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	1,1,1,1,2 maximum: 2, lcm: 2, number: 3
Schur index values of irreducible representations	1,1,1,1,2
Smallest ring of realization for all irreducible representations (characteristic zero)	$\mathbb {Z} [i]$
Smallest ring containing all character values (characteristic zero)	$\mathbb {Z}$
Smallest field of realization for all irreducible representations (characteristic zero)	$\mathbb {Q} (i)$
Smallest field containing all character vales (characteristic zero)	$\mathbb {Q}$ (hence it is a rational group)
Condition for being a splitting field for this group	Any field containing a primitive fourth root of unity is a splitting field. Smallest finite splitting field: field:F5

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 over the complex numbers

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 $i$, and take a basis as $1$ and $j$ 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.

Four-dimensional representation over the real numbers

The quaternion group has no irreducible two-dimensional representation over the reals. However, it has a four-dimensional representation over the reals, which splits over the complex numbers as a direct sum of two copies of the two-dimensional irreducible representation over the complex numbers. This representation is obtained by viewing the Hamiltonian quaternions as a four-dimensional vector space over the real numbers, and writing the matrices for left multiplication by the elements of the quaternion group. The typical choice of basis is $\{1,i,j,k\}$. Note that multiplication by anything other than $\pm 1$ gives a matrix with zeros on the diagonal, hence the character is zero on all elements outside the center.

Note that this representation is actually a representation over the rational numbers.

Representations over other fields

Over fields of characteristic not equal to two, there arise two cases:

• If $-1$ is a square in the field, then there is a two-dimensional irreducible representation over the field, analogous to that over the complex numbers. For finite fields, this happens precisely when the size of the field is $1$ modulo $4$.
• If $-1$ is not a square in the field, then there is a four-dimensional irreducible representation over the field, analogous to that over the real numbers.

Character table

This character table works over characteristic zero:

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

The size-degree-weighted characters are given as follows (size-degree-weighted characters are algebraic integers):

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

Degrees of irreducible representations

Below is a description of the degrees of irreducible representations over fields of characteristic not equal to $2$.

Type of field	Condition on polynomials	Condition on $q$ for finite field of size $q$	Degrees of irreducible representations
has squareroot of $-1$	$x^{2}+1$ splits	$4$ divides $q-1$	1,1,1,1,2
does not have squareroot of $-1$	$x^{2}+1$ does not split	$4$ does not divide $q-1$	1,1,1,1,4

Realization information

Smallest ring of realization

Representation	Smallest ring of realization	Smallest set of elements occurring as entries of matrices
trivial representation	$\mathbb {Z}$ -- ring of integers	$\{1\}$
$i$-kernel	$\mathbb {Z}$ -- ring of integers	$\{1,-1\}$
$j$-kernel	$\mathbb {Z}$ -- ring of integers	$\{1,-1\}$
$k$-kernel	$\mathbb {Z}$ -- ring of integers	$\{1,-1\}$
two-dimensional irreducible (over field containing squareroot of $-1$)	$\mathbb {Z} [i]$ -- Gaussian integers	$\{0,1,-1,i,-i\}$
four-dimensional irreducible (over field not containing squareroot of $-1$)	$\mathbb {Z}$ -- ring of integers	$\{0,1,-1\}$

Orthogonality relations and numerical checks

• Any field in which $-1$ is a square turns out to be a splitting field for the group. For finite fields, any field whose size is $1$ modulo $4$ turns out to be a splitting field for the group. Note that this is in line with the general fact that sufficiently large implies splitting: if a field has primitive $n^{th}$ roots where $n$ is the exponent of the group, it is a splitting field. In this case, $n=4$, and having a primitive fourth root is equivalent to $-1$ having a squareroot.
• The number of irreducible representations over such a field is $5$, which is equal to the number of conjugacy classes.
• The number of one-dimensional irreducible representations equals four, which is the order of the abelianization.
• The sum of squares of degrees is $1^{2}+1^{2}+1^{2}+1^{2}+2^{2}=8$, equal to the order of the group. This confirms the fact that sum of squares of degrees of irreducible representations equals order of group.
• The degrees of irreducible representations all divide $2$, which is the smallest possible index of an abelian normal subgroup. This confirms the fact that degree of irreducible representation divides index of abelian normal subgroup.
• The order of the inner automorphism group (which is $4$) bounds the square of the degree of every irreducible representation. This confirms the fact that order of inner automorphism group bounds square of degree of irreducible representation.
• The character table satisfies the orthogonality relations: in particular, the row orthogonality theorem and the column orthogonality theorem.

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

Since the quaternion 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 quaternion group.
3. The trivial representation on the cyclic subgroup generated by $i$ induces a representation on the whole group that is the sum of a trivial representation and the representation with the $i$-kernel. Analogous statements hold for $j,k$.
4. A representation on $\langle i\rangle$ that sends $i$ to $-1$ induces a representation of the whole group that is the sum of the representations with $j$-kernel and $k$-kernel. Analogous statements hold for $j,k$.
5. A representation on $\langle i\rangle$ that sends $i$ 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 of cyclic subgroups span the space of class functions. This is easy to check for the quaternion group from the points made above. By point (2) or point (5), the two-dimensional irreducible character is in the span. Points (3) and (4) show that all pairwise sums of the four one-dimensional representations are in the span. Taking suitable linear combinations of these yields that all the four one-dimensional representations are in the span.