Orbit separation and stratification by isotropy classes of piezoelectricity tensors

Abstract

We explore an innovative method to compute separating invariants in a real G-variety $V$. A refinement of Seshadri slice Lemma enables us to decompose $V$ into a union of stable subsets $G\cdot Z_1\bigsqcup ...\bigsqcup G\cdot Z_r$. This reduces the problem to separating the orbits in the slices $Z_i$ for their normalizers $N_i<G$. This sequencing allows also to identify efficiently the isotropy class of any point. After the presentation of three types of Seshadri slices for representations of ${\mathrm{SO}}_3\left(R\right)$, we apply the method to the space of piezoelectricity tensors $P\mathrm{iez}$. This provides a separating set of low cardinality and a complete stratification of $P\mathrm{iez}$ by isotropy classes.