Faithful irreducible representation of quaternion group

Representation table
 Suppose $$\alpha, \beta$$ are elements of a field (or more generally, a commutative unital ring) such that $$\alpha^2 + \beta^2 + 1 = 0$$, then the representation can be realized in terms of the entries $$\{ 0,1,-1,\alpha,-\alpha,\beta,-\beta \}$$. The explicit representation involving the Hamiltonian quaternions is the special case of this where $$\alpha$$ is a square root of $$-1$$ and $$\beta = 0$$. Another case is $$\alpha^2 = -2$$, $$\beta = 1$$. A third case of interest is $$\mathbb{Q}(\sqrt{-3})$$, which contains primitive cube roots of unity, where we set $$\alpha$$ and $$\beta$$ as distinct primitive cube roots of unity.

Note that the representation makes sense in all characteristics, but there are some problems interpreting it in characteristic two.



Frobenius-Schur indicator
The Frobenius-Schur indicator of a representation is the inner product of the character of the representation and the indicator character (which is the character that assigns to every element is number of square roots). The Frobenius-Schur indicator can be computed as $$\sum \chi(g^2)$$ where $$\chi$$ is the character of the representation.

For this faithful irreducible representation of the quaternion group, the Frobenius-Schur indicator can be computed as follows:

The inner product is $$-8/8$$ which is $$-1$$. Thus, by the indicator theorem, the representation is a quaternionic representation, i.e., it has a real-valued character but cannot be realized over the real numbers. In this case, the representation in fact has rational character value and cannot be realized over the real numbers.

Realizability information
We see from the above that the representation can be realized over any field where $$-1$$ is a sum of two squares. Are there other fields over which it can be realized?

Finite fields
Since every element of a finite field is expressible as a sum of two squares, we can always, for a finite field, find $$\alpha, \beta$$ such that $$\alpha^2 + \beta^2 = -1$$, so this representation can be realized over any finite field. See below:

Fields over which it cannot be realized
The representation cannot be realized over any formally real field, and in particular over any subfield of the field of real numbers, even though the character of the representation can be realized over such a field. The Schur index of the representation in characteristic zero is 2, which means we need to take a suitable quadratic extension of such a field (for instance, by adjoining the square root of $$-1$$ or $$-2$$ or $$-1-m^2$$ for any $$m$$ in the field), in order to realize the representation.

Over a formally real field, there is a four-dimensional irreducible representation that is not absolutely irreducible, and that, over a quadratic extension of the sort described above, splits into two copies of the two-dimensional irreducible representation discussed on the current page. For more on the four-dimensional irreducible (but not absolutely irreducible) representation, see four-dimensional irreducible representation of quaternion group.

Interpretation as Hamiltonian quaternions
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.

In characteristic two
The general description of the representation can be applied to give a representation in characteristic two, by finding elements $$\alpha,\beta$$ in a field of characteristic two such that $$\alpha^2 + \beta^2 + 1 = 0$$ and constructing the matrices. However, there are some crucial differences:

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

Embeddings in general linear groups and special linear groups
As discussed above, this representation can always be realized over a finite field of characteristic not equal to two. Moreover, for any finite field of characteristic not equal to two, the representation is faithful and gives an embedding of the quaternion group in the general linear group of degree two over the field. In fact, the embedding goes inside the special linear group of degree two, because all matrices have determinant 1. Further, since the representation is unique up to equivalence of representations, and it is the only faithful representation of degree two, this forces the subgroup to be an isomorph-conjugate subgroup in the general linear group (though it's unclear whether the conjugation can be done within the special linear group as well).

We look at some particular finite fields:

Representations of supergroups that restrict to this representation on the quaternion group
Note that by Frobenius reciprocity, it also turns out that these representations are contained in the induced representation from the quaternion group.