Linear representation theory of quaternion group

This article gives specific information, namely, linear representation theory, about a particular group, namely: quaternion group.
View linear representation theory of particular groups | View other specific information about 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. Note, however, that the fields of realization for the representations differ, because one of the representations of the quaternion group has Schur index two.

Item	Value
Degrees of irreducible representations over a splitting field (such as $\mathbb{C}$ or $\overline{\mathbb{Q}}$)	1,1,1,1,2 maximum: 2, lcm: 2, number: 5, sum of squares: 8
Schur index values of irreducible representations	1,1,1,1,2 (characteristic zero) maximum: 2, lcm: 2 1,1,1,1,1 (characteristic other than 0,2)
Smallest ring of realization for all irreducible representations (characteristic zero)	There are multiple candidates. $\mathbb{Z}[i]$ where $i$ is a square root of $-1$, equivalently $\mathbb{Z}[t]/(t^2+1)$, the ring of Gaussian integers is one candidate. Another is $\mathbb{Z}[\sqrt{-2}]$ or $\mathbb{Z}[t]/(t^2+2)$. More generally, any ring of the form $\mathbb{Z}[\alpha ,\beta ]$ where ${\alpha }^2+{\beta }^2=-1$ is a ring of realization for all irreducible representations. In particular, $\mathbb{Z}[-m^2-1]$ works.
Smallest splitting field (i.e., field of realization) for all irreducible representations (characteristic zero)	There are multiple candidates. $\mathbb{Q}(i)$ or $\mathbb{Q}[t]/(t^2+1)$ works, so does $\mathbb{Q}(\sqrt{2}i)$ or $\mathbb{Q}[t]/(t^2+2)$. More generally, $\mathbb{Q}(\alpha ,\beta )$ where ${\alpha }^2+{\beta }^2=-1$ is a splitting field. In particular, $\mathbb{Q}(\sqrt{-1-m^2})$ works for any rational $m$.
Smallest ring containing all character values (characteristic zero)	$\mathbb{Z}$
Smallest field containing all character vales (characteristic zero)	$\mathbb{Q}$ (hence it is a rational group)
Orbit structure of irreducible representations over splitting field under automorphism group	orbit sizes: 1 (degree 1 representation), 3 (degree 1 representations), 1 (degree 2 representation) number: 3
Degrees of irreducible representations over a non-splitting field, e.g., the field of rational numbers or field:F3	1,1,1,1,4
Condition for being a splitting field for this group	Sufficient condition: the characteristic is not two and there exist $\alpha ,\beta$ in the field such that ${\alpha }^2+{\beta }^2+1=0$. In particular, every finite field of characteristic not two is a splitting field, because every element of a finite field is expressible as a sum of two squares and in particular, $-1$ is a sum of two squares in any finite field.
Smallest size splitting field	field:F3, i.e., the field of three elements.

Representations

Trivial representation

The trivial or principal representation is a one-dimensional representation that sends every element of the group to the 1-by-1 matrix 1. This representation makes sense over any field and in fact over any unital ring.

Element	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value
$1$	$(1)$	$t-1$	$t-1$	1
$-1$	$(1)$	$t-1$	$t-1$	1
$i$	$(1)$	$t-1$	$t-1$	1
$-i$	$(1)$	$t-1$	$t-1$	1
$j$	$(1)$	$t-1$	$t-1$	1
$-j$	$(1)$	$t-1$	$t-1$	1
$k$	$(1)$	$t-1$	$t-1$	1
$-k$	$(1)$	$t-1$	<m