${\mathbb{R}}^n$ with generator matrix $B$, ${\mathcal{P}}(RB) = R{\mathcal{P}}(B)R^{-1}$, for $R \in O(n)$ [@zanzotto]) and $M_{{\mathcal{G}}}(t) \in {\mathbb{Z}}({\widetilde{\mathcal{G}}},{\mathbb{R}})$ for all $t$. Hence the Schur rotation is equivalent to a Bain strain transformation between *congruent* lattices, i.e. whose bases are related via a rotation [@conway]. The advantage of the former is that the use of Schur’s Lemma and tools from representation theory can be used in the computation and allow a characterisation of such transitions in a purely group theoretical framework.

Computations and applications {#schur_comp}
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In this section we compute the Schur rotations for the maximal subgroups of the icosahedral group, and discuss applications and specific examples. It follows from Section \[schur\] that the crucial point is the computation of the matrix groups ${\mathcal{Z}}({\widetilde{\mathcal{G}}},{\mathbb{R}}) \cap SO(6)$, where ${\widetilde{\mathcal{G}}}\subseteq {\widetilde{\mathcal{I}}}$ is a representation of ${\mathcal{G}}$ in $B_6$. To this aim, we first focus on the group ${\mathcal{Z}}(\hat{{\mathcal{G}}},{\mathbb{R}}) \cap SO(6)$, which consists of all the rotations in $SO(6)$ that commute with the matrices constituting the *reduced* representation $\hat{{\mathcal{G}}}$. A matrix in this group can be easily computed using Schur’s Lemma [@jones]; the group ${\mathcal{Z}}({\widetilde{\mathcal{G}}},{\mathbb{R}}) \cap SO(6)$ then easily follows since the following holds (see [@artin]): $$\label{t_1}
{\mathcal{Z}}(\hat{{\mathcal{G}}}, {\mathbb{R}}) = {\mathcal{Z}}(R^{-1} {\widetilde{\mathcal{G}}}R, {\mathbb{R}}) = 