gives constraints on the matrices that commute with a representation of a group [@jones]. In Section \[schur\_comp\] we prove the existence and determine the explicit forms of for all the maximal subgroups of the icosahedral group.

Let us consider the path $R_{{\mathcal{G}}}(t) : [0,1] \rightarrow O(6)$ defined by $R_{{\mathcal{G}}}(t) := M_{{\mathcal{G}}}(t)R$. For every $t \in [0,1]$ the matrix $R_{{\mathcal{G}}}(t)$ encodes the basis of a physical space $E^{\parallel}_t$ and an orthogonal space $E^{\perp}_t$ that carry the representations ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ of ${\mathcal{G}}$ as in since $$\label{comm_G}
R_{{\mathcal{G}}}(t)^{-1} {\widetilde{\mathcal{G}}}R_{{\mathcal{G}}}(t) = R^{-1} M_{{\mathcal{G}}}(t)^{-1} {\widetilde{\mathcal{G}}}M_{{\mathcal{G}}}(t) R = R^{-1} {\widetilde{\mathcal{G}}}R = {\mathcal{G}}_1 \oplus {\mathcal{G}}_2.$$ In particular, we have $E^{\parallel}_t = M_{{\mathcal{G}}}(t) E^{\parallel}$ and $E^{\parallel}_0 \equiv E^{\parallel}$, $E^{\parallel}_1 \equiv E^{\parallel}_{{\mathcal{G}}}$ (and similarly for the orthogonal spaces). For $t \in [0,1]$, the projections $\pi^{\parallel}_t : {\mathbb{R}}^6 \rightarrow E^{\parallel}_t$ and $\pi^{\perp}_t : {\mathbb{R}}^6 \rightarrow E^{\perp}_t$ are given by (compare with ): $$\label{pt}
 \left( \begin{array}{c}
\pi^{\parallel}_{t} \\
\pi^{\perp}_{t} 
\end{array} \right) = R_{{\mathcal{G}}}^{-1}(t) = R^{-1}M_{{\mathcal{G}}}(t)^{-1} = 
 \left( \begin{array}{c}
\pi^{\parallel} \\
\pi^{\perp} 
\end{array} \right)M_{{\mathcal{G}}}(t)^{-1} =  \left( \begin{array}{c}
\pi^{\parallel} M_{{\mathcal{G}}}(t)^{-1}\\
\pi^{\perp} M_{{\mathcal{G}}}(t)^{-1} 
\end{array} \right).$$

With this setup, we can define structural transitions between icosahedral quasicrystals that keep the symmetry encoded by ${\mathcal{G}}$ preserved. Specifically, let ${\mathcal{L}}$ be one of the three hypercubic 