a^3,a^5,a^7,ax,a^3x,a^5x,a^7x \}$ $(-1)$ $t + 1$ $t + 1$ -1

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

Elements Matrix Characteristic polynomial Minimal polynomial Trace, character value
$\{ e,a^2,a^4,a^6,ax,a^3x,a^5x,a^7x \}$ $(1)$ $t - 1$ $t - 1$ 1
$\{ a,a^3,a^5,a^7,x,a^2x,a^4x,a^6x \}$ $(-1)$ $t + 1$ $t + 1$ -1

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:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
$\{ e, a^4 \}$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $(t - 1)^2 = t^2 - 2t + 1$ $t - 1$ 2
$\{ a, a^5 \}$ $\begin{pmatrix}0 & -1 \\ 1 & 0 \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0
$\{ a^2, a^6 \}$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $(t + 1)^2 = t^2 + 2t + 1$ $t + 1$ -2
$\{ a^3, a^7 \}$ $\begin{pmatrix}0 & 1 \\ -1 & 0 \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0
$\{ x, a^4x \}$ $\begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0
$\{ ax, a^5x \}$ $\begin{pmatrix}0 & 1 \\ 1 & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0
$\{ a^2x, a^6x \}$ $\begin{pmatrix}-1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0
$\{ a^3x, a^7x \}$ $\begin{pmatrix}0 & -1 \\ -1 & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0

Two-dimensional faithful irreducible representations

Further information: faithful irreducible representation of dihedral group:D16

There are two such representations, and they are related by the group automorphism $a \mapsto a^3, x \mapsto x$, and also by the Galois automorphism $\sqrt{2} \mapsto -\sqrt{2}$ for the extension $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Q}$.

We give each of these representations in three forms. One is a representation as orthogonal matrices (with the generator $a$ mapping to rotation by an odd multiple of $\pi/4$ and $x$ mapping to a reflection), and this representation is realized over the ring $\mathbb{Z}[1/\sqrt{2}]$. The second is as complex unitary matrices. The third epresentation is realized over the smaller subring $\mathbb{Z}[\sqrt{2}]$ but the matrices are no longer orthogonal matrices.

Here is the first representation in all three forms:

The table below is incomplete, it has only 11 of the 16 elements, more will be added later

Element Matrix as real orthogonal Matrix as complex unitary Matrix as real, non-orthogonal, in $\mathbb{Z}[\sqrt{2}]$ Characteristic polynomial Minimal polynomial Trace, character value Determinant
$e$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $(t - 1)^2 = t^2 - 2t + 1$ $t - 1$ 2 1
$a$ $\begin{pmatrix} 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \\\end{pmatrix}$ $\begin{pmatrix} (1 + i)/\sqrt{2} & 0 \\ 0 & (1 - i)/\sqrt{2}\\\end{pmatrix}$ $\begin{pmatrix} 0 & -1 \\ 1 & \sqrt{2} \\\end{pmatrix}$ $t^2 - \sqrt{2} t + 1$ $t^2 - \sqrt{2} t + 1$ $\sqrt{2}$ 1
$a^2$ $\begin{pmatrix} 0 & -1 \\ 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} i & 0 \\ 0 & -i \\\end{pmatrix}$ $\begin{pmatrix} -1 & -\sqrt{2} \\ \sqrt{2} & 1 \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0 1
$a^3$ $\begin{pmatrix} -1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \\\end{pmatrix}$ $\begin{pmatrix} (-1 + i)/\sqrt{2} & 0 \\ 0 & (-1-i)/\sqrt{2}\\\end{pmatrix}$ $\begin{pmatrix} -\sqrt{2} & -1 \\ 1 & 0 \\\end{pmatrix}$ $t^2 + \sqrt{2}t + 1$ $t^2 + \sqrt{2}t + 1$ $-\sqrt{2}$ 1
$a^4$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $(t + 1)^2 = t^2 + 2t + 1$ $t + 1$ -2 1
$a^5$ $\begin{pmatrix} -1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & -1/\sqrt{2} \\\end{pmatrix}$ $\begin{pmatrix} (-1 - i)/\sqrt{2} & 0 \\ 0 & (-1 + i)/\sqrt{2} \\\end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ -1 & -\sqrt{2} \\\end{pmatrix}$ $t^2 + \sqrt{2}t + 1$ $t^2 + \sqrt{2}t + 1$ $-\sqrt{2}$ 1
$a^6$ $\begin{pmatrix} 0 & 1 \\ -1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} -i & 0 \\ 0 & i \\\end{pmatrix}$ $\begin{pmatrix} 1 & \sqrt{2} \\ -\sqrt{2} & -1 \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0 1
$a^7$ $\begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \\\end{pmatrix}$ $\begin{pmatrix} (1 - i)/\sqrt{2} & 0 \\ 0 & (1 + i)/\sqrt{2}\\\end{pmatrix}$ $\begin{pmatrix} \sqrt{2} & 1 \\ -1 & 0 \\\end{pmatrix}$ $t^2 - \sqrt{2}t + 1$ $t^2 - \sqrt{2}t + 1$ $\sqrt{2}$ 1
$x$ $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix}0 & 1 \\ 1 & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
$ax$ $\begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \\\end{pmatrix}$ $\begin{pmatrix} 0 & (1 + i)/\sqrt{2} \\ (1 - i)/\sqrt{2} & 0 \\\end{pmatrix}$ $\begin{pmatrix} -1 & 0 \\ \sqrt{2} & 1 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
$a^2x$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix}0 & i \\ -i & 0 \\\end{pmatrix}$ $\begin{pmatrix} -\sqrt{2} & -1 \\ 1 & \sqrt{2} \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1

Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma
Representation/conjugacy class representative $\{ e \}$ (size 1) $\{ a^4 \}$ (size 1) $\{ a^2, a^6 \}$ (size 2) $\{ a, a^7 \}$ (size 2) $\{ a^3, a^5 \}$ (size 2) $\{ x, a^2x, a^4x, a^6x \}$ (size 4) $\{ ax, a^3x, a^5x, a^7x \}$ (size 4)
trivial 1 1 1 1 1 1 1
$\langle a \rangle$-kernel sign 1 1 1 1 1 -1 -1
$\langle a^2, x \rangle$-kernel sign 1 1 1 -1 -1 1 -1
$\langle a^2, ax \rangle$-kernel sign 1 1 1 -1 -1 -1 1
two-dimensional unfaithful, kernel is center 2 2 -2 0 0 0 0
first faithful irreducible representation 2 -2 0 $\sqrt{2}$ $-\sqrt{2}$ 0 0
second faithful irreducible representation 2 -2 0 $-\sqrt{2}$ $\sqrt{2}$ 0 0

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.

Representation/conjugacy class representative $\{ e \}$ (size 1) $\{ a^4 \}$ (size 1) $\{ a^2, a^6 \}$ (size 2) $\{ a, a^7 \}$ (size 2) $\{ a^3, a^5 \}$ (size 2) $\{ x, a^2x, a^4x, a^6x \}$ (size 4) $\{ ax, a^3x, a^5x, a^7x \}$ (size 4)
trivial 1 1 2 2 2 4 4
$\langle a \rangle$-kernel sign 1 1 2 2 2 -4 -4
$\langle a^2, x \rangle$-kernel sign 1 1 2 -2 -2 4 -4
$\langle a^2, ax \rangle$-kernel sign 1 1 2 -2 -2 -4 4
two-dimensional unfaithful, kernel is center 1 1 -1 0 0 0 0
first faithful irreducible representation 1 -1 0 $\sqrt{2}$ $-\sqrt{2}$ 0 0
second faithful irreducible representation 1 -1 0 $-\sqrt{2}$ $\sqrt{2}$ 0 0

GAP implementation

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( <pc group of size 16 with 4 generators> ),
[ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( <pc group of size
16 with 4 generators> ), [ 1, -1, 1, 1, 1, -1, 1 ] ),
Character( CharacterTable( <pc group of size 16 with 4 generators> ),
[ 1, 1, -1, 1, 1, -1, -1 ] ),
Character( CharacterTable( <pc group of size 16 with 4 generators> ),
[ 1, -1, -1, 1, 1, 1, -1 ] ),
Character( CharacterTable( <pc group of size 16 with 4 generators> ),
[ 2, 0, 0, -2, 2, 0, 0 ] ), Character( CharacterTable( <pc group of size
16 with 4 generators> ), [ 2, 0, E(8)-E(8)^3, 0, -2, 0, -E(8)+E(8)^3 ] ),
Character( CharacterTable( <pc group of size 16 with 4 generators> ),
[ 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 ] ] ] ]