Editing Quaternion group (section)

== Matrix representations ==
[[File:Quaternion group; Cayley table; subgroup of SL(2,C).svg|thumb|Multiplication table of quaternion group as a subgroup of [[Special linear group|SL]](2,[[Complex number|'''C''']]). The entries are represented by sectors corresponding to their arguments: 1 (green), ''i'' (blue), −1 (red), −''i'' (yellow).]]
The two-dimensional irreducible complex [[group representation|representation]] described above gives the quaternion group Q<sub>8</sub> as a subgroup of the [[general linear group]] <math>\operatorname{GL}(2, \C)</math>. The quaternion group is a multiplicative subgroup of the quaternion algebra:

:<math>\H = \R 1 + \R i + \R j + \R k= \C 1+ \C j,</math>

which has a [[regular representation]] <math>\rho:\H \to \operatorname{M}(2, \C)</math> by left multiplication on itself considered as a complex vector space with basis <math>\{1,j\},</math> so that <math>z \in \H</math> corresponds to the <math>\C</math>-linear mapping <math>\rho_z: a + jb \mapsto z\cdot(a + jb).</math> The resulting representation

:<math>\begin{cases} \rho:\mathrm{Q}_8 \to \operatorname{GL}(2,\C)\\ g\longmapsto\rho_g \end{cases}</math>

is given by:

:<math>\begin{matrix}
e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} &
i \mapsto \begin{pmatrix}
         i &          0 \\
         0 & \!\!\!\!-i
\end{pmatrix}&
j \mapsto \begin{pmatrix}
         0 & \!\!\!\!-1 \\
         1 &          0
\end{pmatrix}&
k \mapsto \begin{pmatrix}
         0 & \!\!\!\!-i \\
  \!\!\!-i &          0
\end{pmatrix} \\
\overline{e} \mapsto \begin{pmatrix}
  \!\!\!-1 &          0 \\
         0 & \!\!\!\!-1
\end{pmatrix} &
\overline{i} \mapsto \begin{pmatrix}
  \!\!\!-i &          0 \\
         0 &          i
\end{pmatrix}&
\overline{j} \mapsto \begin{pmatrix}
         0 &          1 \\
  \!\!\!-1 &          0
\end{pmatrix}&
\overline{k} \mapsto \begin{pmatrix}
         0 &          i \\
         i &          0
\end{pmatrix}.
\end{matrix}
</math>

Since all of the above matrices have unit determinant, this is a representation of Q<sub>8</sub> in the [[special linear group]] <math>\operatorname{SL}(2,\C)</math>.<ref>{{harvnb|Artin|1991}}</ref>

A variant gives a representation by [[Unitary matrix|unitary matrices]] (table at right). Let <math>g\in \mathrm{Q}_8</math> correspond to the linear mapping <math>\rho_g:a+bj\mapsto (a + bj)\cdot jg^{-1}j^{-1},</math> so that <math>\rho:\mathrm{Q}_8 \to \operatorname{SU}(2)</math> is given by:

:<math>\begin{matrix}
e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} &
i \mapsto \begin{pmatrix}
         i &          0 \\
         0 & \!\!\!\!-i
\end{pmatrix}&
j \mapsto \begin{pmatrix}
         0 &          1 \\
  \!\!\!-1 &          0
\end{pmatrix}&
k \mapsto \begin{pmatrix}
         0 &          i \\
         i &          0
\end{pmatrix} \\
\overline{e} \mapsto \begin{pmatrix}
  \!\!\!-1 &          0 \\
         0 & \!\!\!\!-1
\end{pmatrix} &
\overline{i} \mapsto \begin{pmatrix}
  \!\!\!-i &          0 \\
         0 &          i
\end{pmatrix}&
\overline{j} \mapsto \begin{pmatrix}
         0 & \!\!\!\!-1 \\
         1 &          0
\end{pmatrix}&
\overline{k} \mapsto \begin{pmatrix}
         0 & \!\!\!\!-i \\
  \!\!\!-i &          0
\end{pmatrix}.
\end{matrix}</math>

It is worth noting that physicists exclusively use a different convention for the <math>\operatorname{SU}(2)</math> matrix representation to make contact with the usual [[Spin (physics)#Pauli matrices|Pauli matrices]]:

:<math>\begin{matrix}
&e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} = \quad\, 1_{2\times2} 
&i \mapsto \begin{pmatrix}
         0 &          \!\!\!-i\! \\
         \!\!-i\!\! &          0
\end{pmatrix} = -i \sigma_x
&j \mapsto \begin{pmatrix}
         0 &          \!\!\!-1\! \\
         1 &           0
\end{pmatrix} = -i \sigma_y
&k \mapsto \begin{pmatrix}
         \!\!-i\!\! &          0 \\
          0 &          i
\end{pmatrix} = -i \sigma_z\\
&\overline{e} \mapsto \begin{pmatrix}
  \!\!-1\! &          0 \\
         0 & \!\!\!-1\!
\end{pmatrix} = -1_{2\times2}
&\overline{i} \mapsto \begin{pmatrix}
         0 &          i \\
         i &          0
\end{pmatrix} = \,\,\,\, i \sigma_x
&\overline{j} \mapsto \begin{pmatrix}
         0 &          1 \\
        \!\!-1\!\! &          0
\end{pmatrix} = \,\,\,\, i \sigma_y
&\overline{k} \mapsto \begin{pmatrix}
         i &           0 \\
         0 &          \!\!\!-i\!
\end{pmatrix} = \,\,\,\, i \sigma_z.
\end{matrix}</math>

This particular choice is convenient and elegant when one describes [[Spinor|spin-1/2 states]] in the <math>(\vec{J}^2, J_z)</math> basis and considers [[Angular momentum operator#Derivation using ladder operators|angular momentum ladder operators]] <math>J_{\pm} = J_x \pm iJ_y.</math>

[[File:Quaternion group; Cayley table; subgroup of SL(2,3).svg|thumb|Multiplication table of the quaternion group as a subgroup of [[:File:SL(2,3); Cayley table.svg|SL(2,3)]]. The field elements are denoted 0, +, −.]]

There is also an important action of Q<sub>8</sub> on the 2-dimensional vector space over the [[finite field]] <math>\mathbb{F}_3 =\{0, 1, -1\}</math> (table at right). A [[Modular representation theory|modular representation]] <math>\rho: \mathrm{Q}_8 \to \operatorname{SL}(2, 3)</math> is given by

:<math>\begin{matrix}
e \mapsto \begin{pmatrix}
         1 &          0 \\
         0 &          1
\end{pmatrix} &
i \mapsto \begin{pmatrix}
         1 &          1 \\
         1 & \!\!\!\!-1
\end{pmatrix} &
j \mapsto \begin{pmatrix}
  \!\!\!-1 &          1 \\
         1 &          1
\end{pmatrix} &
k \mapsto \begin{pmatrix}
         0 & \!\!\!\!-1 \\
         1 &          0
\end{pmatrix} \\
\overline{e} \mapsto \begin{pmatrix}
  \!\!\!-1 &          0 \\
         0 & \!\!\!\!-1
\end{pmatrix} &
\overline{i} \mapsto \begin{pmatrix}
  \!\!\!-1 & \!\!\!\!-1 \\
  \!\!\!-1 &          1
\end{pmatrix} &
\overline{j} \mapsto \begin{pmatrix}
         1 & \!\!\!\!-1 \\
  \!\!\!-1 & \!\!\!\!-1
\end{pmatrix} &
\overline{k} \mapsto \begin{pmatrix}
         0 &          1 \\
  \!\!\!-1 &          0
\end{pmatrix}.
\end{matrix}</math>

This representation can be obtained from the [[Field extension|extension field]]:

:<math> \mathbb{F}_9 = \mathbb{F}_3 [k] = \mathbb{F}_3 1 + \mathbb{F}_3 k,</math>

where <math>k^2=-1</math> and the multiplicative group <math>\mathbb{F}_9^{\times}</math> has four generators, <math>\pm(k\pm1),</math> of order 8. For each <math>z \in \mathbb{F}_9,</math> the two-dimensional <math>\mathbb{F}_3</math>-vector space <math>\mathbb{F}_9</math> admits a linear mapping:

:<math>\begin{cases} \mu_z: \mathbb{F}_9 \to \mathbb{F}_9 \\ \mu_z(a+bk)=z\cdot(a+bk) \end{cases}</math>

In addition we have the [[Frobenius endomorphism|Frobenius automorphism]] <math>\phi(a+bk)=(a+bk)^3</math> satisfying <math>\phi^2 = \mu_1 </math> and <math>\phi\mu_z = \mu_{\phi(z)}\phi.</math> Then the above representation matrices are:

:<math>\begin{align}
\rho(\bar e) &=\mu_{-1}, \\
\rho(i) &=\mu_{k+1}\phi, \\
\rho(j)&=\mu_{k-1} \phi, \\
\rho(k)&=\mu_{k}.
\end{align}</math>

This representation realizes Q<sub>8</sub> as a [[normal subgroup]] of {{nowrap|GL(2, 3)}}. Thus, for each matrix <math>m\in \operatorname{GL}(2,3)</math>, we have a group automorphism

:<math>\begin{cases} \psi_m:\mathrm{Q}_8\to\mathrm{Q}_8 \\ \psi_m(g)=mgm^{-1} \end{cases}</math>

with <math>\psi_I =\psi_{-I}=\mathrm{id}_{\mathrm{Q}_8}.</math> In fact, these give the full automorphism group as:

:<math>\operatorname{Aut}(\mathrm{Q}_8) \cong \operatorname{PGL}(2, 3) = \operatorname{GL}(2,3)/\{\pm I\}\cong S_4.</math>

This is isomorphic to the symmetric group S<sub>4</sub> since the linear mappings <math>m:\mathbb{F}_3^2 \to \mathbb{F}_3^2</math> permute the four one-dimensional subspaces of <math>\mathbb{F}_3^2,</math> i.e., the four points of the [[projective space]] <math>\mathbb{P}^1 (\mathbb{F}_3) = \operatorname{PG}(1,3).</math>

Also, this representation permutes the eight non-zero vectors of <math>\mathbb{F}_3^2,</math> giving an embedding of Q<sub>8</sub> in the [[symmetric group]] S<sub>8</sub>, in addition to the embeddings given by the regular representations.