Linear representation theory of quaternion group

Summary
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.

Summary information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.

Below are representations that are irreducible over a non-splitting field, but split over a splitting field. Again, we assume that the characteristic is not 2.

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.

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 representations are detailed below:

Representation with $$\langle i \rangle$$-kernel:

Representation with $$\langle j \rangle$$-kernel:

Representation with $$\langle k \rangle$$-kernel:

Four-dimensional irreducible representation over a non-splitting field
The quaternion group has no irreducible two-dimensional representation over the reals (see faithful irreducible representation of quaternion group). 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, and all its entries are signed permutation matrices, i.e., matrices with exactly one nonzero entry in every row and every column and that entry is $$\pm 1$$.

This representation is irreducible over any formally real field (is it? verify).

Character table


This character table works over characteristic zero and over any other characteristic not equal to two once we reduce the entries mod the characteristic:

 The size-degree-weighted characters are given as follows, where a size-degree-weighted character value is obtained by multiplying the character value by the size of the conjugacy class and dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers:

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

Splitting field of characteristic not two
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 representation remain invariant under all automorphisms.

The orbit sizes are thus: one size 1 orbit of degree 1 representations (trivial representation), one size 3 orbit of degree 1 representations (sign representations), and one size 1 orbit of degree 2 representations (the faithful irreducible representation).

Non-splitting field of characteristic not two
The automorphism group permutes the three sign representation. The trivial representation and the four-dimensional irreducible representation remain invariant.

The orbit sizes are thus: one size 1 orbit of degree 1 representations (trivial representation), one size 3 orbit of degree 1 representations (sign representations), and one size 1 orbit of degree 4 representations (the faithful irreducible representation that becomes reducible over a splitting field).

Relation with quotients
The quaternion group has six normal subgroups: the whole group, the trivial subgroup, center of quaternion group, and three cyclic maximal subgroups of quaternion group. The irreducible representations with kernel a particular normal subgroup correspond precisely to the faithful irreducible representations of the quotient group; the irreducible representations with kernel containing a particular normal subgroup correspond precisely to the irreducible representations of the quotient group. Information in this regard is presented below:



For a splitting field
Over a splitting field $$K$$ (which could be a finite field or a field such as $$\mathbb{Q}[i]$$ or $$\mathbb{C}$$), we have the decomposition (as a direct product of rings, or equivalently, a direct sum of two-sided ideals). Here $$K[G]$$ is the group ring of the quaternion group over $$K$$:

$$K[G] \cong M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_2(K) \cong K \oplus K \oplus K \oplus K \oplus M_2(K)$$

More generally, the decomposition works over a uniquely 2-divisible ring $$R$$ where all the irreducible representations can be realized:

$$R[G] \cong M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_2(R) \cong R \oplus R \oplus R \oplus R \oplus M_2(R)$$

Note that this does not work over a ring such as $$\mathbb{Z}[i]$$. Although all irreducible representations can be realized over this ring, the absence of unique 2-divisibility is a roadblock.

For a non-splitting field
If $$K$$ is not a splitting field, then we get a decomposition of the form:

$$K[G] \cong M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus L$$

where $$L$$ is a subalgebra of $$M_4(K)$$ that has dimension four over $$K$$, and looks like a quaternion algebra.

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.