Faithful irreducible representation of dihedral group:D8

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:

$$a \mapsto \begin{pmatrix}0 & -1 \\ 1 & 0 \\\end{pmatrix}, \qquad x \mapsto \begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}.$$

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:



Character values and interpretations
The character can be computed using any of the interpretations provided. See below:

Relation with faithful linear representations
In a field of characteristic not equal to two, a linear representation of the dihedral group of order eight is faithful if and only if the faithful irreducible representation (that forms the subject of this page) occurs as one of its irreducible subrepresentations (i.e., one of its summands in its direct sum decomposition by Maschke's theorem) with multiplicity one or higher. We note both directions:


 * Faithful irreducible representation as a subrepresentation implies faithful: The kernel of any subrepresentation contains the kernel of the whole representation, so this follows.
 * Faithful implies contains the faithful irreducible representation as a subrepresentation: The dihedral group of order eight has only one faithful irreducible representation. Moreover, for all other irreducible representations, the kernel of the representation contains the center of dihedral group:D8. Thus, any representation that is a direct sum of these representations has nontrivial kernel (at least the center of dihedral group:D8) and cannot be faithful.

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 of characteristic not two. 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.

Semidirect products
For any linear representation, we can construct an external semidirect product with normal subgroup the additive group of the vector space and acting quotient the group being represented. We describe below some of the semidirect products in the case of finite fields.

Construction of representation as a homomorphism
The representation over characteristic zero can be constructed using:

gap> G := DihedralGroup(8);; phi := Filtered(IrreducibleRepresentations(G),x -> Length(Identity(G)^x)= 2)[1]; Pcgs([ f1, f2, f3 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ]

The set of images of all elements can be computed as matrices. For instance, once the representation is defined as above, the following displays the elements and their matrix images in paired form:

gap> List(Elements(G),x -> [x,x^phi]); [ [ of ..., [ [ 1, 0 ], [ 0, 1 ] ] ], [ f1, [ [ 0, 1 ], [ 1, 0 ] ] ], [ f2, [ [ E(4), 0 ], [ 0, -E(4) ] ] ], [ f3, [ [ -1, 0 ], [ 0, -1 ] ] ], [ f1*f2, [ [ 0, -E(4) ], [ E(4), 0 ] ] ], [ f1*f3, [ [ 0, -1 ], [ -1, 0 ] ] ], [ f2*f3, [ [ -E(4), 0 ], [ 0, E(4) ] ] ], [ f1*f2*f3, [ [ 0, E(4) ], [ -E(4), 0 ] ] ] ]

We can do a similar construction over other fields. For instance, over field:F3, we need to do:

gap> G := DihedralGroup(8);; phi := Filtered(IrreducibleRepresentations(G,GF(3)),x -> Length(Identity(G)^x)= 2)[1]; Pcgs([ f1, f2, f3 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ]

Construction of character
The character can be constructed by searching for the appropriate row in the character table:

gap> C := Filtered(Irr(CharacterTable(DihedralGroup(8))),x -> DegreeOfCharacter(x) = 2)[1]; Character( CharacterTable(  ), [ 2, 0, 0, -2, 0 ] )