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:

$⟨a,x\mid a^4=x^2=e,xa{x}^{-1}=a^{-1}⟩$.

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $\mathbb{C}$ or Failed to parse (syntax error): {\displaystyle \overline{\mathbb{Q})} )	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)	$\mathbb{Z}$
Smallest splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero)	$\mathbb{Q}$ (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 $n=4$, order $2n=8$	linear representation theory of dihedral groups
prime-cube order group:U(3,p)	Case $p=2$, 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	Quotient by kernel (on which it descends to a faithful representation)	Degree	Schur index	What happens by reducing the $\mathbb{Z}$-representation over bad characteristics?
trivial	1	--	any	remains the same	whole group	trivial group	1	1	--
sign representation with kernel cyclic of order four	1	--	any	remains the same	cyclic maximal subgroup of dihedral group:D8: $⟨a⟩$	cyclic group:Z2	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: $⟨a^2,x⟩$ or $⟨a^2,ax⟩$	cyclic group:Z2	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	dihedral group:D8	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 $\left(\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right)$. 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
$e$	$(1)$	$t-1$	$t-1$	1
$a$	$(1)$	$t-1$	$t-1$	1
$a^2$	$(1)$	$t-1$	$t-1$	1
$a^3$	$(1)$	$t-1$	$t-1$	1
$x$	$(1)$	$t-1$	$t-1$	1
$ax$	$(1)$	$t-1$	$t-1$	1
$a^{2}x$	$(1)$	$t-1$	$t-1$	1
$a^{3}x$	$(1)$	$t-1$	$t-1$	1

Sign representations with kernels as the maximal normal subgroups

The dihedral group has three normal subgroups of index two: the subgroup $⟨a⟩$, the subgroup $⟨a^2,x⟩$, and the subgroup $⟨a^2,ax⟩$. For each such subgroup, there is an irreducible one-dimensional representation sending elements in that subgroup to $1$ and elements outside that subgroup to $-1$.

Here is the representation with kernel $⟨a⟩$:

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
$x$	$(-1)$	$t+1$	$t+1$	-1
$ax$	$(-1)$	$t+1$	$t+1$	-1
$a^{2}x$	$(-1)$	$t+1$	$t+1$	-1
$a^{3}x$	$(-1)$	$t+1$	$t+1$	-1

Here is the representation with kernel $⟨a^2,x⟩$:

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
$x$	$(1)$	$t-1$	$t-1$	1
$ax$	$(-1)$	$t+1$	$t+1$	-1
$a^{2}x$	$(1)$	$t-1$	$t-1$	1
$a^{3}x$	$(-1)$	$t+1$	$t+1$	-1

Here is the representation with kernel $⟨a^2,ax⟩$:

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
$x$	$(-1)$	$t+1$	$t+1$	-1
$ax$	$(1)$	$t-1$	$t-1$	1
$a^{2}x$	$(-1)$	$t+1$	$t+1$	-1
$a^{3}x$	$(1)$	$t-1$	$t-1$	1

Two-dimensional irreducible representation

The dihedral group of order eight has a two-dimensional irreducible representation, where the element $a$ acts as a rotation (by an angle of $\pi /2$, and the element $x$ acts as a reflection about the first axis. The matrices are:

$a\mapsto \left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right),x\mapsto \left(\begin{array}{cc}1& 0\\ 0& -1\\ \end{array}\right).$

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
$e$	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	$t-1$	$(t-1{)}^2$	2
$a$	$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$	$t^2+1$	$t^2+1$	0
$a^2$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	$t+1$	$(t+1{)}^2$	-2
$a^3$	$\left(\begin{array}{cc}0& 1\\ -1& 0\\ \end{array}\right)$	$t^2+1$	$t^2+1$	0
$x$	$\left(\begin{array}{cc}1& 0\\ 0& -1\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0
$ax$	$\left(\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0
$a^{2}x$	$\left(\begin{array}{cc}-1& 0\\ 0& 1\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0
$a^{3}x$	$\left(\begin{array}{cc}0& -1\\ -1& 0\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0

Over other fields

These representations generalize to any field of characteristic not equal to $2$.

Character table

This character table works over characteristic zero:

Rep/Conj class	$e$	$a^2$	$\{a,a^{-1}\}$	$\{x,a^{2}x\}$	$\{ax,a^{3}x\}$
Trivial representation	1	1	1	1	1
$⟨a⟩$-kernel	1	1	1	-1	-1
$⟨a^2,x⟩$-kernel	1	1	-1	1	-1
$⟨a^2,ax⟩$-kernel	1	1	-1	-1	1
2-dimensional	2	-2	0	0	0

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.

Realizability information

Smallest ring of realization

Representation	Smallest ring over which it is realized	Smallest set of elements in matrix entries
trivial representation	$\mathbb{Z}$ -- ring of integers	$\{1\}$
$⟨a⟩$-kernel	$\mathbb{Z}$ -- ring of integers	$\{1,-1\}$
$⟨a^2,x⟩$-kernel	$\mathbb{Z}$ -- ring of integers	$\{1,-1\}$
$⟨a^2,ax⟩$-kernel	$\mathbb{Z}$ -- ring of integers	$\{1,-1\}$
two-dimensional irreducible	$\mathbb{Z}$ -- ring of integers	$\{1,0,-1\}$

Smallest ring of realization as orthogonal matrices

Representation	Smallest ring over which it is realized	Smallest set of elements in matrix entries
trivial representation	$\mathbb{Z}$ -- ring of integers	$\{1\}$
$⟨a⟩$-kernel	$\mathbb{Z}$ -- ring of integers	$\{1,-1\}$
$⟨a^2,x⟩$-kernel	$\mathbb{Z}$ -- ring of integers	$\{1,-1\}$
$⟨a^2,ax⟩$-kernel	$\mathbb{Z}$ -- ring of integers	$\{1,-1\}$
two-dimensional irreducible	$\mathbb{Z}$ -- ring of integers	$\{1,0,-1\}$

Orthogonality relations and numerical checks

General statement	Verification in this case
number of irreducible representations equals number of conjugacy classes	Both numbers are equal to 5
sufficiently large implies splitting: if the field has characteristic not dividing the order of the group and has primitive $d^{th}$ roots of unity for $d$ the exponent of the group, it is a splitting field.	In fact, for this group, any field of characteristic not 2 is a splitting field.
number of one-dimensional representations equals order of abelianization	The number of one-dimensional representations equals $4$, which is the order of the abelianization, which is the quotient by center of dihedral group:D8 and is a Klein four-group
sum of squares of degrees of irreducible representations equals group order	$1^2+1^2+1^2+1^2+2^2=8$.
degree of irreducible representation divides order of group	The degrees (1,1,1,1,2) all divide the order 8.
degree of irreducible representation divides index of abelian normal subgroup	The degrees (1,1,1,1,2) all divide the index 2 of the abelian normal subgroups: cyclic maximal subgroup of dihedral group:D8 and Klein four-subgroups of dihedral group:D8.
order of inner automorphism group bounds square of degree of irreducible representation	The degree are all at most 2, the square of which is 4. The inner automorphism group is the quotient by center of dihedral group:D8, and is a Klein four-group of order 4.
row orthogonality theorem and the column orthogonality theorem	can be verified from the character table.

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 $⟨a⟩$. The two sign representations with kernels $⟨a^2,x⟩$ and $⟨a^2,ax⟩$ 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).

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 $⟨a⟩$ 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 $⟨a⟩$ 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. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.