Linear representation theory of quaternion group

This article is about the linear representation theory of the following group: quaternion group

The quaternion group is one of the few examples of a rational group that is not a rational-representation group. In other words, all its characters over the complex numbers are rational-valued, but not every representation of it can be realized over the rationals.

The character table of the quaternion group is the same as that of the dihedral group of order eight.

Character table

Rep/Conj class	1 (identity)	-1	$\pm i$	$\pm j$	$\pm k$
Trivial representation	1	1	1	1	1
$i$-kernel	1	1	1	-1	-1
$j$-kernel	1	1	-1	1	-1
$k$-kernel	1	1	-1	-1	1
2-dimensional	2	-2	0	0	0

Representations

Trivial representation

The trivial representation is a one-dimensional representation that sends every element of the group to the 1-by-1 matrix 1. This representation has analogues over any field.

Sign representations with $i,j,k$-kernels

The quaternion group has three maximal normal subgroups: the cyclic subgroups generated by $i,j,k$ respectively. For each maximal normal subgroup, we obtain a one-dimensional representation with that subgroup as kernel. The representation sends elements inside the subgroup to 1, and elements outside the subgroup to -1.

The sign representations make sense over any field whose characteristic is not two.

The trivial representation and the three sign representations described above, are precisely the one-dimensional representations, or equivalently, they are precisely the irreducible representations that have the commutator subgroup in their kernel.

Two-dimensional representation

The two-dimensional representation of the quaternion group can be described in a number of explicit ways. One such way is by viewing the Hamiltonian quaternions as a two-dimensional right vector space over the complex numbers, and viewing the actions of the elements of the quaternion group on this vector space by left multiplication.

The specific matrices for the representation depend on how we think of the Hamiltonians as a right vector space over the complex numbers. The typical way is to identify $\mathbb{C}$ as the subspace spanned by 1 and $u$, and take a basis as $1<math>andj$ for the vector space. Then, we have:

$1=(1,0);i=(i,0);j=(0,1);k=(0,-i)$

We can now compute the action of the elements $\pm 1,\pm i,\pm j,\pm k$ by left multiplication on this vector space, and write the matrices.

Action of automorphisms

The automorphism group of the quaternion group permutes the three sign representations. In fact, this automorphism group permutes the sign representations in precisely the same way as it permutes the three maximal normal subgroups.

The trivial representation and the two-dimensional representations remain invariant under all automorphisms.

Relation with representations of subgroups

Induced representations from subgroups

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 quaternion group.