$\rho_3' \simeq T_2$
  ----------- ---------------------------------------- ----------------------------------------
     $g_2$     $\frac{1}{2} \left( \begin{array}{ccc}   $\frac{1}{2} \left( \begin{array}{ccc}
                        \tau-1 & 1 & \tau \\                    \tau-1 & -\tau & -1 \\
                       1 & -\tau & \tau-1 \\                    -\tau & -1 & \tau-1 \\
                         \tau & \tau-1 & -1                      -1 & \tau-1 & -\tau 
                        \end{array} \right)$                     \end{array} \right)$

     $g_3$     $\frac{1}{2} \left( \begin{array}{ccc}   $\frac{1}{2} \left( \begin{array}{ccc}
                        \tau & \tau-1 & 1 \\                    -1 & 1-\tau & -\tau \\
                       1-\tau & -1 & \tau \\                    \tau-1 & \tau & -1 \\
                         1 & -\tau & 1-\tau                      \tau & -1 & 1-\tau 
                        \end{array} \right)$                     \end{array} \right)$
  ---------------------------------------------------------------------------------------------

  : \[irreps\_ico\] Explicit forms of the irreps $\rho_3$ and $\rho_3'$ with $\widetilde{{\mathcal{I}}} \simeq \rho_3 \oplus \rho_3'$.

Schur rotations between icosahedral structures {#schur}
==============================================

In this section we define structural transitions between icosahedral quasicrystals and finite point arrays using the Schur rotation method, and prove the connection with the Bain strain method described in [@paolo].

Let ${\widetilde{\mathcal{G}}}$ be a representation of a maximal subgroup ${\mathcal{G}}$ of ${\mathcal{I}}$, which is a matrix subgroup of ${\widetilde{\mathcal{I}}}$ given in . By Proposition \[reps\], there exists a unique crystallographic representation of ${\mathcal{I}}$ in $B_6$, which we denote by $\widetilde{{\mathcal{I}}}_{{\mathcal{G}}}$, such that ${\widetilde{\mathcal{G}}}$ is a subgroup of ${\widetilde{\mathcal{I}}}$ and 