Abstract

Clips is a binary operation between conjugacy classes of closed subgroups of a compact group, which has been introduced to obtain the isotropy classes of a direct sum of representations $E_1\oplus E_2$, if the isotropy classes are known for each factor $E_1$, $E_2$. This allows one, in particular, to compute the isotropy classes of any reducible representation, once one knows the decomposition into irreducible factors and the symmetry classes for the irreducible factors. In the specific case of the three dimensional orthogonal group $O(3)$, clips between some types of subgroups of $O(3)$ have already been calculated. However, until now, clips between type-II and type-III subgroups were missing. Those are encountered in 3D coupled constitutive laws. In this paper, we complete the clips tables by computing the missing clips. As an application, we obtain 25 isotropy classes for the standard $O(3)$ representation on the full 3D piezoelectric law, which involves the three elasticity, piezoelectricity and permittivity constitutive tensors.

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