Faithful irreducible representation of dihedral group:D8

This article describes a particular irreducible linear representation for the following group: dihedral group:D8. The representation is unique up to equivalence of linear representations and is irreducible, at least over its original field of definition in characteristic zero. The representation may also be definable over other characteristics by reducing the matrices modulo that characteristic, though it may behave somewhat differently in these characteristics.
For more on the linear representation theory of the group, see linear representation theory of dihedral group:D8.

Summary

Item	Value
Degree of representation	2
Schur index	1 in all characteristics
Kernel of representation	trivial subgroup, i.e., it is a faithful linear representation in all characteristic except two. In characteristic two, it behaves somewhat differently. See the section #In characteristic two.
Quotient on which it descends to a faithful representation	dihedral group:D8
Set of character values	$\{2,-2,0\}$ (interpreted/reduced modulo the ring or field) Characteristic zero: Ring generated: $\mathbb{Z}$ -- ring of integers, Ideal within ring generated: $2\mathbb{Z}$, Field generated: $\mathbb{Q}$ -- field of rational numbers
Rings of realization	Realized over any unital ring, by composing the representation over $\mathbb{Z}$ with the map induced by the natural homomorphism from $\mathbb{Z}$ to the ring.
Minimal ring of realization (characteristic zero)	$\mathbb{Z}$ -- ring of integers
Minimal ring of realization (characteristic $p^k$)	The ring of integers mod $p^k$, i.e., $\mathbb{Z}/p^k\mathbb{Z}$
Minimal field of realization	Prime field in all cases In characteristic zero, $\mathbb{Q}$, in characteristic $p$, the prime field ${\mathbb{F}}_p$
Size of equivalence class under automorphisms	1
Size of equivalence class under Galois automorphisms	1, because the representation is realized over $\mathbb{Z}$ in characteristic zero and more generally is realized over the prime subfield in any characteristic.
Bad characteristics	2, the representation can be defined but is not uniquely defined and none of the candidate representations is either faithful or irreducible. See #In characteristic two.

Representation table

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:

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

This particular choice of matrices give a representation as orthogonal matrices, and in fact, the representation is as signed permutation matrices (i.e., it takes values in the signed symmetric group of degree two). Thus, it is also a monomial representation.

Below is a description of the matrices based on the above choice as well as another formulation involving complex unitary matrices:

Element	Matrix (orthogonal/monomial/signed permutation matrices)	Matrix as complex unitary	Characteristic polynomial	Minimal polynomial	Trace, character value	Determinant
$e$	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	$(t-1{)}^2=t^2-2t+1$	$t-1$	2	1
$a$	$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$	$\left(\begin{array}{cc}i& 0\\ 0& -i\\ \end{array}\right)$	$t^2+1$	$t^2+1$	0	1
$a^2$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	$(t+1{)}^2=t^2+2t+1$	$t+1$	-2	1
$a^3$	$\left(\begin{array}{cc}0& 1\\ -1& 0\\ \end{array}\right)$	$\left(\begin{array}{cc}-i& 0\\ 0& i\\ \end{array}\right)$	$t^2+1$	$t^2+1$	0	1
$x$	$\left(\begin{array}{cc}1& 0\\ 0& -1\\ \end{array}\right)$	$\left(\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0	-1
$ax$	$\left(\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right)$	$\left(\begin{array}{cc}0& i\\ -i& 0\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0	-1
$a^{2}x$	$\left(\begin{array}{cc}-1& 0\\ 0& 1\\ \end{array}\right)$	$\left(\begin{array}{cc}0& -1\\ -1& 0\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0	-1
$a^{3}x$	$\left(\begin{array}{cc}0& -1\\ -1& 0\\ \end{array}\right)$	$\left(\begin{array}{cc}0& -i\\ i& 0\\ \end{array}\right)$	$t^2-1$	$t^2-1$	0	-1

Embeddings

Since this representation makes sense over finite fields, and it is faithful if the characteristic is not two, it provides an embedding of dihedral group:D8 in the general linear group of degree two over any finite field. In fact, the representation goes to the orthogonal group for the standard dot product, which is one of the two possible orthogonal groups for the finite field (which one it is depends on the congruence class of the size mod 4).

Further, since this is a unique faithful representation of degree two, the embedding is as isomorph-conjugate subgroups inside the general linear group.

Field size $q$	Field information	General linear group of degree two	Orthogonal group	Embedding of dihedral group:D8 in general linear group of degree two	Embedding of dihedral group:D8 in orthogonal group
3	field:F3	general linear group:GL(2,3)	?	D8 in GL(2,3)	?
5	field:F5	general linear group:GL(2,5)	?	D8 in GL(2,5)	?