Faithful irreducible representation of M16

This article describes a particular irreducible linear representation for the following group: M16. The representation is unique up to equivalence of linear representations and is irreducible, at least over its original field of definition in characteristic zero. The representation may also be definable over other characteristics by reducing the matrices modulo that characteristic, though it may behave somewhat differently in these characteristics.
For more on the linear representation theory of the group, see linear representation theory of M16.

We use the group with presentation (here $e$ denotes the identity element):

$G = M_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^5 \rangle$

Summary

This is actually a collection of two faithful irreducible two-dimensional representations of the group $M_{16}$, which form a single orbit under the action of the automorphism group, and also form a single orbit under the action of Galois automorphisms in the field of realization (at least in characteristic zero).

Item Value
degree of representation (dimension of space on which it is realized, or order of matrices) 2
Schur index value of representation 1
Kernel of representation trivial subgroup, i.e., it is a faithful linear representation
Quotient on which it descends to a faithful linear representation M16
Set of character values $\{ 2, -2, 0, 2i, -2i \}$ where $i$ is a square root of $-1$
Characteristic zero: Ring generated -- $\mathbb{Z}[2i]$, Ideal within ring generated -- $2\mathbb{Z} + 2i\mathbb{Z}$, Field generated -- $\mathbb{Q}(i)$
Rings of realization The representation can be realized precisely over those rings that contain a square root of $-1$.
Fields of realization The representation can be realized precisely over those fields that contain a square root of $-1$.
For a finite field with $q$ elements ($q$ odd), this is equivalent to requiring that $4$ divide $q - 1$.
Minimal field of realization In characteristic zero: $\mathbb{Q}(i) = \mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[t]/(t^2 + 1)$
In characteristic $p \equiv 1 \pmod 4$: $\mathbb{F}_p$
In characteristic $p \equiv 3 \pmod 4$: $\mathbb{F}_{p^2}$
Size of equivalence class under automorphisms 2. An automorphism that flips the two representations is $a \mapsto ax, x \mapsto x$.
Size of equivalence class under Galois automorphisms characteristic zero or $p \equiv 3 \pmod 4$: 2. The automorphism interchanging the square roots of -1 over the prime subfield interchanges the two representations.
characteristic $p \equiv 1 \pmod 4$: 1. The two representations cannot be interchanged, because the two square roots of -1 cannot be interchanged as they live in the prime subfield.
Smallest size field of realization (characteristic not two) field:F5

Representation table

Below are the matrices for concrete realizations of these two representation. Here $i$ denotes a square root of $-1$. The same representations can be realized over any field containing a square root of -1, if $i$ is replaced by that square root. Note that the two representations can be obtained from each other by sending $i$ to $-i$. This automorphism makes sense in characteristic zero, and more generally when the square root of $-1$ does not live in the prime field.

First version:

Element Matrix for monomial representation -- entries in $\mathbb{Z}[i]$ Matrix with entries in $\mathbb{Z}[e^{\pi i/4}] = \mathbb{Z}[(1 + i)/\sqrt{2}]$ Characteristic polynomial Minimal polynomial Trace, character value Determinant
$e$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $(t - 1)^2 = t^2 - 2t + 1$ $t - 1$ 2 1
$a$ $\begin{pmatrix} 0 & i \\ 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} (1 + i)/\sqrt{2} & 0 \\ 0 & -(1 + i)/\sqrt{2} \\\end{pmatrix}$ $t^2 - i$ $t^2 - i$ 0 $-i$
$a^2$ $\begin{pmatrix} i & 0 \\ 0 & i \\\end{pmatrix}$ $\begin{pmatrix} i & 0 \\ 0 & i\\\end{pmatrix}$ $(t - i)^2 = t^2 - 2it - 1$ $(t - i)^2 = t^2 - 2it - 1$ $2i$ -1
$a^3$ $\begin{pmatrix} 0 & -1 \\ i & 0 \\\end{pmatrix}$ $\begin{pmatrix} (-1 + i)/\sqrt{2} & 0 \\ 0 & -(-1 + i)/\sqrt{2} \end{pmatrix}$ $t^2 + i$ $t^2 + i$ 0 $i$
$a^4$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $(t + 1)^2 = t^2 + 2t + 1$ $t + 1$ -2 1
$a^5$ $\begin{pmatrix} 0 & -i \\ -1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} -(1 + i)/\sqrt{2} & 0 \\ 0 & (1 + i)/\sqrt{2} \end{pmatrix}$ $t^2 - i$ $t^2 - i$ 0 $-i$
$a^6$ $\begin{pmatrix} -i & 0 \\ 0 & -i \\\end{pmatrix}$ $\begin{pmatrix} -i & 0 \\ 0 & -i \\\end{pmatrix}$ $(t + i)^2 = t^2 + 2it - 1$ $(t + i)^2 = t^2 + 2it - 1$ $-2i$ -1
$a^7$ $\begin{pmatrix} 0 & 1 \\ -i & 0 \\\end{pmatrix}$ $\begin{pmatrix} -(-1 + i)/\sqrt{2} & 0 \\ 0 & (-1 + i)/\sqrt{2} \\\end{pmatrix}$ $t^2 + i$ $t^2 + i$ 0 $i$
$x$ $\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
$ax$ $\begin{pmatrix} 0 & -i \\ 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} 0 & (1 + i)/\sqrt{2} \\ -(1 + i)/\sqrt{2} & 0 \\\end{pmatrix}$ $t^2 + i$ $t^2 + i$ 0 $i$
$a^2x$ $\begin{pmatrix} i & 0 \\ 0 & -i \\\end{pmatrix}$ $\begin{pmatrix} 0 & i \\ i & 0 \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0 -1
$a^3x$ $\begin{pmatrix} 0 & 1 \\ i & 0 \\\end{pmatrix}$ $\begin{pmatrix} 0 & (-1 + i)/\sqrt{2} \\ -(-1 + i)/\sqrt{2} & 0 \\\end{pmatrix}$ $t^2 - i$ $t^2 - i$ 0 $-i$
$a^4x$ $\begin{pmatrix} -1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 0 & -1 \\ -1 & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
$a^5x$ $\begin{pmatrix} 0 & i \\ -1 & 0 \\\end{pmatrix}$ $\begin{pmatrix}0 & -(1 + i)/\sqrt{2} \\ (1 + i)/\sqrt{2} & 0 \\\end{pmatrix}$ $t^2 + i$ $t^2 + i$ 0 $i$
$a^6x$ $\begin{pmatrix} -i & 0 \\ 0 & i \\\end{pmatrix}$ $\begin{pmatrix}0 & -i \\ -i & 0 \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0 -1
$a^7x$ $\begin{pmatrix} 0 & -1 \\ -i & 0 \\\end{pmatrix}$ $\begin{pmatrix} 0 & -(-1 + i)/\sqrt{2} \\ (-1 + i)/\sqrt{2} & 0 \\\end{pmatrix}$ $t^2 - i$ $t^2 - i$ 0 $-i$
Set of values used $\{ 0, 1, -1, i, -i \}$ $\{ 0,\pm 1, \pm i, \pm (1+i)/\sqrt{2},\pm (-1+i)/\sqrt{2} \}$ -- -- $\{ 0,2,-2,2i,-2i \}$ $\{ 1,-1,i,-i \}$
Ring generated by values (characteristic zero) $\mathbb{Z}[i] = \mathbb{Z}[t]/(t^2 + 1)$ -- ring of Gaussian integers $\mathbb{Z}[(1 + i)/\sqrt{2}] = \mathbb{Z}[e^{\pi i/4}]$ -- -- $\mathbb{Z}[2i]$ $= \mathbb{Z}[t]/(t^2 + 4)$ $\mathbb{Z}[i] = \mathbb{Z}[t]/(t^2 + 1)$ -- ring of Gaussian integers