# Algebraically independent generators for the invariant field of SO 3 (R) and O 3 (R) representations R 3 ⊕ H

Evelyne Hubert, Martin Jalard

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Evelyne Hubert, Martin Jalard. Algebraically independent generators for the invariant field of SO 3 (R) and O 3 (R) representations R 3 ⊕ H. 2024. hal-04604969

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# Algebraically independent generators for the invariant field of SO 3 (R) and O 3 (R) representations R 3 ⊕ H

Evelyne Hubert, Martin Jalard

<figure><img src="image_1.png" /></figure>

## To cite this version:

Evelyne Hubert, Martin Jalard. Algebraically independent generators for the invariant field of SO 3 (R) and O 3 (R) representations R 3 ⊕ H. 2025. hal-04604969v3

HAL Id: hal-04604969

https://inria.hal.science/hal-04604969v3

Preprint submitted on 16 Apr 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
Algebraically independent generators for the invariant field of
$SO_3(\mathbb{R})$ and $O_3(\mathbb{R})$ representations $\mathbb{R}^3 \oplus H$

Evelyne Hubert and Martin Jalard*

Inria Côte d'Azur, France

June 7, 2024

Abstract

We consider the representations of the group $SO_3(\mathbb{R})$ that contain the standard representation $\mathbb{R}^3$
as an irreducible component. This is a large class of representations containing the vector spaces of
ternary forms of any odd degree, the space of Piezoelectric tensors and the classical product space
$\mathbb{R}^3 \times \dots \times \mathbb{R}^3$ which appears in multiple physical and engineering situations. For any such representation
we construct a set of algebraically independent rational invariants that separate orbits outside of an
identified hypersurface. As such they generate the field of rational invariants.

Our key ingredient is a constructive use of Seshadri slice lemma. This latter establishes an isomor-
phism between the $SO_3(\mathbb{R})$-invariant field on the full representation $\mathbb{R}^3 \oplus H$ with the invariant field of
$O_2(\mathbb{R})$ acting on a subspace $\mathbb{R} \oplus H$. The $O_2(\mathbb{R})$-invariants are given explicitly as polynomials. We pre-
dict the denominators of the related $SO_3(\mathbb{R})$-invariants and show how to evaluate these latter thanks
to Wigner matrices. The results are then extended to $O_3(\mathbb{R})$ actions and several cases of interest to
applications are detailed as examples.

<footnote>*Contact: martin.jalard@inria.fr</footnote>
Algebraically independent generators for the invariant field of
$SO_3(\mathbb{R})$ and $O_3(\mathbb{R})$ representations $\mathbb{R}^3 \oplus \mathcal{H}$

Evelyne Hubert and Martin Jalard*

Inria Côte d'Azur, France

April 15, 2025

**Abstract**

We consider the representations of the group $SO_3(\mathbb{R})$ that contain the standard representation $\mathbb{R}^3$ as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of piezoelectricity tensors or the classical product space $\mathbb{R}^3 \times \dots \times \mathbb{R}^3$ which appears in multiple physical and engineering situations. For any such representation we construct a set of algebraically independent rational invariants that generates the invariant field.

Our key ingredient is a constructive use of Seshadri slice lemma. This lemma establishes an isomorphism between the $SO_3(\mathbb{R})$-invariant field on the full representation $\mathbb{R}^3 \oplus \mathcal{H}$ with the invariant field of a representation of $O_2(\mathbb{R})$ on a subspace $\mathbb{R} \oplus \mathcal{H}$. We exhibit a set of $O_2(\mathbb{R})$-invariant polynomials generating this latter invariant field. The explicit expressions for the generating rational $SO_3(\mathbb{R})$-invariants can then be obtained with symbolic manipulations. The results are then extended to actions of $O_3(\mathbb{R})$ and several cases of interest to applications are detailed as examples.

<footnote>*Contact: martin.jalard@inria.fr</footnote>
# Contents

<table><tr><td>1 Introduction</td><td>3</td></tr><tr><td>2 Invariants for a class of O<sub>2</sub>(R)-representation</td><td>4</td></tr><tr><td>2.1 Representations of O<sub>2</sub>(C)</td><td>4</td></tr><tr><td>2.2 Invariants for SO<sub>2</sub>(C)</td><td>4</td></tr><tr><td>2.3 Invariants for O<sub>2</sub>(C)</td><td>6</td></tr><tr><td>2.4 Invariants for O<sub>2</sub>(R)</td><td>7</td></tr><tr><td>3 Invariants for SO<sub>3</sub>(R)</td><td>8</td></tr><tr><td>3.1 Representations of SO<sub>3</sub>(R)</td><td>8</td></tr><tr><td>3.2 Seshadri slice</td><td>10</td></tr><tr><td>3.3 Restrictions of the invariants to the slice</td><td>11</td></tr><tr><td>3.4 Evaluating the invariants</td><td>12</td></tr><tr><td>3.5 Separation properties</td><td>14</td></tr><tr><td>3.6 Inverse problem</td><td>15</td></tr><tr><td>4 Generalisation to O<sub>3</sub>(R) representations</td><td>17</td></tr><tr><td>5 Applications</td><td>19</td></tr><tr><td>5.1 Odd degree ternary forms</td><td>19</td></tr><tr><td>5.2 Piezoelectricity tensors</td><td>20</td></tr><tr><td>A Rewriting an invariant in terms of the generators</td><td>21</td></tr></table>
# Contents

<table><tr><td>1 Introduction</td><td>3</td></tr><tr><td>2 Invariants for a class of O<sub>2</sub>(R)-representation</td><td>4</td></tr><tr><td>2.1 Representations of O<sub>2</sub>(C)</td><td>4</td></tr><tr><td>2.2 Invariants for SO<sub>2</sub>(C)</td><td>5</td></tr><tr><td>2.3 Invariants for O<sub>2</sub>(C) and O<sub>2</sub>(R)</td><td>6</td></tr><tr><td>3 Invariants for SO<sub>3</sub>(R)</td><td>7</td></tr><tr><td>3.1 Representations of SO<sub>3</sub>(R)</td><td>8</td></tr><tr><td>3.2 Seshadri slice</td><td>10</td></tr><tr><td>3.3 Restrictions of the invariants to the slice</td><td>11</td></tr><tr><td>3.4 Evaluating the invariants</td><td>12</td></tr><tr><td>3.5 Inverse problem</td><td>14</td></tr><tr><td>4 Generalisation to O<sub>3</sub>(R) representations</td><td>16</td></tr><tr><td>5 Applications</td><td>18</td></tr><tr><td>5.1 Odd degree ternary forms</td><td>18</td></tr><tr><td>5.2 Piezoelectricity tensors</td><td>19</td></tr><tr><td>5.3 3D Moment Invariants</td><td>20</td></tr><tr><td>A Rewriting an invariant in terms of the generators</td><td>23</td></tr></table>
# 1 Introduction

Invariants are useful for classifying objects up to a group of transformations. In this article we consider a large class of representations of $SO_3(\mathbb{R})$ among which some are relevant to domains as continuum mechanics and machine learning. We construct a generating set of rational invariants that is algebraically independent.

Computational Invariant Theory [DK15; Stu93] has long focused on polynomial invariants. In the case of the group $O_3(\mathbb{R})$ or $SO_3(\mathbb{R})$, any two real orbits are separated by a generating set of polynomial invariants. The generating set can nonetheless be very large, of very high degree and difficult to numerically evaluate at a given point robustly. A generating set of rational invariants is typically of smaller cardinality and yet they separate orbits in general position [PV94; Ros56] – this remains true for any group, even for non-reductive groups. Rational invariants can thus prove to be sufficient, and sometimes more relevant, in applications and in connection with other mathematical disciplines [Bür+21; HK07b; Hub12]. A practical and general algorithm to compute a generating set of rational invariants of a group action appeared in [HK07a]; see also [DK15; Hub19; MB99]. The main idea of this construction was refined to provide specific algorithms for group actions relevant to diverse applications [GHP19; HL13]. As corollary of these constructions, the invariant field of these actions is seen to be rational, a problem sometimes referred to as Noether's problem and amply addressed in the literature, as reviewed in [CS05].

In this article we consider the representations of the group $SO_3(\mathbb{R})$ that contain the standard representation $\mathbb{R}^3$ as an irreducible component. This is a large class of representations containing the vector spaces of ternary forms of any odd degree, the space of piezoelectric tensors and, last but not least, the classical product space $\mathbb{R}^3 \times \dots \times \mathbb{R}^3$ that appears in multiple physical and engineering situations. The key constructive refinement we introduce, to exhibit a generating set of rational invariants, is based on Seshadri slice lemma.

Seshadri slice lemma establishes an isomorphism between the $SO_3(\mathbb{R})$-invariant field on the full representation $\mathbb{R}^3 \oplus \mathbb{H}$ with the invariant field of $O_2(\mathbb{R})$ acting on a subspace $\mathbb{R} \oplus \mathbb{H}$. We give explicitly a set of dim $\mathbb{H}$ – 1 algebraically independent invariant polynomials that generate the $O_2(\mathbb{R})$ invariant field. They correspond to algebraically independent rational $SO_3(\mathbb{R})$-invariant and we make explicit what are the denominators of these and how to evaluate them where defined. Conversely, we solve the inverse problem: recover a representative of any real orbit distinguished by the values of the invariants.

The results are then extended to $O_3(\mathbb{R})$ actions and several cases of interest to applications are detailed as examples. As a side product, our construction provides a proof of rationality for the invariant fields of all the representations considered; only *rational stability* is ensured other free $SO_3(\mathbb{R})$-representations [CS05].

In continuum mechanics, if a minimal set of generators of the invariant field were determined for the action of $SO_3(\mathbb{R})$ on the space of elasticity tensors [Des+19], this is the first time a minimal generating set of $SO_3(\mathbb{R})$-rational invariants is given for the space of piezoelectric tensors. Other separating sets of invariants are provided in [Oli14; Che+19], but their cardinalities are drastically larger.

The paper is organized as follows. In Section 2 we make explicit a set of polynomials of small size that form an algebraically independent set of generators for the field of $O_2(\mathbb{R})$-invariant for the large class of $O_2(\mathbb{R})$-representations relevant to our purpose. In Section 3 we introduce the class of $SO_3(\mathbb{R})$-representations of interest to us and show how their rational invariants can be deduced from the $O_2(\mathbb{R})$-invariants just constructed. We show how to evaluate them numerically on real data and examine where they separate orbits. We furthermore solve the inverse problem: given a set of values for the invariants, can we identify a real point where these values are obtained? In Section 4 we extend our result to the construction of $O_3(\mathbb{R})$-invariants. And finally in Section 5 we give explicitly the generating $O_3(\mathbb{R})$ or $SO_3(\mathbb{R})$-invariants for cases of interest in applications. In Appendix A we explain how to rewrite any other $SO_3(\mathbb{R})$-invariant on $\mathbb{R}^3 \oplus \mathbb{H}$ in terms of our generators.

For a real algebraic variety $\mathcal{V}$, we note $\hat{\mathcal{V}}$ its complexification.
# 1 Introduction

Invariants are essential for classifying objects up to a group of transformations. In this article we consider a large class of representations of $SO_3(\mathbb{R})$ among which some are relevant to domains as continuum mechanics and 3D image analysis. We exhibit a set of algebraically independent rational invariants that generates the invariant field.

By a result of Rosenlicht [Ros56, Section 1, Theorem 2] (see also [PV94, Theorem 2.3]), for a rational action of an algebraic group, a generating set of the invariant field separates generic orbits. This is a clear advantage over the algebra of polynomial invariants for either linear actions of non compact groups [Bür+21; HL12; HL13] or rational actions of any algebraic group [HK07b; Hub12]. Even for compact groups, as $SO_3(\mathbb{R})$, where there exist finite sets of polynomial invariants that separate all orbits, the trade off is that their cardinality is often much larger than a generating set of rational invariants.

A general and practical algorithm to compute a generating set of rational invariants is proposed in [HK07a; Hub19] (see also [DK15, Section 4.10]). The construction is based on a choice of cross-section (a.k.a. quasi-section in [PV94]). The construction presented here exploits the existence of a *Seshadri slice* (a.k.a. relative sections in [PV94]), i.e., a subvariety $\mathcal{S}$ of the representation space $\mathcal{V}$, with normalizer $N < G$, such that $G \cdot \mathcal{S}$ is dense in $\mathcal{V}$. A result attributed to Seshadri [Ses62] indicates that the restriction of $\mathcal{V}$ to $\mathcal{S}$ induces an isomorphism between $\mathbb{R}(\mathcal{V})^G$ and $\mathbb{R}(\mathcal{S})^N$. Variations on this result also appeared in [Pop92; NM+65; GV78; CS07]. Seshadri slice lemma is an essential tool for proving the rationality of invariant fields [CS07] but previous constructive use of it also appeared in [GHP19; LRS20].

We consider the representations $\mathcal{V}$ of the group $SO_3(\mathbb{R})$ that contain the standard representation $\mathbb{R}^3$ as an irreducible component; we write $\mathcal{V} = \mathbb{R}^3 \oplus \mathcal{H}$. We identify a Seshadri slice $(\mathcal{S}, N)$ with $\mathcal{S} = \mathbb{R} \oplus \mathcal{H}$ and $N \cong O_2(\mathbb{R})$. Seshadri slice lemma establishes then the field isomorphism $\mathbb{R}(\mathcal{V})^{SO_3(\mathbb{R})} \cong \mathbb{R}(\mathcal{S})^{O_2(\mathbb{R})}$. We exhibit a set of $\dim(\mathcal{H})$ algebraically independent generators for $\mathbb{R}(\mathcal{S})^{O_2(\mathbb{R})}$. They correspond unequivocally to algebraically independent generators of $\mathbb{R}(\mathcal{V})^{SO_3(\mathbb{R})}$. Wigner matrices enable us to compute the explicit expressions of the latters, with respect to spherical coordinates. Symbolic manipulations based on Groebner bases translate them in Cartesian coordinates. The Maple code is available at https://sites.google.com/view/martin-jalard/accueil.

As a side product, we get a constructive proof for the rationality of the invariant fields of the representations considered. Nonetheless this could also be deduced from the rationality of the invariant field of any $SL_2(\mathbb{C})$-representations, proved in [Kat83; Kat84; BK85], thanks to the two-fold covering $SL_2(\mathbb{C}) \to SO_3(\mathbb{C})$. Our approach allows to derive directly the rationality over $\mathbb{R}$, for both representations of $SO_3(\mathbb{R})$ and $O_3(\mathbb{R})$. Furthermore, in [HJ24b], the multihomogeneity of the invariants exhibited serve the base case of an inductive proof of the rationality of the invariant fields for a class of representations of $SO_n(\mathbb{R})$.

The class of representations of $SO_3(\mathbb{R})$ containing $\mathbb{R}^3$ as an irreducible component is quite general, and we apply our method to various examples of interest in mathematics, mechanics and engineering. Firstly, the classical product space $\mathbb{R}^3 \times \dots \times \mathbb{R}^3$ for which generically separating invariants are in demand to compute invariant neural networks [DG24]. We also consider the spaces of homogeneous forms of odd degree<fnref n="1" />, and more particularly the ternary cubics, classically addressed in invariant theory [Stu08]. We next give generators for the action of $SO_3(\mathbb{R})$ for the space of piezoelectricity tensors which stems from continuum mechanics. Other sets of separating invariants were provided in [Oli14; Che+19], but their cardinalities are drastically larger. Finally we provide invariants on the space of moments up to any order which are used in 3D image analysis [FSZ17]. Many such invariants were previously constructed and applied in diverse contexts but there was no firm foundations to their independence and generating properties.

The paper is organized as follows. In Section 2 we construct algebraically independent polynomial invariants generating the field of $O_2(\mathbb{R})$-invariant for representations relevant to our purpose. In Section 3 we show how to obtain explicit generating rational invariants for the corresponding $SO_3(\mathbb{R})$-representations. We next

<footnote n="1">¹The invariants for the action on even degree forms were given in [GHP19].</footnote>
## 2 Invariants for a class of $O_2(\mathbb{R})$-representation

The goal of this article is to construct generators of $SO_3(\mathbb{R})$-invariant fields. For the representations we shall consider these fields are isomorphic to $O_2(\mathbb{R})$-invariant fields. As a preliminary we thus consider, in this section, the relevant representations of $O_2(\mathbb{R})$ (see Section 2.2). For each of these we make explicit a set of algebraically independent polynomial invariants that generates the field of invariants (Theorem 2.8). We describe precisely the open set on which they separate $O_2(\mathbb{R})$-orbits.

### 2.1 Representations of $O_2(\mathbb{C})$

The orthogonal group $O_2(\mathbb{R})$ is often thought of as the one parameter family of linear transformations in the plane given by:

$$O_2(\mathbb{R}) := \left\{ \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \end{pmatrix}, \alpha \in \mathbb{R} \right\} \cup \left\{ \begin{pmatrix} \cos(\alpha) & \sin(\alpha) \\ \sin(\alpha) & -\cos(\alpha) \end{pmatrix}, \alpha \in \mathbb{R} \right\}.$$

If $(x_1, x_{-1})$ are the coordinates for the plane $\mathbb{R}^2$, considering the coordinates

$$\begin{pmatrix} y_1 \\ y_{-1} \end{pmatrix} = P \begin{pmatrix} x_1 \\ x_{-1} \end{pmatrix} \quad \text{where } P := \begin{pmatrix} 1 & i \\ 1 & -i \end{pmatrix}$$

of the $\mathbb{R}$-vector space $\mathcal{V} = \{(y_1, y_{-1}) \in \mathbb{C}^2 | \bar{y}_1 = y_{-1}\}$ we obtain the parametrization

$$O_2(\mathbb{R}) \cong \left\{ \begin{pmatrix} e^{i\alpha} & 0 \\ 0 & e^{-i\alpha} \end{pmatrix}, \alpha \in \mathbb{R} \right\} \cup \left\{ \begin{pmatrix} 0 & e^{i\alpha} \\ -e^{i\alpha} & 0 \end{pmatrix}, \alpha \in \mathbb{R} \right\}.$$

Concomitantly [FH13, Exercise 18.2]

$$O_2(\mathbb{C}) \cong \left\{ g_z^+ = \begin{pmatrix} z & 0 \\ 0 & \frac{1}{z} \end{pmatrix}, z \in \mathbb{C}^* \right\} \cup \left\{ g_z^- = \begin{pmatrix} 0 & \frac{1}{z} \\ z & 0 \end{pmatrix}, z \in \mathbb{C}^* \right\}.$$

**Proposition 2.1.** [GSS88] The irreducible representations of $O_2(\mathbb{C})$ are indexed by $j \in \mathbb{Z}$ with $j \ge -1$. They consist of representations $\phi_j : O_2(\mathbb{C}) \to \text{GL}(\hat{\mathcal{V}}_j)$ where

* for $j \in \{0, -1\}$, $\hat{\mathcal{V}}_j \cong \mathbb{C}$ is one dimensional and

$$\phi_0(g_z^+) = (1), \phi_0(g_z^-) = (1); \quad \phi_{-1}(g_z^+) = (1), \phi_{-1}(g_z^-) = (-1);$$

* for $j \ge 1$, $\hat{\mathcal{V}}_j \cong \mathbb{C}^2$ is two dimensional and

$$\phi_j(g_z^+) = \begin{pmatrix} z^j & 0 \\ 0 & z^{-j} \end{pmatrix} \quad \text{and} \quad \phi_j(g_z^-) = \begin{pmatrix} 0 & z^{-j} \\ z^j & 0 \end{pmatrix}$$

The real trace of $\hat{\mathcal{V}}_j$, for $j \ge 1$, is $\mathcal{V}_j = \{(y_1, y_{-1}) \in \mathbb{C}^2 | \bar{y}_1 = y_{-1}\}$ and the real irreducible representations of $O_2(\mathbb{R})$ are obtained by restricting the above ones to $\mathcal{V}_j$ and $\{z \in \mathbb{C}^* | z \bar{z} = 1\}$.

### 2.2 Invariants for $SO_2(\mathbb{C})$

Let $p, q, m_1, \dots, m_r$ be integers with $m_1 = 1$, $q \ge 1$. We consider the representation

$$\rho : O_2(\mathbb{C}) \to \text{GL}(\hat{\mathcal{V}}) \text{ where } \hat{\mathcal{V}} = \bigoplus_{j=1}^{p} \hat{\mathcal{V}}_0 \oplus \bigoplus_{j=1}^{q} \hat{\mathcal{V}}_{-1} \oplus \bigoplus_{j=1}^{r} \hat{\mathcal{V}}_{m_j} \quad (m_1 = 1, q \ge 1).$$
solve the inverse problem: given a set of values for the invariants, can we identify a real point where these values are obtained? In Section 4 we extend our result to the construction of $O_3(\mathbb{R})$-invariants. In Section 5 we examine closely the mentioned cases of interest. Additionally, in Appendix A, we explain how to rewrite any other $SO_3(\mathbb{R})$-invariant on $\mathbb{R}^3 \oplus \mathcal{H}$ in terms of our generators.

# 2 Invariants for a class of $O_2(\mathbb{R})$-representation

The goal of this article is to construct algebraically independent generators for the invariant field of a representation of $SO_3(\mathbb{R})$. For the representations we shall consider these fields are isomorphic to $O_2(\mathbb{R})$-invariant fields. As a preliminary we thus consider the related representations of $O_2(\mathbb{R})$ for which we make explicit algebraically independent generators for their fields of invariants (Theorem 2.6).

## 2.1 Representations of $O_2(\mathbb{C})$

The orthogonal group $O_2(\mathbb{R})$ is often thought of as the one parameter family of linear transformations in the plane given by:

$$O_2(\mathbb{R}) := \left\{ \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) \\ \sin(\alpha) & \cos(\alpha) \end{pmatrix}, \alpha \in \mathbb{R} \right\} \cup \left\{ \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) \\ -\sin(\alpha) & -\cos(\alpha) \end{pmatrix}, \alpha \in \mathbb{R} \right\}.$$

If $(x_{-1}, x_1)$ are the coordinates for the plane $\mathbb{R}^2$, considering the coordinates

$$\begin{pmatrix} y_{-1} \\ y_1 \end{pmatrix} = P \begin{pmatrix} x_{-1} \\ x_1 \end{pmatrix} \text{ where } P := \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ 1 & -i \end{pmatrix}$$

of the $\mathbb{R}$-vector space $\mathcal{V} = \{(y_{-1}, y_1) \in \mathbb{C}^2 | \bar{y}_1 = y_{-1}\}$ we obtain the parametrization

$$O_2(\mathbb{R}) \cong \left\{ \begin{pmatrix} e^{i\alpha} & 0 \\ 0 & e^{-i\alpha} \end{pmatrix}, \alpha \in \mathbb{R} \right\} \cup \left\{ \begin{pmatrix} 0 & e^{-i\alpha} \\ e^{i\alpha} & 0 \end{pmatrix}, \alpha \in \mathbb{R} \right\}.$$

Concomitantly we have

$$O_2(\mathbb{C}) \cong \left\{ g_z^+ = \begin{pmatrix} z & 0 \\ 0 & \frac{1}{z} \end{pmatrix}, z \in \mathbb{C}^* \right\} \cup \left\{ g_z^- = \begin{pmatrix} 0 & \frac{1}{z} \\ z & 0 \end{pmatrix}, z \in \mathbb{C}^* \right\}.$$

**Proposition 2.1.** [GSS88] The irreducible representations of $O_2(\mathbb{C})$ are indexed by $j \in \mathbb{Z}$ with $j \ge -1$. They consist of representations $\phi_j : O_2(\mathbb{C}) \to \text{GL}(\hat{V}_j)$ where

* for $j \in \{0, -1\}$, $\hat{V}_j \cong \mathbb{C}$ is one dimensional and

$$\phi_0(g_z^+) = (1), \phi_0(g_z^-) = (1); \quad \phi_{-1}(g_z^+) = (1), \phi_{-1}(g_z^-) = (-1);$$

* for $j \ge 1$, $\hat{V}_j \cong \mathbb{C}^2$ is two dimensional and

$$\phi_j(g_z^+) = \begin{pmatrix} z^j & 0 \\ 0 & z^{-j} \end{pmatrix} \quad \text{and} \quad \phi_j(g_z^-) = \begin{pmatrix} 0 & z^{-j} \\ z^j & 0 \end{pmatrix}$$

The real irreducible representations of $O_2(\mathbb{R})$ are obtained by restricting the above ones to the real vector space $\mathcal{V}_j = \{(y_{-1}, y_1) \in \mathbb{C}^2 | \bar{y}_1 = y_{-1}\}$ and $\{z \in \mathbb{C}^* | z \bar{z} = 1\}$.
We endow $\hat{\mathcal{V}}$ with coordinates $(t_1, ..., t_p, d_1, ..., d_q, y_1, y_{-1}, ..., y_r, y_{-r})$. We first proceed to give an algebraically independent generating set for the invariant field of the normal subgroup $SO_2(\mathbb{C})$ and then deduce the invariant field of the group $O_2(\mathbb{C})$.

**Proposition 2.2.** The following set of polynomials is algebraically independent and generates the invariant field $\mathbb{C}(\hat{\mathcal{V}})^{SO_2(\mathbb{C})}$.

$$ \mathfrak{e} := \left\{ \begin{array}{ll} t_j = t_j & 1 \le j \le p \\ \mathfrak{d}_j = d_j & 1 \le j \le q \\ \mathfrak{p}_j = y_j y_{-1}^{m_j}, & 1 \le j \le r \\ \mathfrak{q}_j = y_{-j} y_1^{m_j}, & 2 \le j \le r \end{array} \right\} $$

Note that the omitted polynomial $\mathfrak{q}_1$ would be equal to $\mathfrak{p}_1$. The proof of Proposition 2.2 rests on the two following lemmata. By Lemmas 2.3 and 3.5, polynomials in $\mathfrak{e}$ are invariant and separate orbits in the Zariski open set $\hat{\mathcal{U}}_1$. Thanks to Rosenlicht theorem [Ros56], [PV94, Lemma 2.1], they generate $\mathbb{C}(\hat{\mathcal{V}})^{SO_2(\mathbb{C})}$.

**Lemma 2.3.** Each polynomial in $\mathfrak{e}$ is invariant for $SO_2(\mathbb{C})$

*Proof.* Take $z \in \mathbb{C}^*$ to identify some $g_z^+ \in SO_2(\mathbb{C})$, and $v = (t_1, ..., t_p, d_1, ..., d_q, y_1, y_{-1}, ..., y_r, y_{-r}) \in \hat{\mathcal{V}}$.

* For all $j \le p$, $t_j(\rho(g_z^+)(v)) = t_j = t_j(v)$

* For all $j \le q$, $\mathfrak{d}_j(\rho(g_z^+)(v)) = \det(\rho(g_z^+))d_j = d_j = \mathfrak{d}_j(v)$

* For all $1 \le j \le r$, $\mathfrak{p}_j(\rho(g_z^+)(v)) = z^{m_j} y_j z^{-m_j} y_{-1}^{m_j} = \mathfrak{p}_j(v)$

* For all $2 \le j \le r$, $\mathfrak{q}_j(\rho(g_z^+)(a)) = z^{m_j} y_1^{m_j} z^{-m_j} y_{-m_j} = \mathfrak{q}_j(v)$

□

**Lemma 2.4.** The set $\mathfrak{e}$ separates orbits in $\hat{\mathcal{U}}_1 := \{v \in \hat{\mathcal{V}} | y_1 y_{-1} \neq 0\}$. That is for $u, v \in \hat{\mathcal{U}}_1$ we have

$$ \forall \mathfrak{f} \in \mathfrak{e}, \mathfrak{f}(u) = \mathfrak{f}(v) \Rightarrow \exists g \in SO_2(\mathbb{C}), \rho(g)(u) = v. $$

*Proof.* Take $u = (t_1, ..., t_p, d_1, ..., d_q, u_1, u_{-1}, ..., u_r, u_{-r})$ and $v = (t'_1, ..., t'_p, d'_1, ..., d'_q, v_1, v_{-1}, ..., v_r, v_{-r})$ points in $\hat{\mathcal{U}}_1$.

* For $1 \le j \le p$, $t_j(u) = t_j(v) \Rightarrow t_j = t'_j$.

Similarly, for $1 \le j \le q$, $\mathfrak{d}_j(u) = \mathfrak{d}_j(v) \Rightarrow d_j = d'_j$.

* Since $u_1 \neq 0$ and $v_1 \neq 0$, $z := \frac{v_1}{u_1} \in \mathbb{C}^*$ gives

$$ \begin{align*} \mathfrak{p}_1(u) = \mathfrak{p}_1(v) &\Rightarrow \mathfrak{p}_1(g_z^+(u)) = \mathfrak{p}_1(v) \\ &\Rightarrow \frac{zu_1u_{-1}}{v_1u_{-1}^z} = v_1v_{-1} \\ &\Rightarrow \frac{u_{-1}^z}{v_z} = v_1v_{-1} \\ &\Rightarrow \frac{z}{v} = v_{-1} \end{align*} $$

That is $\phi_1(g_z^+)(u_1, u_{-1}) = (v_1, v_{-1})$.

* Suppose furthermore that for some $2 \le j \le r$, $\mathfrak{p}_j(u) = \mathfrak{p}_j(v)$ and $\mathfrak{q}_j(u) = \mathfrak{q}_j(v)$. One then has:

$$ \begin{align*} \mathfrak{p}_j(u) = \mathfrak{p}_j(v) &\Rightarrow \mathfrak{p}_j(g_z^+(u)) = \mathfrak{p}_j(v) \\ &\Rightarrow z^{m_j} u_j \left(\frac{u_{-1}}{z}\right)^{m_j} = v_j v_{-1}^{m_j} \\ &\Rightarrow z^{m_j} u_j v_{-1}^{m_j} = v_j v_{-1}^{m_j} \\ &\Rightarrow z^{m_j} u_j = v_j \end{align*} $$

Similarly, $\mathfrak{q}(u) = \mathfrak{q}(v) \Rightarrow z^{-m_j} u_{-j} = v_{-j}$. Hence, $\phi_{m_j}(g_z^+)(u_j, u_{-j}) = (v_j, v_{-j})$.

We showed that if all polynomials in $\mathfrak{e}$ coincide on $u$ and $v$, $\rho(g_z^+)(u) = v$ and $u$ and $v$ lie on the same orbit. □
## 2.2 Invariants for SO₂($\mathbb{C}$)

Let $p, n, m_1, ..., m_r$ be positive integers with $1 \le p \le n$ and $m_1 = 1$. We consider the representation

$$ \rho: \mathrm{O}_2(\mathbb{C}) \to \mathrm{GL}(\hat{\mathcal{V}}) \text{ where } \hat{\mathcal{V}} = \bigoplus_{j=1}^{p} \hat{\mathcal{V}}_{-1} \oplus \bigoplus_{j=p+1}^{n} \hat{\mathcal{V}}_0 \oplus \bigoplus_{j=1}^{r} \hat{\mathcal{V}}_{m_j}. $$

We endow $\hat{\mathcal{V}}$ with coordinates $(v_1, ..., v_n, y_{-1}, y_1, ..., y_{-r}, y_r)$. We first proceed to give algebraically independent generators for the invariant field of the normal subgroup $\mathrm{SO}_2(\mathbb{C})$ and then deduce the invariants generating the invariant field of the group $\mathrm{O}_2(\mathbb{C})$.

**Proposition 2.2.** The following set of polynomials is algebraically independent and generates the invariant field $\mathbb{C}(\hat{\mathcal{V}})^{\mathrm{SO}_2(\mathbb{C})}$.

$$ \mathfrak{e} := \left\{ \begin{array}{ll} t_j = v_j & 1 \le j \le n \\ \mathfrak{p}_j = y_j y_{-1}^{m_j}, & 1 \le j \le r \\ \mathfrak{q}_j = y_{-j} y_1^{m_j}, & 2 \le j \le r \end{array} \right\} $$

Note that the omitted polynomial $\mathfrak{q}_1$ would be equal to $\mathfrak{p}_1$. The proof of Proposition 2.2 rests on the two following lemmas which show that polynomials in $\mathfrak{e}$ are invariant (Lemma 2.3) and that they separate generic orbits (Lemma 2.4). By a theorem due to Rosenlicht [Ros56, Section 1 Theorem 2], [PV94, Lemma 2.1], they generate $\mathbb{C}(\hat{\mathcal{V}})^{\mathrm{SO}_2(\mathbb{C})}$.

**Lemma 2.3.** Each polynomial in $\mathfrak{e}$ is invariant for $\mathrm{SO}_2(\mathbb{C})$.

*Proof.* Take $z \in \mathbb{C}^*$ to identify some $g_z^+ \in \mathrm{SO}_2(\mathbb{C})$, and $v = (v_1, ..., v_n, y_{-1}, y_1, ..., y_{-r}, y_r) \in \hat{\mathcal{V}}$.

* For all $1 \le j \le p$, $t_j(\rho(g_z^+)(v)) = \det(g_z^+)v_j = v_j = t_j(v)$

* For all $p+1 \le j \le n$, $t_j(\rho(g_z^+)(v)) = v_j = t_j(v)$

* For all $1 \le j \le r$, $\mathfrak{p}_j(\rho(g_z^+)(v)) = z^{m_j} y_j z^{-m_j} y_{-1}^{m_j} = \mathfrak{p}_j(v)$

* For all $2 \le j \le r$, $\mathfrak{q}_j(\rho(g_z^+)(v)) = z^{m_j} y_1^{m_j} z^{-m_j} y_{-m_j} = \mathfrak{q}_j(v)$. $\square$

**Lemma 2.4.** Consider the set $\mathcal{U} = \{v \in \hat{\mathcal{V}} | y_1 y_{-1} \neq 0\}$. For any $v, v' \in \mathcal{U}$, we have the following separation property:

$$ \forall f \in \mathfrak{e}, f(v) = f(v') \Rightarrow \exists g \in \mathrm{SO}_2(\mathbb{C}), \rho(g)(v') = v. $$

*Proof.* Take $v = (v_1, ..., v_n, y_{-1}, y_1, ..., y_{-r}, y_r)$ and $v' = (v'_1, ..., v'_n, y'_{-1}, y'_1, ..., y'_{-r}, y'_r)$ in $\mathcal{U}$ and define $z := \frac{y_1}{y_1} \in \mathbb{C}^*$.

* For $1 \le j \le n$, $t_j(v) = t_j(v') \Rightarrow v_j = v'_j$.

* We have $\phi_1(g_z^+)(y_{-1}, y_1) = (y'_{-1}, y'_1)$ since

$$ \begin{align*} \mathfrak{p}_1(v) = \mathfrak{p}_1(v') &\Rightarrow \mathfrak{p}_1(g_z^+(y)) = \mathfrak{p}_1(v') \\ &\Rightarrow z z^{-1} y_1 y_{-1} = y'_1 y'_{-1} \\ &\Rightarrow z y'_1 y_{-1} = y'_1 y'_{-1} \\ &\Rightarrow z y_{-1} = y'_{-1}. \end{align*} $$

* Furthermore for $2 \le j \le r$ we have:

$$ \begin{align*} \mathfrak{p}_j(v) = \mathfrak{p}_j(v') &\Rightarrow \mathfrak{p}_j(g_z^+(v)) = \mathfrak{p}_j(v') \\ &\Rightarrow z^{-m_j} y_j (z y_{-1})^{m_j} = y'_j (y'_{-1})^{m_j} \\ &\Rightarrow z^{-m_j} y_j (y'_{-1})^{m_j} = y'_j (y'_{-1})^{m_j} \\ &\Rightarrow z^{-m_j} y_j = y'_j \end{align*} $$

Similarly, $\mathfrak{q}_j(v) = \mathfrak{q}_j(v') \Rightarrow z^{m_j} y_{-j} = y'_{-j}$. Hence, $\phi_{m_j}(g_z^+)(y_{-j}, y_j) = (y'_{-j}, y'_j)$.
## 2.3 Invariants for $O_2(\mathbb{C})$

$SO_2(\mathbb{C})$ is a normal subgroup of $O_2(\mathbb{C})$ with $O_2(\mathbb{C})/SO_2(\mathbb{C}) = \{g_1^+, g_1^-\} \cong C_2$. There is thus a natural action of $C_2$ on $\mathbb{C}(\hat{V})^{SO_2(\mathbb{C})} = \mathbb{C}(\mathfrak{e})$ and

$$ \mathbb{C}(\hat{V})^{O_2(\mathbb{C})} = (\mathbb{C}(\hat{V})^{SO_2(\mathbb{C})})^{C_2} = \mathbb{C}(\mathfrak{e})^{C_2}. $$

The action of $C_2$ on $\mathfrak{e}$ consists of permutations and sign changes. It thus amounts to a linear action of $C_2$ on $\mathcal{W} := \mathrm{Vect}(f | f \in \mathfrak{e})$. We can thus deduce from Proposition 2.2 an algebraically independent generating invariants for $\mathbb{C}(\hat{V})^{O_2(\mathbb{C})}$ thanks to the following easy lemma.

**Lemma 2.5.** Consider $(\mathfrak{r}, \mathcal{W})$ the representation of $C_2 = \{1, -1\}$ on $\mathbb{C}^n$ where

$$ \mathfrak{r}(-1) = \begin{pmatrix} -I_k & 0 \\ 0 & I_{n-k} \end{pmatrix}. $$

If $\mathbb{C}^n$ is endowed with coordinates $(w_1, ..., w_k, w_{k+1}, ..., w_n)$ then $\{w_1w_i, 1 \le i \le k\} \cup \{w_j, k+1 \le j \le n\}$ is an algebraically independent set of polynomials generating $\mathbb{C}(\mathcal{W})^{C_2}$.

Indeed, one checks that this set of invariants separates orbits on $U_2 = \{w \in \mathcal{W} | w_1 \neq 0\}$.

**Proposition 2.6.** The following set of algebraically independent polynomials generates the field $\mathbb{C}(\hat{V})^{O_2(\mathbb{C})}$:

$$ \mathcal{E}_{\mathbb{C}} = \left\{ \begin{array}{l} T_j = \begin{array}{c} t_j, \quad 1 \le j \le p \end{array} \\ D_j = \begin{array}{c} d_j d_1, \quad 1 \le j \le q \end{array} \\ P_j = \begin{array}{c} y_j y_{-1}^{m_j} + y_j y_1^{m_j}, \quad 1 \le j \le r \end{array} \\ Q_j = \begin{array}{c} (y_j y_{-1}^{m_j} - y_{-j} y_1^{m_j}) d_1, \quad 2 \le j \le r \end{array} \end{array} \right\}. $$

*Proof.* The natural action of $C_2$ on $\mathbb{C}(\hat{V})^{SO_2(\mathbb{C})}$ is given by

$$ [g] \cdot \underbrace{P}_{\in C_2} := P \circ g. $$

Take $v = (t_1, ..., t_p, d_1, ..., d_q, y_1, y_{-1}, ..., y_r, y_{-r})$ a point in $\hat{V}$.

* For $1 \le j \le p$, $(g_1^- \cdot t_j)(v) = t_j(\rho(g_1^-)v) = \phi_0(g_1^-)t_j = t_j(v)$;

* For $1 \le j \le q$, $(g_1^- \cdot \mathfrak{d}_j)(v) = \mathfrak{d}_j(\rho(g_1^-)v) = \phi_{-1}(g_1^-)d_j = -\mathfrak{d}_j(v)$;

* For $1 \le j \le r$, $(g_1^- \cdot \mathfrak{p}_j)(v) = \mathfrak{p}_j(\rho(g_1^-)v) = y_{-j} y_1^{m_j} = \mathfrak{q}_j(v)$
and $(g_1^- \cdot \mathfrak{q}_j)(v) = \mathfrak{q}_j(\rho(g_1^-)v) = y_j y_{-1}^{m_j} = \mathfrak{p}_j(v)$.

Since polynomials in $\mathfrak{e}$ are linearly (and actually algebraically) independent, they form a basis of the vector space $\mathcal{W} := \mathrm{Vect}(f | f \in \mathfrak{e})$ and this vector space is invariant by the action of $C_2$ on $\mathfrak{e}$; it is a linear action.

The matrix representation of $g_1^-$ in the basis

$$ \mathfrak{e}' := \left\{ \begin{array}{l} t'_j = t_j, \quad 1 \le j \le p \\ \mathfrak{d}'_j = d_j, \quad 1 \le j \le q \\ \mathfrak{p}'_j = \mathfrak{p}_j + \mathfrak{q}_j, \quad 1 \le j \le r \\ \mathfrak{q}'_j = \mathfrak{p}_j - \mathfrak{q}_j, \quad 2 \le j \le r \end{array} \right\} \text{ of } \mathcal{W} \text{ is } M(g_1^-) = \begin{pmatrix} I_p & 0 & 0 & 0 \\ 0 & -I_q & 0 & 0 \\ 0 & 0 & I_r & 0 \\ 0 & 0 & 0 & -I_{r-1} \end{pmatrix}. $$
We showed that if all polynomials in $\mathfrak{e}$ coincide on $v$ and $v'$, $\rho(g_z^+)(v) = v'$, i.e., $v$ and $v'$ lie on the same orbit. $\square$

## 2.3 Invariants for $O_2(\mathbb{C})$ and $O_2(\mathbb{R})$

Since $O_2(\mathbb{C}) = \text{SO}_2(\mathbb{C}) \times \text{C}_2$, the normal subgroup $\text{C}_2$ acts naturally on $\mathbb{C}(\hat{\mathcal{V}})^{\text{SO}_2(\mathbb{C})} = \mathbb{C}(\mathfrak{e})$ and

$$ \mathbb{C}(\hat{\mathcal{V}})^{\text{O}_2(\mathbb{C})} = (\mathbb{C}(\hat{\mathcal{V}})^{\text{SO}_2(\mathbb{C})})^{\text{C}_2} = \mathbb{C}(\mathfrak{e})^{\text{C}_2}. $$

The action of $\text{C}_2$ on $\mathfrak{e}$ consists of permutations and sign changes. It thus amounts to a linear action of $\text{C}_2$ on $\mathcal{W} := \text{Vect}(f | f \in \mathfrak{e})$. We can thus deduce from Proposition 2.2 a set of algebraically independent invariants generating $\mathbb{C}(\hat{\mathcal{V}})^{\text{O}_2(\mathbb{C})}$ thanks to the following lemma.

**Lemma 2.5.** For a fixed $k \in \{1, \dots, n\}$, consider $(\mathfrak{r}, \mathcal{W})$ the representation of $\text{C}_2 = \{1, -1\}$ on $\mathcal{W} = \mathbb{C}^n$ where

$$ \mathfrak{r}(-1) = \begin{pmatrix} -I_k & 0 \\ 0 & I_{n-k} \end{pmatrix}. $$

If $\mathcal{W}$ is endowed with coordinates $(w_1, \dots, w_k, w_{k+1}, \dots, w_n)$ then $\{w_1w_i, 1 \le i \le k\} \cup \{w_j, k+1 \le j \le n\}$ is an algebraically independent set of polynomials generating $\mathbb{C}(\mathcal{W})^{\text{C}_2}$.

One can indeed check that this set of invariants separates orbits on $\{w \in \mathcal{W} | w_1 \neq 0\}$.

**Theorem 2.6.** The following set of algebraically independent polynomials generates the fields $\mathbb{C}(\hat{\mathcal{V}})^{\text{O}_2(\mathbb{C})}$ and $\mathbb{R}(\mathcal{V})^{\text{O}_2(\mathbb{R})}$:

$$ \mathcal{E} = \left\{ \begin{array}{l} D_j = \qquad v_j v_1, \qquad 1 \le j \le p \\ T_j = \qquad v_j, \qquad p+1 \le j \le n \\ R_j = \qquad y_j y_{-1}^{m_j} + y_{-j} y_1^{m_j}, \qquad 1 \le j \le r \\ I_j = \qquad i v_1 (y_j y_{-1}^{m_j} - y_{-j} y_1^{m_j}), \qquad 2 \le j \le r \end{array} \right\}. $$

*Proof.* The natural action of $\text{C}_2$ on $\mathbb{C}(\hat{\mathcal{V}})^{\text{SO}_2(\mathbb{C})}$ is given by

$$ \underbrace{[g]}_{\in \mathbb{C}_2} \cdot \underbrace{P}_{\in \mathbb{C}(\hat{\mathcal{V}})^{\text{SO}_2(\mathbb{C})}} := P \circ g. $$

Take $v = (v_1, \dots, v_n, y_{-1}, y_1, \dots, y_{-r}, y_r)$ a point in $\hat{\mathcal{V}}$.

* For $1 \le j \le p$, $(g_1^- \cdot t_j)(v) = t_j(\rho(g_1^-)v) = \det(g_1^-) v_j = -t_j(v)$;

* For $p+1 \le j \le n$, $(g_1^- \cdot t_j)(v) = t_j(\rho(g_1^-)v) = v_j = t_j(v)$;

* For $1 \le j \le r$, $(g_1^- \cdot \mathfrak{p}_j)(v) = \mathfrak{p}_j(\rho(g_1^-)v) = y_{-j} y_1^{m_j} = \mathfrak{q}_j(v)$ and $(g_1^- \cdot \mathfrak{q}_j)(v) = \mathfrak{q}_j(\rho(g_1^-)v) = y_j y_{-1}^{m_j} = \mathfrak{p}_j(v)$.

We notice that the linear space $\mathcal{W} := \text{Vect}(\mathfrak{e})$ is stable by the action of $\text{C}_2$. Furthermore, if we consider the basis of $\mathcal{W}$ given by

$$ \left\{ \begin{array}{ll} t_j, & 1 \le j \le n \\ \mathfrak{p}_j + \mathfrak{q}_j, & 1 \le j \le r \\ \mathfrak{p}_j - \mathfrak{q}_j, & 2 \le j \le r \end{array} \right\}, $$

then the matrix corresponding to the action of $g_1^-$ is

$$ \begin{pmatrix} -I_p & 0 & 0 & 0 \\ 0 & I_{n-p} & 0 & 0 \\ 0 & 0 & I_r & 0 \\ 0 & 0 & 0 & -I_{r-1} \end{pmatrix}. $$
By Lemma 2.5, the following is a generating set for $\mathbb{C}(\mathcal{W})_2^\mathbb{C}$:

$$ \mathcal{E}_{\mathbb{C}} := \left\{ \begin{array}{l} T_j = t'_j, \quad 1 \le j \le p \\ D_j = \mathfrak{d}'_j \mathfrak{d}'_1, \quad 1 \le j \le q \\ P_j = \mathfrak{p}'_j, \quad 1 \le j \le r \\ Q_j = \mathfrak{q}'_j \mathfrak{d}'_1, \quad 2 \le j \le r \end{array} \right\} $$

To check the algebraic independence of $\mathcal{E}_{\mathbb{C}}$, note that $\#\mathcal{E}_{\mathbb{C}} = p+q+2r-1 = \dim(\hat{\mathcal{V}}) - \dim(\mathcal{O}_2(\mathbb{C}))$. However, the transcendence degree of the field $\mathbb{C}(\hat{\mathcal{V}})^{\mathcal{O}_2(\mathbb{C})}$ is equal to $\dim(\hat{\mathcal{V}}) - \max_{v \in \mathcal{V}} (\dim(\mathcal{O}_2(\mathbb{C}) \cdot v))$ [PV94, Section 2.3]. Hence $\#\mathcal{E}_{\mathbb{C}} \le \deg_{tr}(\mathbb{C}(\hat{\mathcal{V}})^{\mathcal{O}_2(\mathbb{C})})$ while generating this field. Thus $\mathcal{E}_{\mathbb{C}}$ is algebraically independent. $\square$

The explicit expressions of the elements of $\mathcal{E}_{\mathbb{C}}$ allow us to determine where $\mathcal{E}_{\mathbb{C}}$ fails to separate orbits.

**Proposition 2.7.** Polynomials in $\mathcal{E}_{\mathbb{C}}$ separate the orbits of $\mathcal{O}_2(\mathbb{C})$ in $\hat{\mathcal{U}} = \{v \in \hat{\mathcal{V}} | d_1 y_1 y_{-1} \neq 0\}$.

## 2.4 Invariants for $\mathcal{O}_2(\mathbb{R})$

One observes that the restrictions of elements in $\mathcal{E}_{\mathbb{C}}$ on the real trace

$$ \mathcal{V} = \{ (t_1, \dots, t_p, d_1, \dots, d_q, y_1, y_{-1}, \dots, y_r, y_{-r}) \in \hat{\mathcal{V}} \mid t_1, \dots, t_p, d_1, \dots, d_q \in \mathbb{R}; \bar{y}_i = y_{-i}, 1 \le i \le n \} $$

of $\hat{\mathcal{V}}$ are either real or pure imaginary. One can thus determine generating invariants for $\mathbb{R}(\mathcal{V})^{\mathcal{O}_2(\mathbb{R})}$.

**Theorem 2.8.** The following set of algebraically independent polynomials generates both $\mathbb{R}(\mathcal{V})^{\mathcal{O}_2(\mathbb{R})}$ or $\mathbb{C}(\hat{\mathcal{V}})^{\mathcal{O}_2(\mathbb{C})}$.

$$ \mathcal{E} = \left\{ \begin{array}{l} T_j = t_j, \qquad 1 \le j \le p \\ D_j = d_i d_1, \qquad 1 \le j \le q \\ R_j = P_j = 2\Re(y_{-1}^{m_j} y_j), \qquad 1 \le j \le r \\ I_j = i Q_j = -2\Im(y_{-1}^{m_j} y_j) d_1, \qquad 2 \le j \le r \end{array} \right\} $$

*Proof.* On $\mathcal{V}$, $\mathfrak{p}_j = y_j y_{-1}^{m_j} = \bar{y}_{-j} \bar{y}_1^{m_j} = \bar{q}_j$. Hence, $\mathfrak{p}_j + \mathfrak{q}_j = 2\Re(\mathfrak{p}_j)$ and $\mathfrak{p}_j - \mathfrak{q}_j = 2i\Im(\mathfrak{p}_j)$. Thus $R_j$ and $I_j$ are real and they generate $\mathbb{R}(\mathcal{V})^{\mathcal{O}_2(\mathbb{R})}$ over $\mathbb{R}$ and $\mathbb{C}(\hat{\mathcal{V}})^{\mathcal{O}_2(\mathbb{C})}$ over $\mathbb{C}$. $\square$

Since we just multiplied polynomials by a constant, polynomials in $\mathcal{E}$ still separate orbits in

$$ \mathcal{U} = \{v \in \mathcal{V} | dy_1 y_{-1} \neq 0\}. $$

The polynomials of Theorem 2.8 can be translated into real coordinates through the base change $P^{-1}$. It involves an expansion by the Newton binomial which makes the complex i disappear. Nevertheless the expressions become rather involved as shown in the following example. This explains why we mostly work with complex coordinates throughout the article.

**Example 2.9.** For the 8-dimensional linear representation $S_T = 2\mathcal{V}_{-1} \oplus \mathcal{V}_1 \oplus \mathcal{V}_2 \oplus \mathcal{V}_3$, an algebraically independent generating set of $\mathbb{R}(S_T)^{\mathcal{O}_2(\mathbb{R})}$ is

$$ \mathcal{E}_T = \left\{ \begin{array}{l} D_1 = d_1^2 \\ D_2 = d_1 d_2 \\ R_1 = 2y_1 y_{-1} \\ R_2 = y_1^2 y_{-2} + y_{-1}^2 y_2 \\ R_3 = y_1^3 y_{-3} + y_{-1}^3 y_3 \\ I_2 = i d_1 (y_1^2 y_{-2} - y_{-1}^2 y_2) \\ I_3 = i d_1 (y_1^3 y_{-3} - y_{-1}^3 y_3) \end{array} \right\} = \left\{ \begin{array}{l} D_1 = d_1^2 \\ D_2 = d_1 d_2 \\ R_1 = 2(x_1^2 + x_{-1}^2) \\ R_2 = 4x_{-2}x_{-1}x_1 - 2x_{-1}^2x_2 + 2x_1^2x_2 \\ R_3 = -2x_{-3}x_{-1}^3 + 6x_{-3}x_{-1}x_1^2 - 6x_{-1}^2x_1x_3 + 2x_1^3x_3 \\ I_2 = 2d_1(x_{-2}x_{-1}^2 - x_{-2}x_1^2 + 2x_{-1}x_1x_2) \\ I_3 = 2d_1(3x_{-3}x_{-1}^2x_1 - x_{-3}x_1^3 - x_{-1}^3x_3 + 3x_{-1}x_1^2x_3) \end{array} \right\} $$
By Lemma 2.5, the following is a generating set for $\mathbb{C}(\mathcal{W})_2^\mathbb{C}$:

$$ \mathcal{E} = \left\{ \begin{array}{ll} D_j = \mathfrak{t}_1 \mathfrak{t}_j, & 1 \le j \le p \\ T_j = \mathfrak{t}_j, & p+1 \le j \le n \\ R_j = \mathfrak{p}_j + \mathfrak{q}_j, & 1 \le j \le r \\ I_j = \mathfrak{i} \mathfrak{t}_1 (\mathfrak{p}_j - \mathfrak{q}_j), & 2 \le j \le r \end{array} \right\} $$

Note that $\#\mathcal{E} = n + 2r - 1 = \dim(\hat{\mathcal{V}}) - \dim(\mathcal{O}_2(\mathbb{C}))$. However, the transcendence degree of the field $\mathbb{C}(\hat{\mathcal{V}})^{\mathcal{O}_2(\mathbb{C})}$ is equal to $\dim(\hat{\mathcal{V}}) - \max_{v \in \mathcal{V}} (\dim(\mathcal{O}_2(\mathbb{C}) \cdot v))$ [PV94, Section 2.3]. Hence $\#\mathcal{E} \le \text{tr. deg} (\mathbb{C}(\hat{\mathcal{V}})^{\mathcal{O}_2(\mathbb{C})})$ while generating this field. Thus $\mathcal{E}$ is algebraically independent.

The coefficient i in $I_j$ is added to obtain real polynomials. Indeed, recall that on the real trace

$$ \mathcal{V} = \{ v \in \hat{\mathcal{V}} | v_1, \dots, v_n \in \mathbb{R}; \bar{y}_i = y_{-i}, 1 \le i \le n \} $$

we have $\mathfrak{p}_j = y_j y_{-1}^{m_j} = \bar{y}_{-j} \bar{y}_1^{m_j} = \bar{q}_j$. Hence, $\mathfrak{p}_j + \mathfrak{q}_j = 2\Re(\mathfrak{p}_j)$ and $\mathfrak{p}_j - \mathfrak{q}_j = 2i\Im(\mathfrak{p}_j)$. Thus the restrictions of the polynomials of $\mathcal{E}$ to $\mathcal{V}$ are real. Since $\mathcal{O}_2(\mathbb{R})$ is Zariski dense in $\mathcal{O}_2(\mathbb{C})$, they also generate $\mathbb{R}(\mathcal{V})^{\mathcal{O}_2(\mathbb{R})}$ over $\mathbb{R}$. $\square$

The polynomials of $\mathcal{E}$ can be translated into real coordinates through the change of basis $P^{-1}$ defined in Section 2.1. It involves an expansion according to Newton binomial formula which makes the complex i disappear. The expressions become rather involved though, as shown in the following example. This explains why we mostly work with complex coordinates throughout the article.

**Example 2.7.** For the 8-dimensional linear representation $S_T = 2\mathcal{V}_{-1} \oplus \mathcal{V}_1 \oplus \mathcal{V}_2 \oplus \mathcal{V}_3$, an algebraically independent generating set of $\mathbb{R}(S_T)^{\mathcal{O}_2(\mathbb{R})}$ is

$$ \mathcal{E}_T = \left\{ \begin{array}{l} D_1 = v_1^2 \\ D_2 = v_1 v_2 \\ R_1 = 2y_1 y_{-1} \\ R_2 = y_1^2 y_{-2} + y_{-1}^2 y_2 \\ R_3 = y_1^3 y_{-3} + y_{-1}^3 y_3 \\ I_2 = \mathrm{id}_1 (y_1^2 y_{-2} - y_{-1}^2 y_2) \\ I_3 = \mathrm{id}_1 (y_1^3 y_{-3} - y_{-1}^3 y_3) \end{array} \right\} = \left\{ \begin{array}{l} D_1 = v_1^2 \\ D_2 = v_1 v_2 \\ R_1 = x_1^2 + x_{-1}^2 \\ R_2 = \frac{\sqrt{2}}{2} (2x_{-2}x_{-1}x_1 - x_{-1}^2x_2 + x_1^2x_2) \\ R_3 = -\frac{1}{2}x_{-3}x_{-1}^3 + \frac{3}{2}x_{-3}x_{-1}x_1^2 - \frac{3}{2}x_{-1}^2x_1x_3 + \frac{1}{2}x_1^3x_3 \\ I_2 = \frac{\sqrt{2}}{2}d_1 (x_{-2}x_{-1}^2 - x_{-2}x_1^2 + 2x_{-1}x_1x_2) \\ I_3 = -\frac{1}{2}d_1 (3x_{-3}x_{-1}^2x_1 - x_{-3}x_1^3 - x_{-1}^3x_3 + 3x_{-1}x_1^2x_3) \end{array} \right\} $$

### 3 Invariants for $\mathrm{SO}_3(\mathbb{R})$

We now consider representations of $\mathrm{SO}_3(\mathbb{R})$ on

$$ \mathcal{V} = \mathbb{R}^3 \oplus \mathcal{H}, $$

where $\mathcal{H}$ is an arbitrary representation of $\mathrm{SO}_3(\mathbb{R})$. We introduce the Seshadri slice Lemma and identify a slice $(S, N)$ where $S = \mathbb{R} \oplus \mathcal{H}$ and $N = \mathcal{O}_2(\mathbb{R})$. From the generating set of $\mathbb{R}(S)^N$ computed in the previous section, we derive an algebraically independent generating set for $\mathbb{R}(\mathcal{V})^{\mathrm{SO}_3(\mathbb{R})}$. We show how to evaluate these invariants thanks to Wigner matrices. We shall furthermore solve the inverse problem: given values for these invariants provide, firstly, conditions for the existence of a real orbit where the invariants have these values and, secondly, an element in this orbit if it exists.
### 3 Invariants for SO₃(R)

In this section we consider the representation of SO₃(R) on

$$ \mathcal{H} = \mathcal{H}_1 \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} $$

where $\mathcal{H}_d$, for $d \in \mathbb{N}$, are the irreducible representations of SO₃(R), where $\mathcal{H}_1 \cong \mathbb{R}^3$. We shall characterize an algebraically independent set of rational invariants that generates the invariant field $\mathbb{R}(\mathcal{H})^{SO_3(\mathbb{R})}$. We make clear where these invariants have singularities, on which open set they separate orbits and how to evaluate them. We shall furthermore solve the inverse problem: given values of for these invariants provide firstly conditions for the existence of a (real) orbit where the invariants have these values and secondly an element in this orbit if it exists. All these results rest on a constructive use of Seshadri slice lemma. In the present case this latter asserts that the restriction to $\mathcal{S} = \mathbb{R} \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j}$ provides an isomorphism between $\mathbb{R}(\mathcal{H})^{SO_3(\mathbb{R})}$ and $\mathbb{R}(\mathcal{S})^{O_2(\mathbb{R})}$. We can therefore apply the results of previous section to characterize the invariants of SO₃(R) on $\mathcal{H}$.

In Section 3.1 $\mathcal{H}_d$ is made explicit as the space of harmonic ternary forms of degree $d$. The spherical harmonics form bases that decompose the $\mathcal{H}_d$ into irreducible representations of $O_2(\mathbb{R})$. In Section 3.2 we recall Seshadri's lemma and exhibit a slice $\mathcal{S}$ for $\mathcal{H}$. In Section 3.3 we characterize an algebraically independent set of invariants that generates $\mathbb{R}(\mathcal{H})^{SO_3(\mathbb{R})}$. They are determined by their restriction to $\mathcal{S}$. In Section 3.4 and 3.6 we show how this is sufficient to evaluate these SO₃(R)-invariants on $\mathcal{H} \setminus \mathcal{Q}$, where $\mathcal{Q}$ is the variety of an invariant quadratic polynomial, solve the inverse problem.

#### 3.1 Representations of SO₃(R)

The irreducible representations $\mathcal{H}_d$, $d \in \mathbb{N}$, of SO₃(R) are given explicitly in terms of spaces of homogeneous harmonic polynomials of degree $d$. The spherical harmonics form bases for these that decompose them into irreducible $O_2(\mathbb{R})$ representations. In this section we recall these notions and introduce the notations we shall use.

$\mathcal{H}_d$ is the vector space of degree $d$ real harmonic homogeneous polynomials on $\mathbb{R}^3$:

$$ \mathcal{H}_d := \left\{ P \in \mathbb{R}[U, V, W]_d \mid \frac{\partial^2 P}{\partial U^2} + \frac{\partial^2 P}{\partial V^2} + \frac{\partial^2 P}{\partial W^2} = 0 \right\}. $$

The standard action of SO₃(R) on $\mathbb{R}^3$ induces, by composition, a linear action on the polynomial ring: $\forall P \in \mathbb{R}[U, V, W], g \in SO_3(\mathbb{R}), g \cdot P = P \circ g^{-1}$. This action stabilises the vector space $\mathcal{H}_d$ for any $d \in \mathbb{N}$. It thus induces the representations $\rho_d: SO_3(\mathbb{R}) \to GL(\mathcal{H}_d)$ for $d \in \mathbb{N}$. They are the only irreducible representations of SO₃(R) [GSS88]. Among them, $(\rho_0, \mathcal{H}_0)$ is the trivial representation and $(\rho_1, \mathcal{H}_1)$ is the equivalent to the standard representation on $\mathbb{R}^3$. The class of representations we consider are thus all the representations where the multiplicity of $(\mathcal{H}_1, \rho_1)$ is strictly positive. This is specific and yet covers a number of relevant cases.

The spherical harmonics form bases of the spaces $\mathcal{H}_d$. They are homogeneous polynomials in $\mathbb{R}[U, V, W]$ that decompose the spaces $\mathcal{H}_d$ into irreducible representations of $O_2(\mathbb{R})$. Their expressions are nontrivial better known in spherical coordinates than in Cartesian coordinates. We thus introduce spherical coordinates $(r, \theta, \varphi)$ in $\mathbb{R}^3$ so that

$$ u = r \cos(\theta), \quad v = r \sin(\theta) \cos(\phi), \quad w = r \sin(\theta) \sin(\phi). $$

**Theorem 3.1.** [Dai13] For $1 \le m \le d$ note $P_d^m$ the $(d, m)^{th}$ associated Legendre polynomial. The following
## 3.1 Representations of SO$_3(\mathbb{R})$

We first classify the irreducible representations of SO$_3(\mathbb{R})$. They are given in terms of spaces of homogeneous harmonic polynomials of degree $d$. We next introduce the bases of spherical harmonics, which allow to decompose the spaces of harmonic polynomials into irreducible representations of the subgroup $O_2(\mathbb{R})$.

**Definition 3.1.** The vector space of degree $d$ harmonic homogeneous polynomials on $\mathbb{R}^3$ is

$$ \mathcal{H}_d := \left\{ P \in \mathbb{R}[U, V, W]_d \;\middle|\; \frac{\partial^2 P}{\partial U^2} + \frac{\partial^2 P}{\partial V^2} + \frac{\partial^2 P}{\partial W^2} = 0 \right\}. $$

The standard action of SO$_3(\mathbb{R})$ on $\mathbb{R}^3$ induces, by composition, a linear action on the polynomial ring:

$$ \forall P \in \mathbb{R}[U, V, W], g \in \text{SO}_3(\mathbb{R}), g \cdot P = P \circ g^{-1}. $$

This action stabilises the vector spaces $\mathcal{H}_d$. It thus induces the representations $\rho_d : \text{SO}_3(\mathbb{R}) \to \text{GL}(\mathcal{H}_d)$ which are the only irreducible representations of SO$_3(\mathbb{R})$ [GSS88]. Among them, $(\rho_0, \mathcal{H}_0)$ is the trivial representation and $\mathcal{H}_1$ is equivalent to the standard representation on $\mathbb{R}^3$. The class of representations we consider are thus all the representations containing $(\mathcal{H}_1, \rho_1)$.

The spherical harmonics form bases of the spaces $\mathcal{H}_d$. They are homogeneous polynomials in $\mathbb{R}[U, V, W]_d$ that decompose the spaces $\mathcal{H}_d$ into irreducible representations of $O_2(\mathbb{R})$. Their expressions are nonetheless better known in spherical coordinates than in Cartesian coordinates. We thus introduce spherical coordinates $(r, \theta, \varphi)$ in $\mathcal{H}_1$:

$$ u = r \sin(\theta) \cos(\varphi) $$

$$ v = r \sin(\theta) \sin(\varphi) $$

$$ w = r \cos(\theta) $$

<figure><img src="image_3.png" /></figure>

**Theorem 3.2.** [Wig59, p. 15.3] For $0 \le m \le d$, note $c_{m,d} = \sqrt{\frac{(2d+1)(d-m)!}{4\pi(d+m)!}}$ a normalisation constant and $P_d^m = (1-t^2)^{\frac{m}{2}} \frac{1}{2^d d!} \frac{\partial^{d+m}}{\partial(t)^{d+m}} (t^2-1)^d$ is the $(d,m)^{th}$ associated Legendre polynomial. A basis for $\hat{\mathcal{H}}_d$ is given by $\{Y_d^{-m}, \dots, Y_d^d\} \subset \mathbb{C}[U, V, W]_d$ where

$$ Y_d^{-m}(r \sin(\theta) \cos(\varphi), r \sin(\theta) \sin(\varphi), r \cos(\theta)) = i^{-d} c_{m,d} r^d e^{-im\varphi} P_{m,d}(\cos(\theta)), \text{ for } 1 \le m \le d, $$

$$ Y_d^0(r \sin(\theta) \cos(\varphi), r \sin(\theta) \sin(\varphi), r \cos(\theta)) = c_d^0 r^d P_d^0(\cos(\theta)), $$

$$ Y_d^m(r \sin(\theta) \cos(\varphi), r \sin(\theta) \sin(\varphi), r \cos(\theta)) = i^d c_{m,d} r^d e^{im\varphi} P_{m,d}(\cos(\theta)), \text{ for } 1 \le m \le d. $$

We slightly modified the basis of [Wig59] to fit better the irreducible representations of $O_2(\mathbb{R})$ defined in Section 2.1. More precisely, we multiplied each $Y_d^m$ by $i^d(-1)^m$ and each $Y_d^{-m}$ by $i^{-d}$. With the change of basis $P$ introduced in Section 2.1 we obtain the real spherical harmonics:

$$ X_d^{-m} = \frac{1}{\sqrt{2}} (Y_d^{-m} + Y_d^m), \quad X_d^0 = Y_d^0, \quad X_d^m = \frac{i}{\sqrt{2}} (Y_d^{-m} - Y_d^m) $$

so that

$$ X_d^{-m}(r \sin(\theta) \cos(\varphi), r \sin(\theta) \sin(\varphi), r \cos(\theta)) = \sqrt{2} c_{m,d} r^d \cos(m\varphi + \frac{d\pi}{2}) P_{m,d}(\cos(\theta)), $$

$$ X_d^0(r \sin(\theta) \cos(\varphi), r \sin(\theta) \sin(\varphi), r \cos(\theta)) = c_d^0 r^d P_d^0(\cos(\theta)), $$

$$ X_d^m(r \sin(\theta) \cos(\varphi), r \sin(\theta) \sin(\varphi), r \cos(\theta)) = \sqrt{2} c_{m,d} r^d \sin(m\varphi + \frac{d\pi}{2}) P_{m,d}(\cos(\theta)). $$
set of functions of $(r, \theta, \varphi)$ is a polynomial basis for $\hat{\mathcal{H}}_d$:

$$ \hat{\mathcal{B}}_d := \left\{ \begin{array}{ll} Y_d^m = & i^{m+d} r^d e^{im\varphi} P_d^m(\cos(\theta)), & 1 \le m \le d \\ Y_d^{-m} = & i^{-m-d} r^d e^{-im\varphi} P_d^m(\cos(\theta)), & 1 \le m \le d \\ Y_d^0 = & r^d P_d^0(\cos(\theta)) & \end{array} \right\} $$

With respect to this basis, we put $h_j = (y_{-d_j, j}, y_{-d_j+1, j}, \dots, y_{d_j-1, j}, y_{d_j, j})$ coordinates in $\hat{\mathcal{H}}_{d_j}$, $h = (h_1, \dots, h_n)$ coordinates in $\bigoplus_{j=1}^n \hat{\mathcal{H}}_{d_j}$. When $\theta, \phi$ and $r$ are in $\mathbb{R}$ we have $Y_d^{-m} = \bar{Y}_d^m$, for any $0 \le m \le d$. With the change $P^{-1}$ of basis introduced in Section 2.1 we obtain the real harmonic basis:

$$ \begin{aligned} X_d^m &= \frac{1}{2}(Y_d^m + Y_d^{-m}) = r^d \cos(m\varphi + (m+d)\frac{\pi}{2}) P_d^m(\cos(\theta)) \\ X_d^{-m} &= \frac{1}{2i}(Y_d^m - Y_d^{-m}) = r^d \sin(m\varphi + (m+d)\frac{\pi}{2}) P_d^m(\cos(\theta)) \end{aligned} $$

Hence, natural real coordinates can be defined on $\mathcal{H}_{d_j}$ with $h_j = (x_{-d_j, j}, \dots, x_{d_j, j})$.

**Example 3.2.** The vector space $\mathcal{H}_1 = \mathbb{R}[U, V, W]_1$ is endowed with the real basis $\mathcal{B}_1 = \{X_1^1 = -W, X_1^0 = U, X_1^{-1} = V\}$.

The real vector space $\mathcal{H}_3$ is endowed with the basis $\mathcal{B}_3 = \{X_3^m, -3 \le m \le 3\}$. We can compute their expressions in Cartesian coordinates up to a normalisation constant:

$$ \mathcal{B}_3 = \left\{ \begin{array}{ll} X_3^3 = 3W^2V - V^3 & X_3^2 = UVW \\ X_3^0 = U(2U^2 - 3V^2 - 3W^2) & X_3^{-2} = U(W^2 - V^2) \\ X_3^{-3} = 3WV^2 - W^3 & X_3^{-1} = -V(4U^2 - V^2 - W^2) \end{array} \right\} $$

**Proposition 3.3.** Write $n := p+q$ such that $d_j$ is even for $1 \le j \le p$, and odd for $p+1 \le j \le p+q$. For the restriction to $O_2(\mathbb{C})$, subgroup of $SO_3(\mathbb{R})$, the representation on $\bigoplus_{j=1}^n \hat{\mathcal{H}}_{d_j}$ is isomorphic to

$$ \bigoplus_{j=1}^{p} \hat{\mathcal{V}}_{0} \oplus \bigoplus_{j=1}^{q} \hat{\mathcal{V}}_{-1} \oplus \bigoplus_{j=1}^{n} \bigoplus_{m=1}^{d_j} \hat{\mathcal{V}}_{m} $$

where the $\hat{\mathcal{V}}_m$ are the irreducible representations of $O_2(\mathbb{R})$ introduced in Section 2.1.

*Proof.* Consider the subgroup of rotations of axis $U$. Its standard representation writes in spherical coordinates:

$$ \forall \alpha \in \mathbb{R}, \forall r, \theta, \varphi, \left\{ \begin{array}{l} g_{e^{i\alpha}}^{+}(r, \theta, \varphi) = (r, \theta, \varphi + \alpha) \\ g_{e^{-i\alpha}}^{\bar{}}(r, \theta, \varphi) = (r, \pi - \theta, -\varphi - \alpha) \end{array} \right. $$

Consider the induced action on harmonic polynomials. Let $d$ be a positive integer and $1 \le m \le d$. For all $(r, \theta, \varphi) \in \mathbb{R}^3$, and $\alpha$ real,

$$ \begin{aligned} \rho_d(g_{e^{i\alpha}}^{+}) (Y_d^m(r, \theta, \varphi)) &= Y_d^m(g_{e^{-i\alpha}}^{+}(r, \theta, \varphi)) \\ &= Y_d^m(r, \theta, \varphi - \alpha) \\ &= i^{m+d} r^d e^{im(\varphi+\alpha)} P_d^m(\cos(\theta)) \\ &= e^{-im\alpha} Y_d^m(r, \theta, \varphi) \end{aligned} $$

$$ \begin{aligned} \rho_d(g_{e^{i\alpha}}^{-}) (Y_d^m(r, \theta, \varphi)) &= Y_d^m(g_{e^{-i\alpha}}^{-}(r, \theta, \varphi)) \\ &= Y_d^m(r, \pi - \theta, -\varphi + \alpha) \\ &= i^{m+d} r^d e^{-im(\varphi-\alpha)} P_d^m(-\cos(\theta)) \end{aligned} $$
**Example 3.3.** The bases of real spherical harmonics for $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ are:

$$ \mathcal{B}_1 = \left\{ X_1^{-1} = -\sqrt{\frac{3}{4\pi}} V, X_1^0 = \sqrt{\frac{3}{4\pi}} W, X_1^1 = \sqrt{\frac{3}{4\pi}} U \right\}, $$

$$ \mathcal{B}_2 = \left\{ \begin{array}{l} X_2^{-2} = \frac{\sqrt{15}}{4\sqrt{\pi}} (V^2 - U^2), \quad X_2^{-1} = -\frac{\sqrt{15}}{2\sqrt{\pi}} UW, \quad X_2^0 = \frac{\sqrt{5}}{4\sqrt{\pi}} (2W^2 - U^2 - V^2), \\ X_2^1 = -\frac{\sqrt{15}}{2\sqrt{\pi}} VW, \quad X_2^2 = -\frac{\sqrt{15}}{2\sqrt{\pi}} UV \end{array} \right\}, $$

$$ \mathcal{B}_3 = \left\{ \begin{array}{l} X_3^{-3} = \frac{\sqrt{70}}{8\sqrt{\pi}} (3U^2 - V^2) V, \quad X_3^{-2} = \frac{\sqrt{105}}{2\sqrt{\pi}} UVW, \quad X_3^{-1} = \frac{\sqrt{42}}{8\sqrt{\pi}} (4W^2 - U^2 - V^2) V, \\ X_3^0 = \frac{\sqrt{7}}{4\sqrt{\pi}} (2W^2 - 3U^2 - 3V^2) W, \\ X_3^1 = \frac{\sqrt{42}}{8\sqrt{\pi}} (U^2 + V^2 - 4W^2) U, \quad X_3^2 = \frac{\sqrt{105}}{4\sqrt{\pi}} (V^2 - U^2) W, \quad X_3^3 = \frac{\sqrt{70}}{8\sqrt{\pi}} U (3V^2 - U^2) \end{array} \right\}. $$

With respect to these bases, we put $(y_{-d_j,j}, y_{-d_j+1,j}, \dots, y_{d_j-1,j}, y_{d_j,j})$ coordinates in $\hat{\mathcal{H}}_{d_j}$, and $(x_{-d_j,j}, \dots, x_{d_j,j})$ coordinates in $\mathcal{H}_{d_j}$. Spherical harmonics fit very well to the restricted representation of $\mathcal{O}_2(\mathbb{C})$. To show it, we introduce Euler angles:

**Definition 3.4.** A rotation $g \in \mathcal{SO}_3(\mathbb{R})$ is determined by Euler angles $(\alpha, \beta, \gamma)$ where $\alpha \in [0, 2\pi[, \beta \in [0, \pi]$ and $\gamma \in [0, 2\pi[$. It is the composition of rotations of axes $w, v, w$ and angles $(\alpha, \beta, \gamma)$:

$$ g(\alpha, \beta, \gamma) := \begin{pmatrix} \cos(\gamma) & -\sin(\gamma) & 0 \\ \sin(\gamma) & \cos(\gamma) & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \cos(\beta) & 0 & \sin(\beta) \\ 0 & 1 & 0 \\ -\sin(\beta) & 0 & \cos(\beta) \end{pmatrix} \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) & 0 \\ \sin(\alpha) & \cos(\alpha) & 0 \\ 0 & 0 & 1 \end{pmatrix} $$

**Proposition 3.5.** Let $d_1, \dots, d_n$ be $n \ge 1$ non negative integers, sorted so that the first $p$ are odd, and the others are even. The representation of $\mathcal{O}_2(\mathbb{R})$ as a subgroup of $\mathcal{SO}_3(\mathbb{R})$ on $\bigoplus_{j=1}^n \mathcal{H}_{d_j}$ is isomorphic to

$$ \bigoplus_{j=1}^{p} \mathcal{V}_{-1} \oplus \bigoplus_{j=p+1}^{n} \mathcal{V}_{0} \oplus \bigoplus_{j=1}^{n} \bigoplus_{m=1}^{d_j} \mathcal{V}_{m}. $$

*Proof.* Consider the embedding $\mathcal{O}_2(\mathbb{R}) \hookrightarrow \mathcal{SO}_3(\mathbb{R})$ given by

$$ \forall \alpha \in \mathbb{R}, \left\{ \begin{array}{l} g_{e^{i\alpha}}^{+} \rightarrow g(\alpha, 0, 0) \\ g_{e^{i\alpha}}^{-} \rightarrow g(\alpha, \pi, 0) \end{array} \right. $$

It normalises the $W$-axis and its action on spherical coordinates $(r, \theta, \varphi)$ is

$$ \left\{ \begin{array}{l} g_{e^{i\alpha}}^{+}(r, \theta, \varphi) = (r, \theta, \varphi + \alpha). \\ g_{e^{i\alpha}}^{-}(r, \theta, \varphi) = (r, \pi - \theta, \pi - \varphi - \alpha). \end{array} \right. $$

We compute the induced action on $Y_d^m$ for $d \ge 1$ and $1 \le m \le d$. For all $(r, \theta, \varphi) \in \mathbb{R}^3$, and $\alpha$ real,

$$ \begin{align*} \rho_d(g_{e^{i\alpha}}^{+}) (Y_d^m(r, \theta, \varphi)) &= Y_d^m(g_{e^{-i\alpha}}^{+}(r, \theta, \varphi)) \\ &= Y_d^m(r, \theta, \varphi - \alpha) \\ &= i^d c_{m,d} r^d e^{im(\varphi - \alpha)} P_{m,d}(\cos(\theta)) \\ &= e^{-im\alpha} Y_d^m(r, \theta, \varphi) \end{align*} $$

$$ \begin{align*} \rho_d(g_{e^{i\alpha}}^{-}) (Y_d^m(r, \theta, \varphi)) &= Y_d^m(g_{e^{-i\alpha}}^{-}(r, \theta, \varphi)) \\ &= Y_d^m(r, \pi - \theta, \pi - \varphi - \alpha) \\ &= i^d c_{m,d} r^d e^{im(\pi - \varphi - \alpha)} P_{m,d}(-\cos(\theta)). \end{align*} $$
Using Legendre associated polynomials parity [Ami11] $P_d^m(-t) = (-1)^{d+m}P_d^m(t)$ we obtain

$$
\begin{align*}
\rho_d(g_{e^{i\alpha}}^{-}) (Y_d^m(r, \theta, \varphi)) &= (-1)^{d+m} i^{m+d} r^d e^{-im(\varphi-\alpha)} P_d^m(\cos(\theta)) \\
&= i^{-d-m} r^d e^{-im(\varphi-\alpha)} P_d^m(\cos(\theta)) \\
\rho_d(g_{e^{i\alpha}}^{-}) (Y_d^m) &= e^{im\alpha} Y_d^{-m}
\end{align*}
$$

Thus, the $O_2(\mathbb{C})$ action on $\mathrm{Vect}(Y_d^m, Y_d^{-m})$ is isomorphic to $\hat{\mathcal{V}}_m$. Meanwhile for $m=0$, the corresponding

$$
\text{action on } \mathrm{Vect}(Y_d^0) \text{ is } \left\{ \begin{array}{ll} \phi_{-1} & \text{when } d \text{ is odd.} \\ \phi_0 & \text{when } d \text{ is even.} \end{array} \right. \quad \text{We have } \rho_j \sim \left\{ \begin{array}{ll} \phi_{-1} \oplus \bigoplus_{m=1}^{d_j} \phi_m & \text{when } d_j \text{ is odd.} \\ \phi_0 \oplus \bigoplus_{m=1}^{d_j} \phi_m & \text{when } d_j \text{ is even.} \end{array} \right. \quad \square
$$

## 3.2 Seshadri slice

In this section we identify a Seshadi slice $(S, N)$, where $N \cong O_2(\mathbb{R})$, for $\mathcal{H} = \mathcal{H}_1 \oplus \bigoplus_{j=1}^n \mathcal{H}_{d_j}$ that establishes an isomorphism between $\mathbb{R}(\mathcal{H})^{SO_3(\mathbb{R})}$ and $\mathbb{R}(S)^N$.

Seshadri slice lemma is instrumental in proving the rationality of the invariant field of some algebraic group actions [CS05]. And indeed, in Section 3.3 we make explicit a set of algebraically independent polynomial invariants generating $\mathbb{R}(S)^N$; $\mathbb{R}(S)^N$ is thus rational and so is therefore $\mathbb{R}(\mathcal{H})^{SO_3(\mathbb{R})}$. We shall nonetheless go further. In Section 3.4 we show how to evaluate the corresponding invariants of $\mathbb{R}(\mathcal{H})^{SO_3(\mathbb{R})}$ at any point in $\mathcal{H}$. The evaluation process also provides a way to give the explicit expressions of these invariants after some symbolic manipulations.

**Theorem 3.4.** Seshadri slice lemma [CS05, Theorem 3.1]. Let $G$ be a real algebraic group acting on a vector space $\mathcal{X}$. Let $S \subset \mathcal{X}$ be a subvector space and $N < G$ the normalizer of $S$ in $G$. Assume that:

* The closure of $G \times S$ is the whole of $\mathcal{X}$.

* There is an non empty $N$-stable Zariski open subset $\mathcal{Z} \subset S$ such that for all $g \in \hat{G}$ and $s \in \hat{Z}$ satisfying $g(s) \in \hat{Z}$, there exists $h \in \hat{N}$ such that $h(s) = g(s)$.

Then, the restriction to $\mathcal{Z}$ of rational functions on $\mathcal{X}$ induces a field isomorphism $\mathbb{R}(\mathcal{X})^G \cong \mathbb{R}(S)^N$.

**Definition 3.5.** Such a pair $(S, N)$ is called a Seshadri slice for $G \curvearrowleft \mathcal{X}$.

For $P \in \mathbb{R}(S)^N$ there thus exists a unique $\tilde{P} \in \mathbb{R}(\mathcal{X})^G$ such that the restriction of $\tilde{P}$ to $\mathcal{Z}$ equals $P$; We shall write $\tilde{P}|_{\mathcal{Z}} = P$.

**Proposition 3.6.** Consider the subspace $S_1 = \{\lambda U, \lambda \in \mathbb{R}\} \subset \mathcal{H}_1$ and

$$S := S_1 \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} \subset \mathcal{H}$$

Then, $(S, N)$ is a Seshadri slice with normalizer $N \cong O_2(\mathbb{R})$

$$N = \left\{ \begin{pmatrix} \det(g) & 0 & 0 \\ 0 & 0 & g \end{pmatrix}, g \in O_2(\mathbb{R}) \right\} \subset \mathrm{SO}_3(\mathbb{R})$$

<figure><img src="image_2.png" /></figure>

The normalizer $N$ of $S$ thus consists of the rotations in $\mathbb{R}^3$ that leave the $U$ axis invariant. That includes the rotations around the $U$ axis and the rotations by $\pi$ around any axis in the $(V, W)$ plane.
But for any $1 \le m \le d$ we have $P_{m,d}(-t) = (-1)^{d+m} P_{m,d}(t)$. We thus obtain

$$
\begin{align*}
\rho_d(g_{e^{i\alpha}}^{-}) (Y_d^m(r, \theta, \varphi))
  &= i^d c_{m,d} r^d (-1)^m e^{im(-\varphi-\alpha)} (-1)^{d+m} P_{m,d}(\cos(\theta)) \\
  &= i^{-d} c_{m,d} r^d e^{im(-\varphi-\alpha)} P_{m,d}(\cos(\theta)) \\
  &= e^{-im\alpha} Y_d^{-m}(r, \theta, \varphi).
\end{align*}
$$

Similarly we check that

$$
\left\{
\begin{array}{l}
\rho_d(g_{e^{i\alpha}}^{+}) (Y_d^{-m}) = e^{im\alpha} Y_d^{-m}, \\
\rho_d(g_{e^{i\alpha}}^{-}) (Y_d^{-m}) = e^{im\alpha} Y_d^{m}.
\end{array}
\right.
$$

Thus, the representation of $O_2(\mathbb{C})$ on $\mathrm{Vect}(Y_d^{-m}, Y_d^m)$ is isomorphic to $\hat{\mathcal{V}}_m$. Meanwhile for $m=0$, the corresponding action on $\mathrm{Vect}(Y_d^0)$ is $\phi_{-1}$ when $d$ is odd and $\phi_0$ when $d$ is even. $\square$

3.2 Seshadri slice

In this section we introduce Seshadri slice lemma and identify a Seshadi slice ($\mathcal{S}$, $\mathbb{N}$) for the representation of $\mathrm{SO}_3(\mathbb{R})$ on $\mathcal{V} = \mathbb{R}^3 \oplus \mathcal{H}$. This establishes an isomorphism between $\mathbb{R}(\mathcal{V})^{\mathrm{SO}_3(\mathbb{R})}$ and $\mathbb{R}(\mathcal{S})^\mathbb{N}$ that will allow us to compute generating invariants for $\mathbb{R}(\mathcal{V})^{\mathrm{SO}_3(\mathbb{R})}$. For $\mathcal{A}$ a real algebraic variety, we note $\hat{\mathcal{A}}$ its complexification, i.e., its Zariski closure over $\mathbb{C}$.

**Theorem 3.6.** Seshadri slice lemma [CS07, Theorem 3.1]. Let $G$ be a real algebraic group acting on a vector space $\mathcal{X}$. Let $\mathcal{S} \subset \mathcal{X}$ be a subvector space and $\mathbb{N} < G$ the normalizer of $\mathcal{S}$ in $G$. Assume that:

• The closure of $G \cdot S$ is the whole of $\mathcal{X}$.

• There is an non empty $N$-stable Zariski open subset $\mathcal{Z} \subset \mathcal{S}$ such that for all $g \in \hat{G}$ and $s \in \hat{\mathcal{Z}}$ satisfying $g \cdot s \in \hat{\mathcal{Z}}$, there exists $h \in \hat{N}$ such that $h \cdot s = g \cdot s$.

Then, the restriction to $\mathcal{Z}$ of rational functions on $\mathcal{X}$ induces a field isomorphism $\mathbb{R}(\mathcal{X})^{\mathcal{G}} \cong \mathbb{R}(\mathcal{S})^{\mathcal{N}}$.

**Definition 3.7.** Such a pair $(\mathcal{S}, \mathbb{N})$ is called a Seshadri slice for $G \curvearrowright \mathcal{X}$.

For $P \in \mathbb{R}(\mathcal{S})^{\mathbb{N}}$, we shall write $\tilde{P}$ the unique rational invariant in $\mathbb{R}(\mathcal{X})^{\mathcal{G}}$ such that for generic $s \in \mathcal{S}$,

$$
P(s) = \tilde{P}(s).
$$

For some set of invariants $\mathcal{E} \subset \mathbb{R}(\mathcal{S})^{\mathbb{N}}$, we note $\tilde{\mathcal{E}} = \{\tilde{P}, P \in \mathcal{E}\} \subset \mathbb{R}(\mathcal{V})^{\mathcal{G}}$.

**Proposition 3.8.** Consider the subspace

$$
\mathcal{S} := \mathbb{R}W \oplus \mathcal{H} \subset \mathcal{V}
$$

Then, $(\mathcal{S}, \mathbb{N})$ is a Seshadri slice with normalizer

$$
N = \left\{ \left( \begin{array}{c|c} g & 0 \\ \hline 0 & 0 \end{array} \right) , g \in \mathcal{O}_2(\mathbb{R}) \right\} \subset \mathrm{SO}_3(\mathbb{R}).
$$

<figure><img src="image_4.png" /></figure>

The normalizer $N \cong \mathrm{O}_2(\mathbb{R})$ of $\mathcal{S}$ consists of the rotations in $\mathbb{R}^3$ that leave the $W$ axis invariant. That includes the rotations around the $W$ axis and the rotations by $\pi$ around any axis in the $(U, V)$ plane. The open subset $\mathcal{Z}$ in $\mathcal{S}$ that is relevant in the proof and later on in the article is $\mathcal{Z} = \mathbb{R}^*W \oplus \mathcal{H}$.
The open subset $\mathcal{Z}$ in $\mathcal{S}$ that is relevant in the proof and later on in the article is

$$ \mathcal{Z} = \{p_1 U + \sum_{j=1}^{n} P_{d_j} \in \mathcal{S} \text{ with } p_1 \neq 0\}. $$

*Proof.* We prove the two conditions to be a Seshadri slice:

* Let $P := P_1 + \sum_{j=1}^{n} P_{d_j}$ be a polynomial in $\mathcal{H}$. Since $(\rho_1, \mathcal{H}_1)$ is isomorphic to the standard representation, the orbits in $\mathcal{H}_1$ are the concentric spheres $r \mathbb{S}^2$ of radius $r \in \mathbb{R}^+$. Hence there exists a rotation $g \in \mathrm{SO}_3(\mathbb{R})$ such that $\rho_1(g)(P_1) = \|P_1\|U \in \mathcal{S}_1$. That is, $\rho(g)(P) \in \mathcal{S}_1 \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} = \mathcal{S}$.

* We take the non empty subset $\mathcal{Z} = \{p_1 U + \sum_{j=1}^{n} P_{d_j} \in \mathcal{S} \text{ with } p_1 \neq 0\} \subset \mathcal{S}$. $\mathcal{Z}$ is N-stable and its complexification is $\hat{\mathcal{Z}} = \{p_1 U + \sum_{j=1}^{n} P_{d_j} \in \hat{\mathcal{S}} \text{ with } p_1 \neq 0\} \subset \hat{\mathcal{S}}$. Then for any $P \in \hat{\mathcal{Z}}$ an element $g \in \mathrm{SO}_3(\mathbb{C})$ such that $\rho(g)(P) \in \hat{\mathcal{Z}}$ satisfies $\rho_1(g) = \begin{pmatrix} \lambda & a & b \\ 0 & * & * \\ 0 & * & * \end{pmatrix}$ with $\lambda, a, b \in \mathbb{C}$. However we have $\rho_1(g)^t = g^t \in \mathrm{SO}_3(\mathbb{C})$ and $\rho_1(g)^t \rho_1(g) = I_3$.

$$ \Rightarrow \quad \rho_1(g)^t \rho_1(g) = \begin{pmatrix} \lambda & 0 & 0 \\ a & * & * \\ b & * & * \end{pmatrix} \begin{pmatrix} \lambda & a & b \\ 0 & * & * \\ 0 & * & * \end{pmatrix} = \begin{pmatrix} \lambda^2 & \lambda a & \lambda b \\ \lambda a & * & * \\ \lambda b & * & * \end{pmatrix} = I_3 $$

This is $\lambda = \pm 1$, $a = 0$ and $b = 0$. Then, $g \in \hat{\mathbb{N}}$. $\square$

<figure><img src="image_3.png" /></figure>

### 3.3 Restrictions of the invariants to the slice

We saw that

$$ \mathcal{S} := \mathcal{S}_1 \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} \subset \mathcal{H}_1 \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} =: \mathcal{H} $$

with normalizer $N \cong \mathrm{O}_2(\mathbb{R})$ is a Seshadri slice for $(\mathcal{H}, \mathrm{SO}_3(\mathbb{R}))$. By Theorem 3.3 we know that

$$ \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} \cong \bigoplus_{j=1}^{p} \mathcal{V}_0 \oplus \bigoplus_{j=1}^{q} \mathcal{V}_{-1} \oplus \bigoplus_{j=1}^{n} \bigoplus_{m=1}^{d_j} \mathcal{V}_m $$

where the $\mathcal{V}_m$ are the irreducible representations of $\mathrm{O}_2(\mathbb{R})$ of Section 2.1. This decomposition is furthermore explicitly given by a basis of spherical harmonics as shown in introduced in Proposition 3.3. The action of $N$ on $\mathcal{S}_1$ is furthermore the determinantal representation $\phi_{-1}$. We thus can write

$$ \mathcal{S} \cong \mathcal{V}_{-1} \oplus \bigoplus_{j=1}^{p} \hat{\mathcal{V}}_0 \oplus \bigoplus_{j=1}^{q} \mathcal{V}_{-1} \oplus \bigoplus_{j=1}^{n} \bigoplus_{m=1}^{d_j} \mathcal{V}_m. $$

with coordinates $(y_0, z_{m,j}, 1 \le j \le r, -d_j \le m \le d_j)$. The group action $\mathrm{O}_2(\mathbb{R}) \cong N \backslash \mathcal{S}$ is a representation satisfying the hypotheses of Theorem 2.8. We apply it to obtain generators:
Proof. We prove the two conditions to be a Seshadri slice:

• Take $V = H_1 + H$ in $\mathcal{V} = \mathcal{H}_1 \oplus \mathcal{H}$. Since $(\rho_1, \mathcal{H}_1)$ is isomorphic to the standard representation, the orbits in $\mathcal{H}_1$ are the concentric spheres. Hence there exists a rotation $g \in \text{SO}_3(\mathbb{R})$ such that $\rho_1(g)(H_1) = \|H_1\|W$. That is, $\rho(g)(V) \in S$.

• $\mathcal{Z}$ is $N$-stable and its complexification is $\mathcal{Z} = \mathbb{C}^*W \oplus \hat{\mathcal{H}}$. Then for any $V \in \hat{\mathcal{Z}}$, an element $g \in \text{SO}_3(\mathbb{C})$ such that $\rho(g)(V) \in \hat{\mathcal{Z}}$ satisfies

$$ \rho_1(g) = \begin{pmatrix} * & * & 0 \\ * & * & 0 \\ a & b & c \end{pmatrix} $$

with $a, b, c \in \mathbb{C}$. However we have $\rho_1(g)^t = g^t \in \text{SO}_3(\mathbb{C})$ and $\rho_1(g)^t \rho_1(g) = I_3$. This gives

$$ \rho_1(g)^t \rho_1(g) = \begin{pmatrix} * & * & a \\ * & * & b \\ 0 & 0 & c \end{pmatrix} \begin{pmatrix} * & * & 0 \\ * & * & 0 \\ a & b & c \end{pmatrix} \begin{pmatrix} * & * & ac \\ * & * & bc \\ ac & bc & c^2 \end{pmatrix} = I_3. $$

Therefore $a = b = 0$ and $c = \pm 1$. We proved that $g \in \hat{\mathbb{N}}$. $\square$

### 3.3 Restrictions of the invariants to the slice

From now on, we note $\mathcal{V} = \mathcal{H}_1 \oplus \bigoplus_{j=1}^n \mathcal{H}_{d_j}$. We saw that the subvariety

$$ S := \mathbb{R}W \oplus \bigoplus_{j=1}^n \mathcal{H}_{d_j} $$

with normalizer $N \cong \text{O}_2(\mathbb{R})$ forms a Seshadri slice for the representation of $\text{SO}_3(\mathbb{R})$ on $\mathcal{V}$. By Proposition 3.5 we know that

$$ \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} \cong \bigoplus_{j=1}^{p} \mathcal{V}_{-1} \oplus \bigoplus_{j=p+1}^{n} \mathcal{V}_{0} \oplus \bigoplus_{j=1}^{n} \bigoplus_{m=1}^{d_j} \mathcal{V}_{m} $$

where the $\mathcal{V}_m$ are the irreducible representations of $\text{O}_2(\mathbb{R})$ of Section 2.1. The action of $N$ on $\mathbb{R}W$ is furthermore the determinantal representation $\phi_{-1}$. The representation of $N \cong \text{O}_2(\mathbb{R})$ on $S$ decomposes then into

$$ S \cong \mathcal{V}_{-1} \oplus \bigoplus_{j=1}^{p} \mathcal{V}_{-1} \oplus \bigoplus_{j=p+1}^{n} \mathcal{V}_{0} \oplus \bigoplus_{j=1}^{n} \bigoplus_{m=1}^{d_j} \mathcal{V}_{m}. $$

This decomposition satisfies the hypotheses of Section 2. We can thus apply Theorem 2.6 to get the following set of algebraically independent invariants generating $\mathbb{R}(S)^N$:

$$ \mathcal{E} = \left\{ \begin{array}{lclcl} D_j & = & y_{0,0}y_{0,j}, & 0 \le j \le p \\ T_j & = & y_{0,j}, & p+1 \le j \le n \\ R_{mj} & = & y_{m,j}y_{-1,1}^m + y_{-m,j}y_{1,1}^m, & 1 \le j \le n, & 1 \le m \le d_j \\ I_{mj} & = & iy_{0,0}(y_{m,j}y_{-1,1}^m - y_{-m,j}y_{1,1}^m), & 1 \le j \le n, & 1 \le m \le d_j, (j,m) \ne (1,1) \end{array} \right\} $$

**Theorem 3.9.** The set $\tilde{\mathcal{E}}$ is algebraically independent and generates both $\mathbb{R}(\mathcal{V})^{\text{SO}_3(\mathbb{R})}$ and $\mathbb{C}(\hat{\mathcal{V}})^{\text{SO}_3(\mathbb{C})}$.

Since invariants in $\tilde{\mathcal{E}}$ are algebraically independent, they provide a constructive proof of the rationality of the invariant field as a corollary:
**Theorem 3.7.** The following set of algebraically independent polynomials generates both $\mathbb{R}(S)^N$ and $\mathbb{C}(\hat{S})^{\hat{N}}$.

$$ \mathcal{E} = \left\{ \begin{array}{rcl} T_j(y_0, h) & = & y_{0,j}, \qquad 1 \le j \le p \\ D_0(y_0, h) & = & y_0^2, \\ D_j(y_0, h) & = & y_0 y_{0,j}, \qquad p+1 \le j \le n \\ R_{mj}(y_0, h) & = & y_{m,j} y_{-1,1}^m + y_{-m,j} y_{1,1}^m, \qquad 1 \le j \le n, \quad 1 \le m \le d_j \\ I_{mj}(y_0, h) & = & i y_0 (y_{m,j} y_{-1,1}^m - y_{-m,j} y_{1,1}^m), \qquad 1 \le j \le n, \quad 1 \le m \le d_j, \quad (j,m) \ne (1,1) \end{array} \right\} $$

**Corollary 3.8.** Let $\rho: \mathrm{SO}_3(\mathbb{R}) \to \mathrm{GL}(\mathcal{H})$ be a representation including at least once the irreducible representation $(\rho_1, \mathcal{H}_1)$. Then, $\mathrm{R}(\mathcal{H})^{\mathrm{SO}_3(\mathbb{R})}$ is rational.

*Proof.* The invariant field $\mathbb{R}(S)^N$ is rational (Theorem 2.8). According to Seshadri slice lemma, $\mathrm{R}(\mathcal{H})^{\mathrm{SO}_3(\mathbb{R})}$ is rational. $\square$

It was already known that an almost free $\mathrm{SO}_n(\mathbb{C})$ action (as $\mathrm{SO}_3(\mathbb{C})$ representation on $\hat{\mathcal{H}}$ whenever $n > 0$) is stably rational [CS05, Proposition 4.12], this being a weaker property than rational. Our result nonetheless also applies to actions which are not almost free.

**Example 3.9.** The representation $\mathcal{S}_T = 2\mathcal{V}_1 \oplus \mathcal{V}_1 \oplus \mathcal{V}_2 \oplus \mathcal{V}_3$ introduced in Example 2.9 is the slice corresponding to the representation $\mathcal{T} = \mathcal{H}_1 \oplus \mathcal{H}_3$. For this $\mathrm{SO}_3(\mathbb{C})$ action, the restriction to $\mathcal{S}$ of an algebraically independent generating invariants on the slice is

$$ \mathcal{E}_{\mathcal{T}} = \left\{ \begin{array}{l} D_1 = y_0^2 \\ D_2 = y_0 y_{0,1} \\ R_1 = 2y_{1,1} y_{-1,1} \\ R_2 = y_{1,1}^2 y_{-2,1} + y_{-1,1}^2 y_{2,1} \\ R_3 = y_{1,1}^3 y_{-3,1} + y_{-1,1}^3 y_{3,1} \\ I_2 = i y_0 (y_{1,1}^2 y_{-2,1} - y_{-1,1}^2 y_{2,1}) \\ I_3 = i y_0 (y_{1,1}^3 y_{-3,1} - y_{-1,3}^3 y_{-3,1}) \end{array} \right\} $$

## 3.4 Evaluating the invariants

In previous section we characterized the invariants of the action of $\mathrm{SO}_3(\mathbb{R})$ on $\mathcal{H}$ by their restriction to the slice $\mathcal{S}$. In this section we address how to evaluate these invariants at any other point of $\mathcal{H}$. We propose to first find a rotation that takes this point to the slice. Such an intersection of the orbit of the point and the slice is not unique, and the rotation to take it there even less so. Yet the evaluation of the slice invariants at any such point of intersection gives the same values. As we are interested by evaluation on the real points of $\mathcal{H}$ we can use of spherical coordinates and describe the rotations by Euler angles.

**Definition 3.10.** With respect to the Euclidean basis $(u, v, w)$, a rotation $g \in \mathrm{SO}_3(\mathbb{R})$ is determined by Euler angles $(\alpha, \beta, \gamma)$ where $\alpha \in [0, 2\pi[$, $\beta \in [0, \pi]$ and $\gamma \in [0, 2\pi[$. It is the composition of rotations of axes $u, w, u$ and angles $(\alpha, \beta, \gamma)$:

$$ g(\alpha, \beta, \gamma) := \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\gamma) & \sin(\gamma) \\ 0 & -\sin(\gamma) & \cos(\gamma) \end{pmatrix} \begin{pmatrix} \cos(\beta) & \sin(\beta) & 0 \\ -\sin(\beta) & \cos(\beta) & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\alpha) & \sin(\alpha) \\ 0 & -\sin(\alpha) & \cos(\alpha) \end{pmatrix} $$

Euler angles are particularly effective to define a rotation $g \in \mathrm{SO}_3(\mathbb{R})$ mapping a point in $\mathcal{H}$ to the slice:

**Lemma 3.11.** Let $y+h \in \mathcal{H}_1 \oplus \bigoplus_{j=1}^n \mathcal{H}_{d_j}$ a point in $\mathcal{H}$, with $y \ne 0$. Note $y = (r, \theta, \varphi)$ in spherical coordinates. Then, for all $\gamma \in \mathbb{R}$, one has $\rho(g(\varphi, \theta, \gamma))(y+h) \in \mathcal{Z} \subset \mathcal{S}$.
**Corollary 3.10.** Let $\rho: \mathrm{SO}_3(\mathbb{R}) \to \mathrm{GL}(\mathcal{V})$ be a representation including at least once the irreducible representation $(\rho_1, \mathcal{H}_1)$. Then, $\mathrm{R}(\mathcal{V})^{\mathrm{SO}_3(\mathbb{R})}$ is rational.

The rationality of the invariant field is a problem well addressed in invariant theory [Noe83; CS07; Dol85; PV94]. The rationality of the invariant field of any representation of $\mathrm{SL}_2(\mathbb{C})$ was proved in [Kat83; Kat84; BK85]. For irreducible representations of even degree, Katsylo uses Seshadri slices (that he calls (G,N)-sections) very similar to the one we identified in Proposition 3.8. One could thus deduce the rationality for representations of $\mathrm{SO}_3(\mathbb{C})$ thanks to the two fold covering $\mathrm{SL}_2(\mathbb{C}) \to \mathrm{SO}_3(\mathbb{C})$. Nevertheless, the explicit generating set provided by our constructive proof is valuable. It allows us to derive directly the rationality of the real invariant fields (for representations of $\mathrm{SO}_3(\mathbb{R})$ and $\mathrm{O}_3(\mathbb{R})$). Furthermore [HJ24b] uses the multi-homogeneity of the obtained set to show the rationality of the invariant fields of some representations of the larger orthogonal groups $\mathrm{SO}_n(\mathbb{R})$.

Let us mention that a method based on the "no-name lemma" [CS07, Corollary 3.9], is used to prove stable rationality [CS07; BK86]. This is a weaker property, stating that a purely transcendental extension of the invariant field is rational. In this way, the invariant field of an almost free $\mathrm{SO}_n(\mathbb{C})$ representation is stably rational [CS07, Proposition 4.12].

**Example 3.11.** The $\mathrm{O}_2(\mathbb{R})$-representation $S_T = 2\mathcal{V}_{-1} \oplus \mathcal{V}_1 \oplus \mathcal{V}_2 \oplus \mathcal{V}_3$ introduced in Example 2.7 is the slice corresponding to the $\mathrm{SO}_3(\mathbb{R})$-representation $T = \mathcal{H}_1 \oplus \mathcal{H}_3$. The restrictions to $S$ of a generating set of $\mathrm{SO}_3(\mathbb{R})$-invariants are:

$$ \mathcal{E}_T = \left\{ \begin{array}{l} D_0 = y_{0,0}^2 \\ D_1 = y_{0,0}y_{0,1} \\ R_{11} = 2y_{1,1}y_{-1,1} \\ R_{21} = y_{1,1}^2 y_{-2,1} + y_{-1,1}^2 y_{2,1} \\ R_{31} = y_{1,1}^3 y_{-3,1} + y_{-1,1}^3 y_{3,1} \\ I_{21} = i y_{0,0} (y_{1,1}^2 y_{-2,1} - y_{-1,1}^2 y_{2,1}) \\ I_{31} = i y_{0,0} (y_{1,1}^3 y_{-3,1} - y_{-1,1}^3 y_{3,1}) \end{array} \right\} $$

**Example 3.12.** Consider the representation $\mathbb{R}^3 \oplus \dots \oplus \mathbb{R}^3 \cong \mathcal{H}_1^{n+1}$. The associated slice is then $S_{\mathcal{H}_1^{n+1}} = \mathbb{R}W \oplus \mathcal{H}_1^n$ and Theorem 3.9 gives the restrictions to the slice of a generating set of $\mathrm{SO}_3(\mathbb{R})$-invariants.

$$ \mathcal{E}_{\mathcal{H}_1^{n+1}} = \left\{ \begin{array}{lcl} D_j & = & y_{0,0}y_{0,i}, \qquad 0 \le j \le n \\ R_j & = & y_{1,1}y_{-1,j} + y_{-1,1}y_{1,j}, \qquad 1 \le j \le n \\ I_j & = & iy_{0,0} (y_{1,1}y_{-1,j} - y_{-1,1}y_{1,j}), \qquad 2 \le j \le n \end{array} \right\} $$

We note $(r_0, \dots, r_n)$ a $(n+1)$-tuple of vectors in $\mathcal{H}_1^{n+1} \cong (\mathbb{R}^3)^{n+1}$. The scalar products $(r_i, r_j)$, $0 \le i \le j \le n$ and the determinants $\det(r_i|r_j|r_k)$, $0 \le i < j < k \le n$, generate the invariant algebra $\mathrm{R}[\mathcal{H}_1^n]^{\mathrm{SO}_3(\mathbb{R})}$ [Wey46, Theorem 2.9.A]. Examining their restrictions to the slice, we can find their relationships with the $D_j, R_j$, and $I_j$ above. By Theorem 3.6 we deduce that the corresponding $\mathrm{SO}_3(\mathbb{R})$-invariants are:

$$ \tilde{D}_j = (r_0, r_i), \qquad 0 \le j \le n, $$

$$ \tilde{R}_j = 2 \left( (r_1, r_j) - \frac{(r_0, r_j)^2}{(r_0, r_0)} \right), \qquad 1 \le j \le n, $$

$$ \tilde{I}_j = \det(r_0|r_1|r_j), \qquad 2 \le j \le n. $$

## 3.4 Evaluating the invariants

In the previous section we characterized the invariants of the action of $\mathrm{SO}_3(\mathbb{R})$ on $\mathcal{V} = \mathbb{R}^3 \oplus \mathcal{H}$ by their restrictions to the slice $S = \mathbb{R} \oplus \mathcal{H}$. In this section we address how to evaluate these invariants at any other point $v \in \mathcal{V}$. We first find a rotation $g$ that takes $v$ to $S$ and next evaluate our invariants at $g \cdot v$. Neither $g$ nor the point of intersection of the orbit with the slice is unique, yet the values of the invariants is the same
Proof. The representation $\rho_1 : \text{SO}_3(\mathbb{R}) \to \text{GL}(\mathcal{H}_1)$ is isomorphic to the canonical $\text{SO}_3(\mathbb{R})$ representation on $\mathbb{R}^3$. Hence for $g \in \text{SO}_3(\mathbb{R})$ we have $\rho_1(g)(y) \in \mathcal{S}_1 \Leftrightarrow g(y) = (r, 0, 0) \Leftrightarrow g^t(r, 0, 0) = y$. Meanwhile, each rotation in $\{g(\varphi, \theta, \gamma), \gamma \in \mathbb{R}\}$ maps $(r, 0, 0)$ to $y$. $\square$

We thus need to address how to compute $\rho(g(\varphi, \theta, \gamma))$. Wigner matrices provide a solution as they make explicit the action of $\text{SO}_3(\mathbb{R})$ on $\mathcal{H}_d$:

**Definition 3.12.** [Wig59, p. 15.27]. Let $d \in \mathbb{N}$ and $(\alpha, \beta, \gamma)$ some Euler angles. The $d^{th}$ Wigner matrix is the square matrix of size $(2d+1) \times (2d+1)$ defined by

$$W_{(\alpha,\beta,\gamma)}^{d} = \left( \begin{array}{c} e^{i\lambda\alpha} w_{m,l}^{d}(\beta) e^{im\gamma} \\ \hline -d \le m,l \le d \end{array} \right)$$

where

$$w_{m,l}^{d}(\beta) := c_{m,l} \sum_{s \in \Gamma_{mm'}^{d}} \frac{\delta(m,l)(-1)^{s} i^{l-m}}{(d+m-s)!s!(l-m+s)!(d-l-s)!} \cos\left(\frac{\beta}{2}\right)^{2d+m-l-2s} \sin\left(\frac{\beta}{2}\right)^{l-m+2s}.$$

$\Gamma_{mm'}^d$ is the set of integers $s$ such that $d+m-s$, $s$, $l-m+s$ and $d-l-s$ are non negative, and

$$\delta(m,l) = \begin{cases} -1 & \text{if } m > 0 \text{ and } l > 0. \\ -1 & \text{if } m \le 0 \text{ and } l \le 0. \\ 1 & \text{otherwise.} \end{cases}$$

Note that we slightly modified the coefficients of the functions $w_{m,l}^d$ compared to [Wig59] that uses a different normalisation of the spherical harmonics.

**Proposition 3.13.** Let $d \in \mathbb{N}$ and $(\alpha, \beta, \gamma)$ some Euler angles defining the rotation $g(\alpha, \beta, \gamma) \in \text{SO}_3(\mathbb{R})$. Then, for all vectors $Y = (y_d, ..., y_d) \in \mathcal{H}_d$, $\rho_d(g(\alpha, \beta, \gamma))(Y) = W_{(\alpha,\beta,\gamma)}^d Y$.

As the third angle $\gamma$ is basically irrelevant for the rotations taking a point in $\mathcal{H}$ to the slice $\mathcal{S}$ we can give the expressions of the $\text{SO}_3(\mathbb{R})$ invariants as follows.

**Theorem 3.14.** The set of functions of $(y+h) \in \text{SO}_3(\mathbb{R}) \times \mathcal{Z}$ with $y = (r, \theta, \varphi)$ in spherical coordinates

$$\tilde{\mathcal{E}} = \left\{ \begin{array}{l} \tilde{T}_j(y+h) = \sum_{l=-d_j}^{d_j} e^{il\varphi} w_{l,0}^{d_j}(\theta) y_{l,j} \\ \tilde{D}_0(y+h) = r^2 \\ \tilde{D}_j(y+h) = r \sum_{l=-d}^{d} e^{il\varphi} w_{l,0}^{d_j}(\theta) y_{l,j} \\ \tilde{R}_{mi}(y+h) = \left( \sum_{l=-d_j}^{d_j} e^{il\varphi} w_{l,m}^{d_j}(\theta) y_{l,j} \right) \left( \sum_{l=-d_1}^{d_1} e^{il\varphi} w_{l,-1}^{d_1}(\theta) y_{l,1} \right)^m \\ \qquad + \left( \sum_{l=-d_j}^{d_j} e^{il\varphi} w_{l,-m}^{d_j}(\theta) y_{l,j} \right) \left( \sum_{l=-d_1}^{d_1} e^{il\varphi} w_{l,1}^{d_1}(\theta) y_{l,1} \right)^m \\ \tilde{I}_{mi}(y+h) = \text{ir} \left( \sum_{l=-d_j}^{d_j} e^{il\varphi} w_{l,m}^{d_j}(\theta) y_{l,j} \right) \left( \sum_{l=-d_1}^{d_1} e^{il\varphi} w_{l,-1}^{d_1}(\theta) y_{l,1} \right)^m \\ \qquad - \text{ir} \left( \sum_{l=-d_j}^{d_j} e^{il\varphi} w_{l,-m}^{d_j}(\theta) y_{l,j} \right) \left( \sum_{l=-d_1}^{d_1} e^{il\varphi} w_{l,1}^{d_1}(\theta) y_{l,1} \right)^m \end{array} \right\}$$

is rational and algebraically independent. It furthermore generates $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$.
for any choice among those. As we are interested by evaluation on the real points, we introduce spherical coordinates and Euler angles, and then use Wigner matrices.

**Lemma 3.13.** Let $H_1 + H \in \mathcal{H}_1 \oplus \mathcal{H}$ be a point in $\mathcal{V}$, with $H_1 \neq 0$. There is $(r, \theta, \varphi) \in \mathbb{R}_{>0} \times [0, \pi[ \times [0, 2\pi[$ such that $H_1 = r \sin(\theta) \cos(\varphi) U + r \sin(\theta) \sin(\varphi) V + r \cos(\theta) W$. Then one has

$$ \rho(g(-\varphi, -\theta, 0))((H_1 + H)) \in \mathcal{Z} \subset \mathcal{S}. $$

*Proof.* The gradient map $\nabla : \mathcal{H}_1 \to \mathbb{R}^3$ is an isomorphism of $SO_3(\mathbb{R})$-representations. Thus

$$ \rho_1(g(-\varphi, -\theta, 0))(H_1) = rW. $$

This implies $\rho(g(-\varphi, -\theta, 0))(H_1 + H) \in \mathcal{Z}$.

<figure><img src="image_5.png" /></figure>

We thus have determined a rotation $g(-\varphi, -\theta, 0) \in SO_3(\mathbb{R})$ that takes a point in $\mathcal{V}$ to the slice. We now need to make explicit the action of such a rotation on $\mathcal{V}$. For each integer $d$, the matricial representation of $\rho_d(g(-\varphi, -\theta, 0))$ with respect to the basis of spherical harmonics is provided by Wigner matrices.

**Definition 3.14.** [Wig59, p. 15.27]. Let $d \in \mathbb{N}$ and $(\alpha, \beta, \gamma)$ Euler angles. The $d^{th}$ Wigner matrix is the square matrix of size $(2d+1) \times (2d+1)$ defined by

$$ W_{(\alpha, \beta, \gamma)}^{d} = \left( \begin{array}{c} e^{-im\gamma} w_{m,m'}^{d}(-\beta) e^{-im'\alpha} \\ \hline -d \le m, m' \le d \end{array} \right) $$

where

$$ w_{m,m'}^{d}(\beta) := \delta(m,m') \sum_{s=\max(0,m'-m)}^{\min(d-m,d+m')} \frac{(-1)^s \sqrt{(d+m')!(d-m')!(d+m)!(d-m)!}}{(d+m'-s)!s!(m-m'+s)!(d-m-s)!} \cos\left(\frac{\beta}{2}\right)^{2d+m'-m-2s} \sin\left(\frac{\beta}{2}\right)^{m-m'+2s}, $$

$$ \text{and } \delta(m, m') = i^d(\operatorname{sgn}(m') - \operatorname{sgn}(m))(-1)^{\frac{m'+|m'|-m-|m'|}{2}}. $$

We used the convention $\operatorname{sgn}(0) = 0$. The coefficient $\delta(m, m')$ comes from the slight change of basis between Theorem 3.2 and [Wig59]. This requires to conjugate by a diagonal matrix.

**Proposition 3.15.** [Wig59] Let $d \in \mathbb{N}$ and $(\alpha, \beta, \gamma)$ be the Euler angles defining the rotation $g(\alpha, \beta, \gamma) \in SO_3(\mathbb{R})$. For $H = y_{-d}Y_d^{-d} + \dots + y_dY_d^d \in \mathcal{H}_d$, $\rho_d(g(\alpha, \beta, \gamma))(H) = \tilde{y}_{-d}Y_d^{-d} + \dots + \tilde{y}_dY_d^d$ where

$$ \begin{pmatrix} \tilde{y}_{-d} \\ \vdots \\ \tilde{y}_d \end{pmatrix} = W_{(\alpha, \beta, \gamma)}^d \begin{pmatrix} y_{-d} \\ \vdots \\ y_d \end{pmatrix}. $$

As the third angle $\gamma$ is irrelevant for the rotations taking a point in $\mathcal{V}$ to the slice $\mathcal{S}$ we can give the expressions of the $SO_3(\mathbb{R})$ invariants as follows.

**Theorem 3.16.** Consider the representation of $SO_3(\mathbb{R})$ on $\mathcal{V} = \mathcal{H}_1 \oplus \bigoplus_{j=1}^n \mathcal{H}_{d_j}$. For $H_1 \in \mathcal{H}_1 \setminus \{0\}$ there is $(r, \theta, \varphi) \in \mathbb{R}_{>0} \times [0, \pi[ \times [0, 2\pi[$ such that $H_1 = r \sin(\theta) \cos(\varphi) U + r \sin(\theta) \sin(\varphi) V + r \cos(\theta) W$ so that its coordinates in the basis of spherical harmonics are

$$ y_{-1,0} = i \sqrt{\frac{2\pi}{3}} r \sin(\theta) e^{i\varphi}, \quad y_{0,0} = \sqrt{\frac{4\pi}{3}} r \cos(\theta), \quad y_{1,0} = -i \sqrt{\frac{2\pi}{3}} r \sin(\theta) e^{-i\varphi}. $$

Consider the large matrix $W(-\varphi, -\theta, 0) = \operatorname{Diag}(W_{(-\varphi, -\theta, 0)}^{d_j}, 1 \le j \le n)$. The set of rational functions $\tilde{\mathcal{E}} := \{P \circ W(-\varphi, -\theta, 0), P \in \mathcal{E}\}$ is algebraically independent and generates $\mathbb{R}(\mathcal{V})^{SO_3(\mathbb{R})}$.
Proof. We build a large matrix $W_{(\varphi,\theta,0)} = \text{Diag}(W_{(\varphi,\theta,0)}^{d_j}, 1 \le j \le n)$. Take $\tilde{P}$ the invariant function on $\mathcal{H}$ whose restriction to $\mathcal{S}$ is $P$. Proposition 3.13 claims that $\tilde{P}(y+h) = P(W_{(\varphi,\theta,0)}(y+h))$. We thus evaluated polynomials of $\tilde{\mathcal{E}}$ at $W_{(\varphi,\theta,0)}(y+h)$.

Polynomials in $\mathcal{E}$ are rational, algebraically independent, and they furthermore generate $\mathbb{R}(\mathcal{S})^N$. Meanwhile, Seshadri slice Lemma (Theorem 3.4) claims that the restriction to $\mathcal{S}$ is a field isomorphism between $\mathbb{R}(\mathcal{S})^N$ and $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$, or $\mathbb{C}(\hat{\mathcal{S}})^{\hat{N}}$ and $\mathbb{C}(\hat{\mathcal{H}})^{\hat{N}}$. Hence, extensions of polynomials in $\tilde{\mathcal{E}}$ are rational, algebraically independent, and they furthermore generate $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$. $\square$

**Example 3.15.** Consider the representation $\mathbb{R}^3 \oplus \mathbb{R}^3 \oplus \mathbb{R}^3 \oplus \mathbb{R}^3 \cong \mathcal{H}_1^4$. The associated slice is then $\mathcal{S}_{\mathcal{H}_1^4} = \mathcal{S}_1 \oplus \mathcal{H}_1^3$ and Theorem 3.7 gives the generating set

$$ \mathcal{E}_{\mathcal{H}_1^4} = \left\{ \begin{array}{l} D_0 = y_0^2 \\ D_1 = y_0 y_{0,1} \quad R_1 = y_{1,1} y_{-1,1} \\ D_2 = y_0 y_{0,2} \quad R_2 = y_{1,1} y_{-1,2} + y_{-1,1} y_{1,2} \quad I_2 = i y_0 (y_{1,1} y_{-1,2} - y_{-1,1} y_{1,2}) \\ D_3 = y_0 y_{0,3} \quad R_3 = y_{1,1} y_{-1,3} + y_{-1,1} y_{1,3} \quad I_3 = i y_0 (y_{1,1} y_{-1,3} - y_{-1,1} y_{1,3}) \end{array} \right\} $$

In this simple example translating with Maple the invariants given by Theorem 3.14 in spherical coordinates into Cartesian coordinates leads to expressions that can still each fit on one or two lines:

$$ \begin{aligned} \tilde{D}_0 &= 2y_{-1}y_1 + y_0^2 \\ \tilde{D}_1 &= y_{-1}y_{1,1} + y_0y_{0,1} + y_1y_{-1,1} \\ \tilde{D}_2 &= y_{-1}y_{1,2} + y_0y_{0,2} + y_1y_{-1,2} \\ \tilde{D}_3 &= y_{-1}y_{1,3} + y_0y_{0,3} + y_1y_{-1,3} \end{aligned} $$

$$ \tilde{R}_1 = \frac{-y_{-1}^2 y_{1,1}^2 - 2y_{-1} y_0 y_{0,1} y_{1,1} + 2y_{-1} y_1 y_{-1,1} y_{1,1} + 2y_{-1} y_1 y_{0,1}^2 + 2y_0^2 y_{-1,1} y_{1,1} - 2y_0 y_1 y_{-1,1} y_{0,1} - y_1^2 y_{-1,1}^2}{2y_{-1} y_1 + y_0^2} $$

$$ \tilde{R}_2 = \frac{-y_{-1}^2 y_{1,1} y_{1,2} + ((y_{-1,1} y_{1,2} + y_{-1,2} y_{1,1} + 2y_{0,1} y_{0,2}) y_1 - y_0 (y_{0,1} y_{1,2} + y_{0,2} y_{1,1})) y_{-1}}{2y_{-1} y_1 + y_0^2} + \frac{-y_1^2 y_{-1,1} y_{-1,2} - y_0 (y_{-1,1} y_{0,2} + y_{-1,2} y_{0,1}) y_1 + y_0^2 (y_{-1,1} y_{1,2} + y_{-1,2} y_{1,1})}{2y_{-1} y_1 + y_0^2} $$

$$ \tilde{R}_3 = \frac{-y_{-1}^2 y_{1,1} y_{1,3} + ((y_{-1,1} y_{1,3} + y_{-1,3} y_{1,1} + 2y_{0,1} y_{0,3}) y_1 - y_0 (y_{0,1} y_{1,3} + y_{0,3} y_{1,1})) y_{-1}}{2y_{-1} y_1 + y_0^2} + \frac{-y_1^2 y_{-1,1} y_{-1,3} - y_0 (y_{-1,1} y_{0,3} + y_{-1,3} y_{0,1}) y_1 + y_0^2 (y_{-1,1} y_{1,3} + y_{-1,3} y_{1,1})}{2y_{-1} y_1 + y_0^2} $$

$$ \tilde{I}_2 = i (-y_{-1} y_{0,1} y_{1,2} + y_{-1} y_{0,2} y_{1,1} + y_0 y_{-1,1} y_{1,2} - y_0 y_{-1,2} y_{1,1} - y_1 y_{-1,1} y_{0,2} + y_1 y_{-1,2} y_{0,1}) $$

$$ \tilde{I}_3 = i (-y_{-1} y_{0,1} y_{1,3} + y_{-1} y_{0,3} y_{1,1} + y_0 y_{-1,1} y_{1,3} - y_0 y_{-1,3} y_{1,1} - y_1 y_{-1,1} y_{0,3} + y_1 y_{-1,3} y_{0,1}) $$

## 3.5 Separation properties

We computed a set of invariants $\mathcal{E}$ generating the invariant field $\mathbb{C}(\hat{\mathcal{S}})^{\hat{N}}$. By Proposition 2.7, it separates orbits in

$$ \hat{U} = \{ v \in \hat{\mathcal{S}} \mid y_0 y_{1,1} y_{-1,1} \neq 0 \}, $$

which is included in $\hat{\mathcal{Z}} = \{v \in \hat{\mathcal{S}} \mid y_0 \neq 0\}$. Here we deduce separation property for the set of rational invariants $\hat{\mathcal{E}}$ in $\mathcal{H}$.

**Proposition 3.16.** Rational functions in $\tilde{\mathcal{E}}$ separate orbits in the open set $\text{SO}_3(\mathbb{C}) \times \hat{U}$.

Proof. Take $u_1$ and $u_2$ two points in $\text{SO}_3(\mathbb{C}) \times \hat{U}$ which are not in the same orbit. They have $s_1$ and $s_2$ representants of their respective orbits in $\hat{U}$. Since $\mathcal{E}$ separates orbits in $\mathcal{U}$, there is $f \in \mathcal{E}$ such that $f(s_1) \neq f(s_2)$. Since $\mathcal{U} \subset \mathbb{Z}$, $\tilde{f}(u_1) \neq \tilde{f}(u_2)$. $\square$
*Proof.* Take $\tilde{P}$ the invariant function on $\mathcal{V}$ whose restriction to $\mathcal{S}$ is $P$, and $(H_1+H)$ a point in $\mathcal{V} = \mathcal{H}_1 \oplus \mathcal{H}$.
Choose $(-\varphi, -\theta, 0)$ according to Lemma 3.13 so that $\rho(g(-\varphi, -\theta, 0))(H_1+H) \in \mathcal{S}$. Then $\tilde{P}(H_1+H) =
P(g(-\varphi, -\theta, 0)(H_1+H))$. By Proposition 3.15 this latter is $P \circ W(-\varphi, -\theta, 0)(H_1+H)$.

Polynomials in $\mathcal{E}$ are algebraically independent and generate $\mathbb{R}(\mathcal{S})^\mathbb{N}$. Meanwhile, Seshadri slice Lemma (Theorem 3.6) asserts that the restriction to $\mathcal{S}$ provides a field isomorphism between $\mathbb{R}(\mathcal{V})^{\mathrm{SO}_3(\mathbb{R})}$ and $\mathbb{R}(\mathcal{S})^\mathbb{N}$, or $\mathbb{C}(\hat{V})^{\mathrm{SO}_3(\mathbb{C})}$ and $\mathbb{C}(\hat{S})^{\mathbb{N}}$. Hence, extensions of polynomials in $\tilde{\mathcal{E}}$ are rational, algebraically independent, and they furthermore generate $\mathbb{R}(\mathcal{V})^{\mathrm{SO}_3(\mathbb{R})}$. $\square$

Polynomials in $\mathcal{E}$ are multihomogeneous with respect to the decomposition of $\mathcal{S}$ into irreducible representations of $O_2(\mathbb{R})$. Rational invariants in $\tilde{\mathcal{E}}$ inherit this multihomogeneity:

**Proposition 3.17.** *Rational functions in* $\tilde{\mathcal{E}}$ *are multihomogeneous with respect to the decomposition of* $\mathcal{V}$ *into irreducible representations of* $\mathrm{SO}_3(\mathbb{R})$.

The multihomogeneity of a generating set allows [HJ24b] to derive the rationality of the invariant fields of
representations of a direct product $\mathrm{SO}_3(\mathbb{R}) \times \mathbb{C}_2^{n-3}$, $n \in \mathbb{N}$. Seshadri slices give then the rationality for some
representations of $\mathrm{SO}_n(\mathbb{R})$.

Theorem 3.16 provides explicit expressions of the invariants in spherical coordinates. One can obtain
the expression in Cartesian coordinates with symbolic manipulation and rewriting with Groebner bases.
Examples of these computations are given in Example 5.3 for ternary cubics. Note in particular that
$\tilde{D}_0 = y_{0,0}^2 + 2y_{-1,0}y_{1,0}$, or in real coordinates $x_{-1,0}^2 + x_{0,0}^2 + x_{1,0}^2$, is an irreducible polynomial. Its variety
over $\mathbb{C}$ contains the points whose orbit does not intersect $\hat{S}$. As proved in [HJ24a], powers of $\tilde{D}_0$ are the only
possible denominators that can appear in $\tilde{\mathcal{E}}$. This appears in the explicit computations in Example 5.3.

**Proposition 3.18.** *The set* $\hat{V} \setminus (\mathrm{SO}_3(\mathbb{C}) \cdot \hat{Z})$ *is* $\mathcal{V}(\tilde{D}_0)$, *the variety where* $\tilde{D}_0$ *vanishes.*

*Proof.* Consider $(H_1 + H) \in \mathbb{C}^3 \oplus \hat{H}$. Assume $g \in \mathrm{SO}_3(\mathbb{C})$ is such that $\rho_1(g)(H_1) \in \mathrm{CW}$ so that $\rho(g)(H_1 + H) \in \mathcal{S}$. Then $\rho(g)(H_1 + H) \notin \hat{Z}$ iff $D_0(\rho(g)(H_1 + H)) \neq 0$, and equivalently $\tilde{D}_0(H_1 + H) \neq 0$.

Conversely for any $(H_1 + H) \in \mathbb{C}^3 \oplus \hat{H}$ we can find $g \in \mathrm{SO}_3(\mathbb{C})$ such that $\rho(g)(H_1 + H) \in \mathcal{S}$. Write
$H_1 = uU + vV + wW$. If both $u$ and $v$ are zero we can take $g$ to be the identity. Otherwise we can choose
$z_1, z_2 \in \mathbb{C}^3$ that form a basis of the kernel of the linear form over $\mathbb{C}^3$ $z \to (u, v, w) z$ and furthermore satisfy
$z_1^T z_1 = 1, z_2^T z_2 = 1$, and $z_1^T z_2 = 0$. One can then determine $z_3$ such that $g = (z_1, z_2, z_3)^T \in \mathrm{SO}_3(\mathbb{R})$. This $g$
is such that $\rho(g)(H_1 + H) \in \mathcal{S}$. $\square$

**3.5 Inverse problem**

With the functions in $\tilde{\mathcal{E}}$ we construct the rational map $\Phi : \mathcal{V} \to \mathbb{R}^{\#\tilde{\mathcal{E}}}$ that separates orbits almost everywhere.
The converse problem occurs: for a point $z \in \mathbb{R}^{\#\tilde{\mathcal{E}}}$, can we determine if $z$ is in the image $\Phi(\mathcal{V})$? If so, can
we find a preimage $v \in \mathcal{V}$ with $\Phi(v) = z$?

Any point in $\mathcal{V}$ has a representant of its orbit in $\mathcal{S}$. Thus $\Phi(\mathcal{V}) := \Phi(\mathcal{S})$. Since the restrictions of the functions in $\tilde{\mathcal{E}}$ to the slice are polynomial functions, $\Phi(\mathcal{V})$ is a semialgebraic set, i.e., a finite union of set in $\mathbb{R}^{\#\tilde{\mathcal{E}}}$ defined by polynomial equalities and inequalities.

**Proposition 3.19.** *Consider the real vector $z \in \mathbb{R}^{\#\tilde{\mathcal{E}}}$ defined by*

$$
z = \begin{pmatrix}
d_i, & 0 \le i \le p \\
t_j, & p+1 \le j \le n \\
r_{mj}, & 1 \le j \le n, & 1 \le m \le d_j \\
i_{mj}, & 1 \le j \le n, & 1 \le m \le d_j \text{ and } (m,j) \ne (1,1)
\end{pmatrix}.
$$
Recall that $y_0^2 = D_0$ and $y_{1,1}y_{-1,1} = \frac{1}{2}R_{11}$. The non separated orbits are thus pointed out by $\tilde{D}_0 = 0$ or $\tilde{R}_{11} = 0$. We also deduce a set of polynomials satisfying the same separation property. For any $P \in \mathcal{E}$, the rational function $\tilde{P}$ is well defined on $\text{SO}_3(\mathbb{C}) \times \tilde{\mathcal{Z}}$. We can determine $\tilde{\mathcal{H}} \setminus \text{SO}_3(\mathbb{C}) \times \tilde{\mathcal{Z}}$ as an irreducible hypersurface.

**Proposition 3.17.** Put $(u, v, w)$ the natural coordinates on $\hat{\mathcal{H}}_1 \subset \hat{\mathcal{H}}$, and note $Q = u^2 + v^2 + w^2 \in \mathbb{C}[\hat{\mathcal{H}}]$. Then, $\hat{\mathcal{H}} \setminus (\text{SO}_3(\mathbb{C}) \times \hat{\mathcal{Z}}) = \mathcal{V}(Q)$.

*Proof.* Let $z = (u, v, w)$ be a non zero point in $\mathcal{H}_1$. Take $g = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$ a matrix in $\text{SO}_3(\mathbb{C})$. The form $v \mapsto Q(v, z)$ is linear and its kernel $K_z$ is an hyperplane. Note that $g(z) = (r, 0, 0) \Leftrightarrow \begin{cases} Q(z, v_2) = 0. \\ Q(z, v_3) = 0. \end{cases}$ By dimension then, $v_2, v_3$ generate $K_z$ and $\forall w \in K_z$, $Q(v_1, w) = 0$.

* Suppose that $Q(z, z) = 0$ and $g(z) = (*)$ or $0$). Then, $z \in K_z$. Thus $Q(z, v_1) = 0$ and $v_2 \in K_z$. Then, $Q(v_1, v_1) = 0$ and that is a contradiction.

* Suppose that $Q(z, z) \neq 0$. Take then $r$ a complex number such that $r^2 = Q(z, z)$. With $v_2, v_3$ two orthonormal vectors in $K_z$ and $v_1 = \frac{\tilde{z}}{r}$, one has $Q(v_1, v_1) = 1$ and $Q(v_1, v_2) = Q(v_1, v_3) = 0$. This is $g(z) = (r, 0, 0)$. $\square$

**Corollary 3.18.** Let $P \in \mathbb{C}[\hat{S}]^{\mathbb{N}}$. Then, the denominator of $\tilde{P} \in \mathbb{C}(\hat{\mathcal{H}})^{\text{SO}_3(\mathbb{C})}$ is equal to $Q^k$, for some $k \in \mathbb{N}$.

*Proof.* On an irreducible affine variety, any rational function can be written as a quotient of two relatively prime polynomials $\frac{P}{Q}$ [CLO15]. Then, it is well defined exactly outside $\mathcal{V}(Q)$. $\square$

**Example 3.19.** In Example 3.15 we can observe that the polynomial $\tilde{D}_0$ is equal to the quadratic form $Q$ introduced in Proposition 3.17. It is the Euclidean norm on the real trace. Corollary 3.18 claims that the denominator of each rational function in $\tilde{\mathcal{E}}$ is thus a power of $\tilde{D}_0$.

As denominators are only powers of $Q = \tilde{D}_0$, one can select only the numerators of the elements of $\tilde{\mathcal{E}}$. It provides a set generating the invariant field composed with polynomials.

**Corollary 3.20.** Write $\tilde{\mathcal{E}} = \{P_1 = Q, \frac{P_1}{Q^{k_i}}, 1 \le i \ge \#\mathcal{E}\}$ with exponents $k_i$. Then $\mathcal{P} := \{Q, P_i, 2 \le i \le \#\mathcal{E}\}$ is a set of algebraically independent polynomials generating $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$. Moreover, it still separates orbits in $\tilde{\mathcal{U}}$.

*Proof.* Polynomials in $\mathcal{P}$ generate each generator in $\tilde{\mathcal{E}}$, and thus $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$. By cardinality they are algebraically independent.

Take two point $u_1, u_2$ in $\tilde{\mathcal{U}}$ which are not in the same orbit. Then $P_1 = Q$ does not vanish at $u_1$ and $u_2$. Suppose that $Q$ does not separate orbits of $u_1$ and $u_2$. Then, there is some $2 \le i \le \#\mathcal{E}$ such that $\frac{P_1}{Q^{k_i}}(u_1) \neq \frac{P_1}{Q^{k_i}}(u_2)$. By multiplication by $Q(u_1)^{k_i} = Q(u_2)^{k_i} \neq 0$ one has $P_i(u_1) \neq P_i(u_2)$. $\square$

## 3.6 Inverse problem

Functions of $\mathcal{E}$ enable to construct the rational map

$$ \Phi_{\tilde{\mathcal{E}}} : \begin{cases} \mathcal{H} \to \mathbb{R}^{\#\tilde{\mathcal{E}}} \\ h \mapsto (P(h), P \in \tilde{\mathcal{E}}) \end{cases} $$
Then, $z \in \Phi(\mathcal{V})$ iff one of the following conditions holds:

* $d_0 > 0$ and $r_{11} > 0$

* $d_0 > 0$ and $\forall j, m, r_{mj} = i_{mj} = 0$

* $d_0 = 0$ and $r_{11} > 0$ and $\forall j, m, i_{mj} = 0$ and $\forall j, d_j = 0$

* $d_0 = 0$ and $\forall j, m, r_{mj} = i_{mj} = 0$ and $\forall j, d_j = 0$.

The following proof is constructive; Its gives a primage in $\mathcal{S}$ when it exists.

**Proof.** Take $z \in \mathbb{R}^{\#\tilde{\mathcal{E}}}$ a vector as in Proposition 3.19. The slice $\mathcal{S}$ intersects all real orbits. Hence, there exists some real preimage $v \in \mathcal{V}$ iff there exists also a preimage in $\mathcal{S}$. Take then $v$ a real point in the slice.

Firstly, $D_0(H_1) = y_{0,0}^2 \ge 0$ and $R_{11}(H_1) = 2y_{1,1}y_{-1,1} = 2y_{1,1}\overline{y_{1,1}} \ge 0$. Then $\Phi_{\tilde{\mathcal{E}}}(v) = z \Rightarrow d_0 \ge 0$ and $r_{11} \ge 0$. We choose $y_{0,0} = \sqrt{d_0}$ and $y_{1,1}, y_{-1,1}$ conjugate complex numbers of module $\frac{1}{2}\sqrt{r_{11}}$.

Suppose that nor $d_0$ or $r_{11}$ vanish. Since $y_{1,1} \ne 0$ and $y_{0,0} \ne 0$ one can choose for $m \le j, y_{-m,j} = \frac{y_{0,0}r_{mj}-i i_{mj}}{2y_{0,0}y_{1,1}^m}$ and $y_{m,j} = \overline{y_{-m,j}} = \frac{y_{0,0}r_{mj}+i i_{mj}}{2y_{0,0}y_{-1,1}^m}$. That way,

$$
\begin{align*}
R_{mj}(H_1) &= y_{1,1}^m y_{-mj} + y_{-1,1}^m y_{m,j} \\
&= y_{1,1}^m \frac{y_{0,0} r_{mj} - i i_{mj}}{2 y_{0,0} y_{1,1}^m} + y_{-1,1}^m \frac{y_{0,0} r_{mj} + i i_{mj}}{2 y_{0,0} y_{-1,1}^m} \\
&= r_{mj} \\
I_{mj}(H_1) &= i y_{0,0} (y_{1,1}^m y_{-m,j} - y_{-1,1}^m y_{m,j}) \\
&= i y_{0,0} \left( y_{1,1}^m \frac{y_{0,0} r_{mj} - i i_{mj}}{2 y_{0,0} y_{1,1}^m} - y_{-1,1}^m \frac{y_{0,0} r_{mj} + i i_{mj}}{2 y_{0,0} y_{-1,1}^m} \right) \\
&= i_{mj}.
\end{align*}
$$

Take finally $y_{0,j} = \frac{d_j}{y_{0,0}}$ for $0 \le j \le p$ and $y_{0,j} = t_j$ for $p+1 \le j \le n$. Then $\Phi_{\tilde{\mathcal{E}}}(v) = z$.

Suppose that $R_{11}(H_1) = r_{11} = 0$. Then $y_{1,1} = y_{-1,1} = 0$ and $\Phi_{\tilde{\mathcal{E}}}(v) = z \Rightarrow \forall j, m, r_{mj} = i_{mj} = 0$. Take just $y_{0,j} = \frac{d_j}{y_{0,0}}$ for $0 \le j \le p$ and $y_{0,j} = t_j$ for $p+1 \le j \le n$ to have $\Phi_{\tilde{\mathcal{E}}}(v) = z$.

Suppose that $D_0(H_1) = d_0 = 0$. Then $y_{0,0} = 0$ and $\Phi_{\tilde{\mathcal{E}}}(v) = z \Rightarrow \begin{cases} \forall j, & i_{mj} = y_{0,0} \mathfrak{F}(y_{1,1}y_{-m,j}) = 0. \\ \forall j, & d_j = y_{0,0}y_{0i} = 0. \end{cases}$ One can then choose, for all $j \le n$ and $1 \le m \le d_j$, $y_{-m,j} = \frac{r_{mj}}{y_{1,1}^m}$ and $y_{m,j} = \overline{y_{-m,j}} = \frac{r_{mj}}{y_{-1,1}^m}$ to have $R_{mj}(v) = r_{mj}$. Take just $y_{0,j} = t_j$ for $j \le p$ to have $\Phi_{\tilde{\mathcal{E}}}(v) = z$.

Suppose that $R_{11}(H_1) = r_{11} = 0$ and $D_0(H_1) = d_0 = 0$. Then $y_{1,1} = y_{-1,1} = 0$ and $y_{0,0} = 0$. One has then

$$
\Phi_{\tilde{\mathcal{E}}}(v) = z \Rightarrow \begin{cases} \forall j, & i_{mj} = r_{mj} = 0. \\ \forall j, & d_j = y_{0,0}y_{0i} = 0. \end{cases} \quad \text{Take just } y_{0,j} = t_j \text{ for } j \le p \text{ to have } \Phi_{\tilde{\mathcal{E}}}(v) = z. \quad \square
$$

The above inequalities define a semialgebraic set $\Phi(\mathcal{V})$ birationally equivalent to the orbit space $\mathcal{V}/\text{SO}_3(\mathbb{R})$. We furthermore determine a subset birationally equivalent to the stratum of principal orbit type:

**Proposition 3.20.** If $\tilde{D}_0(v) > 0$ and $\tilde{R}_{11}(v) > 0$, then the stabilizer $\text{Stab}(v)$ is trivial.

**Proof.** Take $v$ a point in $\mathcal{V}$ and $s$ a point of its orbit lying on the slice. Write $(y_{0,0}, 0, 0, y_{1,1}, \dots, y_{-1,1})$ the coordinates of $s$. Suppose that $\tilde{D}_0(v) = D_0(s) > 0$ and $\tilde{R}_{11}(v) = R_{11}(s) > 0$. That is, $y_{0,0}, y_{1,1}, y_{-1,1}$ are non zero. Meanwhile, $\text{SO}_3(\mathbb{R})$ action on $\mathcal{H}_1$ is dual to the canonical action of $\text{SO}_3(\mathbb{R})$ on $\mathbb{R}^3$. A rotation $g \in \text{SO}_3(\mathbb{R})$ stabilizes $(0, y_{0,0}, 0)$ in $\mathcal{H}_1$ iff

$$
g \in \left\{ \begin{pmatrix} \cos(\alpha) & -\sin(\alpha) & 0 \\ \sin(\alpha) & \cos(\alpha) & 0 \\ 0 & 0 & 1 \end{pmatrix}, \alpha \in \mathbb{R} \right\} \cong \text{SO}_2(\mathbb{R}).
$$
that separates orbits almost everywhere. Note $\mathcal{X}$ the image $\Phi_{\tilde{\mathcal{E}}}(\mathcal{H})$. Then the converse problem occurs: for a point $z \in \mathbb{R}^{\#\tilde{\mathcal{E}}}$, how can we determine if $z \in \mathcal{X}$? And if the answer is yes, can we find a preimage $h \in \mathcal{H}$ with $\Phi_{\tilde{\mathcal{E}}}(h) = z$?

Any point in $\mathcal{H}$ has a representant of its orbit in $\mathcal{S}$, or it is a singular point for $\Phi_{\tilde{\mathcal{E}}}$. Thus $\mathcal{X} := \Phi_{\tilde{\mathcal{E}}}(\mathcal{S})$. Since functions in $\tilde{\mathcal{E}}$ are polynomials on the slice, $\mathcal{X}$ is semialgebraic. That is, it is a finite union of set in $\mathbb{R}^{\#\tilde{\mathcal{E}}}$ defined by polynomial equalities and inequalities:

**Proposition 3.21.** Let $z = \begin{pmatrix} t_i, & 1 \le i \le p \\ d_j, & 0 \le j \le q \\ r_{mj}, & 1 \le j \le n, & 1 \le m \le d_j \\ i_{mj}, & 1 \le j \le n, & 1 \le m \le d_j \end{pmatrix}$ and $(m, j) \ne (1, 1) \in \mathbb{R}^{\#\tilde{\mathcal{E}}}$ a real vector.

Then, $z \in \mathcal{X}$ iff one of the following conditions holds:

* $d_0 > 0$ and $r_{11} > 0$

* $d_0 > 0$ and $\forall j, m, r_{mj} = i_{mj} = 0$

* $d_0 = 0$ and $r_{11} > 0$

and $\forall j, m, i_{mj} = 0$

and $\forall j, d_j = 0$

* $d_0 = 0$ and $\forall j, m, r_{mj} = i_{mj} = 0$

and $\forall j, d_j = 0$

The following proof is constructive, giving thus a preimage in $\mathcal{S}$ at the same time.

*Proof.* Take $z \in \mathbb{R}^{\#\tilde{\mathcal{E}}}$ a vector as in Proposition 3.21. The slice $\mathcal{S}$ intersects all real orbits. Hence, there exists some real preimage $h \in \mathcal{H}$ iff there exists also a preimage in $\mathcal{S}$. Take then $h$ a real point in the slice.

Firstly, $D_0(y_0, h) = y_0^2 \ge 0$ and $R_{11}(y_0, h) = 2y_{1,1}y_{-1,1} = 2y_{1,1}\overline{y_{1,1}} \ge 0$. Then $\Phi_{\tilde{\mathcal{E}}}(h) = z \Rightarrow d_0 \ge 0$ and $r_{11} \ge 0$. We choose $y_0 = \sqrt{d_0}$ and $y_{11}, y_{-11}$ conjugated complex numbers of module $\frac{1}{2}\sqrt{r_{11}}$.

Suppose that nor $d_0$ or $r_{11}$ vanish. Since $y_{1,1} \ne 0$ and $y_0 \ne 0$ one can choose for $m \le j, y_{-m,j} = \frac{y_0r_{mj}-i i_{mj}}{2y_0y_{1,1}^m}$ and $y_{m,j} = \overline{y_{-m,j}} = \frac{y_0r_{mj}+i i_{mj}}{2y_0y_{1,1}^m}$. That way,

$$ 
\begin{align*}
R_{mj}(y_0, h) &= y_{1,1}^m y_{-m,j} + y_{-1,1}^m y_{m,j} \\
&= y_{1,1}^m \frac{y_0 r_{mj} - i i_{mj}}{2 y_0 y_{1,1}^m} + y_{-1,1}^m \frac{y_0 r_{mj} + i i_{mj}}{2 y_0 y_{-1,1}^m} \\
&= r_{mj} \\
I_{mj}(y_0, h) &= i y_0 (y_{1,1}^m y_{-m,j} - y_{-1,1}^m y_{m,j}) \\
&= i y_0 (y_{1,1}^m \frac{y_0 r_{mj} - i i_{mj}}{2 y_0 y_{1,1}^m} - y_{-1,1}^m \frac{y_0 r_{mj} + i i_{mj}}{2 y_0 y_{-1,1}^m}) \\
&= i_{mj}
\end{align*}
$$

Take finally $y_{0,j} = t_j$ for $j \le p$ and $y_{0,j} = \frac{d_j}{y_0}$ for $j \le q$. Then $\Phi_{\tilde{\mathcal{E}}}(h) = z$.

Suppose that $R_{11}(y_0, h) = r_{11} = 0$. Then $y_{1,1} = y_{-1,1} = 0$ and $\Phi_{\tilde{\mathcal{E}}}(h) = z \Rightarrow \forall j, m, r_{mj} = i_{mj} = 0$. Take just $y_{0,j} = t_j$ for $j \le p$ and $y_{0,j} = \frac{d_j}{y_0}$ for $j \le q$ to have $\Phi_{\tilde{\mathcal{E}}}(h) = z$.

Suppose that $D_0(y_0, h) = d_0 = 0$. Then $y_0 = 0$ and $\Phi_{\tilde{\mathcal{E}}}(h) = z \Rightarrow \begin{cases} \forall j, & i_{mj} = y_0 \Im(y_{11}y_{-m,j}) = 0. \\ \forall j, & d_j = y_0 y_{0i} = 0. \end{cases}$. One can then choose, for all $j \le n$ and $1 \le m \le d_j$, $y_{-m,j} = \frac{r_{mj}}{y_{1,1}^m}$ and $y_{m,j} = \overline{y_{-m,j}} = \frac{r_{mj}}{y_{-1,1}^m}$ to have $R_{mj}(h) = r_{mj}$. Take just $y_{0,j} = t_j$ for $j \le p$ to have $\Phi_{\tilde{\mathcal{E}}}(h) = z$.

Suppose that $R_{11}(y_0, h) = r_{11} = 0$ and $D_0(y_0, h) = d_0 = 0$. Then $y_{1,1} = y_{-1,1} = 0$ and $y_0 = 0$. One has then
$$ \Phi_{\tilde{\mathcal{E}}}(h) = z \Rightarrow \begin{cases} \forall j, & i_{mj} = r_{mj} = 0. \\ \forall j, & d_j = y_0 y_{0i} = 0. \end{cases} \text{ Take just } y_{0,j} = t_j \text{ for } j \le p \text{ to have } \Phi_{\tilde{\mathcal{E}}}(h) = z. \square $$

These inequalities are precious to study the orbit set, giving a semialgebraic subset $\mathcal{X}$ related to $\mathcal{H}/\text{SO}_3(\mathbb{R})$. The values of generating polynomials given by $\Phi_{\tilde{\mathcal{E}}}$ allows almost everywhere to determine when a point $h \in \mathcal{H}$ is of principal orbit type, namely points whose stabilizer in $\text{SO}_3(\mathbb{R})$ is $\text{Stab}(h) = \{I_3\}$:
This subgroup is normal in N and its action on $Vect(y_{1,-1}, y_{1,1})$ is isomorphic to $V_1$. Its only point with a non trivial stabilizer is $(0,0)$. Then, $\mathrm{Stab}(s) = \{I_3\}$. But, the stabilizers $\mathrm{Stab}(v)$ and $\mathrm{Stab}(s)$ are conjugate in $\mathrm{SO}_3(\mathbb{R})$. Thus, $\mathrm{Stab}(v) = \{I_3\}$. $\square$

However, the converse of Proposition 3.20 is not true: consider the point with $y_{0,1} = y_{-1,2} = y_{1,2} = 1$, all the other coordinates being zero. Then $\mathrm{Stab}(v)$ is trivial while $\tilde{D}_0(v) = \tilde{R}_{11}(v) = 0$. On the other hand all the orbits with non trivial isotropy groups lie in $V(\tilde{D}_0)$ and are mapped to zero. Our generating set has thus important limitations to stratify the orbit space. In [HJ24a] we design a strategy that goes beyond the present result by finding other Seshadri slices in the remaining variety $V(\tilde{D}_0)$. This allows to compute step by step a complete separating system, and to stratify $V$ according to the orbit types.

# 4 Generalisation to $O_3(\mathbb{R})$ representations

We consider the representation of the group $O_3(\mathbb{R}) \triangleright \mathrm{SO}_3(\mathbb{R})$ on

$$ \mathcal{H} := \mathcal{H}_1 \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} \oplus \bigoplus_{j=1}^{n^{\sharp}} \mathcal{H}_{e_j}^{\sharp}. $$

where $d_1, ..., d_n, e_1, ..., e_n$ are integers. Both $\mathcal{H}_d^{\sharp}$ and $\mathcal{H}_d$ are the space of harmonic homogeneous polynomials of degree $d$. The representation $\rho_d: O_3(\mathbb{R}) \to \mathrm{GL}(\mathcal{H}_d)$ is defined as for $\mathrm{SO}_3(\mathbb{R})$ in Section 3.1. The representation $\rho_d^{\sharp}: O_3(\mathbb{R}) \to \mathrm{GL}(\mathcal{H}_d^{\sharp})$ is obtained as

$$ \rho_d^{\sharp} : \left\{ \begin{array}{rcl} O_3(\mathbb{R}) & \to & \mathrm{GL}(\mathcal{H}_d^{\sharp}) \\ g & \mapsto & \mathrm{det}(g) \rho_d(g) \end{array} \right. $$

The representations $(\rho_d, \mathcal{H}_d)$ and $(\rho_d^{\sharp}, \mathcal{H}_d^{\sharp})$, $d \in \mathbb{N}$, are the only irreducible representations of $O_3(\mathbb{R})$ [GSS88]. For simplicity we assume that $d_1, ..., d_n$ are sorted decreasingly. A new Seshadri slice $(S, N^{\sharp})$ can be identified with the same subspace $S$ as in Proposition 3.8, and with the normalizer $N^{\sharp} \triangleright N$, generated by $N$ and $-I_3$. This forms the direct product $N^{\sharp} = N \times C_2$. We consider the polynomials:

$$ \begin{align*} D_j^{\sharp} &= \begin{array}{c} y_{0,0}y_{0,j}^{\sharp} \\ y_{0,j}^{\sharp} \end{array} & 1 \le j \le p^{\sharp} \\ T_j^{\sharp} &= \begin{array}{c} y_{0,j}^{\sharp} \\ y_{m,j}^{\sharp}y_{-1,1}^{m} + y_{-m,j}^{\sharp}y_{1,1}^{m} \end{array} & p^{\sharp} + 1 \le j \le n^{\sharp} \\ R_{mj}^{\sharp} &= \begin{array}{c} y_{m,j}^{\sharp}y_{-1,1}^{m} + y_{-m,j}^{\sharp}y_{1,1}^{m} \\ y_{m,j}^{\sharp}y_{-1,1}^{m} - y_{-m,j}^{\sharp}y_{1,1}^{m} \end{array} & j \le n^{\sharp} & -e_j \le m \le e_j \\ I_{mj}^{\sharp} &= \begin{array}{c} y_{0,0} \\ y_{m,j}^{\sharp}y_{-1,1}^{m} - y_{-m,j}^{\sharp}y_{1,1}^{m} \end{array} & j \le n^{\sharp} & -e_j \le m \le e_j \end{align*} $$

**Theorem 4.1.** Take $P$ a polynomial in $\mathbb{R}(S)^N$ as follows:

* If $n^{\sharp} \ge 1$, choose $P := T_j^{\sharp}$ (for some $e_j$ even) or $P := D_j^{\sharp}$ (for some $e_j$ odd).

* Else, if $d_1 \ge 2$, choose $P := I_{d_1 1}$.

* Else, if $n \ge 2$ and $d_2 = 1$, choose $P := I_{12}$.

* Else, choose $P := 1$.
**Proposition 3.22.** If $\tilde{D}_0(h) > 0$ and $\tilde{R}_{11}(h) > 0$, then the stabilizer $\text{Stab}(h)$ is trivial.

*Proof.* Take $h$ a point in $\mathcal{H}$ and $s$ a point of its orbit lying on the slice. Write $(y_0, 0, 0, y_{1,1}, \dots, y_{-1,1})$ the coordinates of $s$ in $\mathcal{S}$. Suppose that $\tilde{D}_0(h) = D_0(s) > 0$ and $\tilde{R}_{11}(h) = R_{11}(s) > 0$. That is, $y_0, y_{1,1}, y_{-1,1}$ are non zero. Meanwhile, $\text{SO}_3(\mathbb{R})$ action on $\mathcal{H}_1$ is dual to the canonic action of $\text{SO}_3(\mathbb{R})$ on $\mathbb{R}^3$. A rotation $g \in \text{SO}_3(\mathbb{R})$ stabilizes $(y_0, 0, 0)$ in $\mathcal{H}_1$ iff $g \in \begin{Bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{Bmatrix}, \theta \in \mathbb{R}$ $\cong \text{SO}_2(\mathbb{R})$. This subgroup is normal in $\mathbb{N}$ and its action on $\text{Vect}(y_{1,1}, y_{-1,1})$ is isomorphic to $\mathcal{V}_1$. Its only point with a non trivial stabilizer is $(0, 0)$. Then, $\text{Stab}(s) = \{\text{Id}\}$. But, the stabilizers $\text{Stab}(h)$ and $\text{Stab}(s)$ are conjugated in $\text{SO}_3(\mathbb{R})$. Thus, $\text{Stab}(h) = \{I_3\}$. $\square$

The inequalities in the above proposition do not capture the whole of the principal stratum though. Indeed the converse of Proposition 3.22 is not true: consider the point with $y_{0,1} = y_{-1,2} = y_{1,2} = 1$, all the other coordinates being zero. Then $\text{Stab}(h)$ is trivial while $\tilde{D}_0(h) = \tilde{R}_{11}(h) = 0$.

On the other hand all the orbits with non trivial isotropy groups lie in $\mathcal{V}(Q)$ and are mapped to zero. Our separating set has thus important limitations to caputre and stratify the orbit space. One can nonetheless design a strategy that goes beyond the present result by finding other Seshadri slices in the remaining variety $\mathcal{V}(Q)$. It allows to compute step by step a complete separating system, and to stratify the orbit space. This is examined in for the space of piezoelectric tensors in [HJ24b].

# 4 Generalisation to $\text{O}_3(\mathbb{R})$ representations

We consider the representations of the group $\text{O}_3(\mathbb{R}) \triangleright \text{SO}_3(\mathbb{R})$

$$ \mathcal{H} := \mathcal{H}_1 \oplus \bigoplus_{j=1}^{n} \mathcal{H}_{d_j} \oplus \bigoplus_{j=1}^{n^{\sharp}} \mathcal{H}_{e_j}^{\sharp}. $$

where $d_1, \dots, d_n, e_1, \dots, e_{n^{\sharp}}$ are integers. Both $\mathcal{H}_d^{\sharp}$ and $\mathcal{H}_d$ can be thought of as the space of harmonic homogeneous polynomials of degree $d$. The representation on $\mathcal{H}_d$ is the restriction to $\mathcal{H}_d$ of the representation of $\text{O}_3(\mathbb{R})$ on the polynomial ring $\mathbb{R}[U, V, W]$ obtained by composition with the standard representation of $\text{O}_3(\mathbb{R})$ on $\mathbb{R}^3$. We call it $\rho_d: \text{O}_3(\mathbb{R}) \to \text{GL}(\mathcal{H}_d)$ as was done for $\text{SO}_3(\mathbb{R})$ in Section 3.1. The $\text{O}_3(\mathbb{R})$ representation $\rho_d^{\sharp}$ on $\mathcal{H}_d^{\sharp}$ is defined by

$$ \rho_d^{\sharp} : \begin{cases} \text{O}_3(\mathbb{R}) & \to \text{GL}(\mathcal{H}_d^{\sharp}) \\ g & \mapsto \det(g) \rho_d(g) \end{cases} $$

The representations $(\rho_d, \mathcal{H}_d)$ and $(\rho_d^{\sharp}, \mathcal{H}_d^{\sharp})$, $d \in \mathbb{N}$, are the only irreducible representations of $\text{O}_3(\mathbb{R})$ [GSS88].

For simplicity we assume that $d_1, \dots, d_n$ are sorted decreasingly. A new Seshadri slice $(\mathcal{S}, \mathbb{N}^{\sharp})$ can be identified with the same subspace $\mathcal{S}$ (Proposition 3.2) and with the normalizer $\mathbb{N}^{\sharp} \triangleright \mathbb{N}$, generated by $\mathbb{N}$ and $-I_3$. It forms the direct product $\mathbb{N}^{\sharp} = \mathbb{N} \times \mathbb{C}_2$. We create the polynomials

$$ \begin{align*} T_j^{\sharp} &= t_j^{\sharp} & j \le p^{\sharp} \\ D_j^{\sharp} &= d_j^{\sharp} & j \le q^{\sharp} \\ R_{mj}^{\sharp} &= y_{m,j}^{\sharp} y_{-1,1}^{m} + y_{-m,j}^{\sharp} y_{1,1}^{m} & j \le n^{\sharp} & -e_j \le m \le e_j \\ I_{mj}^{\sharp} &= i y_0 \left( y_{m,j}^{\sharp} y_{-1,1}^{m} - y_{-m,j}^{\sharp} y_{1,1}^{m} \right) & j \le n^{\sharp} & -e_j \le m \le e_j \end{align*} $$

**Theorem 4.1.** Take $P$ a polynomial in $\mathbb{R}(\mathcal{S})^\mathbb{N}$ as follows:
Then, the following set of algebraically independent polynomials generates both $\mathbb{R}(S)^{\mathbb{N}^z}$ or $\mathbb{C}(\hat{S})^{\hat{\mathbb{N}}^z}$:

$$ \mathcal{E}^{\dagger} = \left\{ \begin{array}{l} D_j^{\dagger} = D_j, \quad d_j \text{ odd} \\ T_j^{\dagger} = T_j, \quad d_j \text{ even} \\ R_{mj}^{\dagger} = \begin{cases} R_{mj}, & d_1 m + d_j \text{ even} \\ PR_{mj}, & d_1 m + d_j \text{ odd} \end{cases} \\ I_{mj}^{\dagger} = \begin{cases} PI_{mj}, & d_1 m + d_j \text{ even} \\ I_{mj}, & d_1 m + d_j \text{ odd} \end{cases} \end{array} \right\} \cup \left\{ \begin{array}{l} D_j^{\ddagger} = PD_j^{\ddagger}, \quad e_j \text{ odd} \\ T_j^{\ddagger} = PT_j^{\ddagger}, \quad e_j \text{ even} \\ R_{mj}^{\ddagger} = \begin{cases} PR_{mj}^{\ddagger}, & d_1 m + e_j \text{ even} \\ R_{mj}^{\ddagger}, & d_1 m + e_j \text{ odd} \end{cases} \\ I_{mj}^{\ddagger} = \begin{cases} I_{mj}^{\ddagger}, & d_1 m + e_j \text{ even} \\ PI_{mj}^{\ddagger}, & d_1 m + e_j \text{ odd} \end{cases} \end{array} \right\} $$

*Proof.* The direct product $\hat{\mathbb{N}}^{\mathbb{N}} = \hat{\mathbb{N}} \times \mathbb{C}_2$ gives $\mathbb{C}(\hat{S})^{\hat{\mathbb{N}}^z} = (\mathbb{C}(\hat{S})^{\hat{\mathbb{N}}})^{\mathbb{C}_2}$. Theorem 3.9 gives the same minimal set of generators for $\mathbb{C}(\hat{S})^{\hat{\mathbb{N}}}$:

$$ \mathcal{E} = \left\{ \begin{array}{l} D_j, \quad d_j \text{ odd} \\ T_j, \quad d_j \text{ even} \\ R_{mj}, \quad 1 \le j \le n, m \le d_j \\ I_{mj}, \quad 1 \le j \le n, m \le d_j, (j, m) \ne (1, 1) \end{array} \right\} \cup \left\{ \begin{array}{l} T_j^{\ddagger}, \quad e_j \text{ even} \\ D_j^{\ddagger}, \quad e_j \text{ odd} \\ R_{mj}^{\ddagger}, \quad 1 \le j \le n^{\ddagger}, m \le e_j \\ I_{mj}^{\ddagger}, \quad 1 \le j \le n^{\ddagger}, m \le e_j \end{array} \right\} $$

Take then the symmetry $\sigma = -I_3 \in \hat{\mathbb{N}}^{\mathbb{N}}$. In spherical coordinates it acts as

$$ \forall (r, \theta, \varphi), \sigma((r, \theta, \varphi)) = (r, \pi - \theta, \varphi + \pi). $$

By composition, for a vector $Y_{d_j}^m \in \mathcal{H}_{d_j}$, $\sigma$ acts as

$$ \begin{aligned} \rho_{d_j}(\sigma)(Y_{d_j}^m)((r, \theta, \varphi)) &= Y_{d_j}^m(\sigma((r, \theta, \varphi))) \\ &= Y_{d_j}^m(r, \pi - \theta, \varphi + \pi) \\ &= r^d c_d^m e^{im(\varphi+\pi)} P_{d_j}^m(-\cos(\theta)) \\ &= (-1)^{d_j} Y_{d_j}^m((r, \theta, \varphi)) \end{aligned} $$

and in a sharped representation $\mathcal{H}_{d_j}^{\ddagger}$,

$$ \begin{aligned} \rho_{d_j}^{\ddagger}(\sigma)(Y_{d_j}^m)((r, \theta, \varphi)) &= -Y_{d_j}^m(\sigma((r, \theta, \varphi))) \\ &= -Y_{d_j}^m(r, \pi - \theta, \varphi + \pi) \\ &= -r^d c_d^m e^{im(\varphi+\pi)} P_{d_j}^m(-\cos(\theta)) \\ &= (-1)^{d_j+1} Y_{d_j}^m((r, \theta, \varphi)) \end{aligned} $$

It thus has an induced action on each elements in $\mathcal{E}$:

$$ \left\{ \begin{array}{l} \sigma(D_j) = D_j, \quad d_j \text{ odd} \\ \sigma(D_j^{\ddagger}) = -D_j^{\ddagger}, \quad e_j \text{ odd} \\ \sigma(T_j) = T_j, \quad d_j \text{ even} \\ \sigma(T_j^{\ddagger}) = -T_j^{\ddagger}, \quad e_j \text{ even} \end{array} \right\} \quad \text{and} \quad \left\{ \begin{array}{l} \sigma(R_{mj}) = (-1)^{d_j+d_1 m} R_{mj}, \\ \sigma(R_{mj}^{\ddagger}) = (-1)^{1+e_j+d_1 m} R_{mj}^{\ddagger} \\ \sigma(I_{mj}) = (-1)^{1+d_j+d_1 m} I_{mj} \\ \sigma(I_{mj}^{\ddagger}) = (-1)^{e_j+d_1 m} I_{mj}^{\ddagger} \end{array} \right. $$

We next apply Lemma 2.5. We need a polynomial $P \in \mathcal{E}$ flipped by $\sigma$. Whenever $n^{\ddagger} \ge 1$, one can choose $P := T_j^{\ddagger}$ (for $e_j$ even) or $P := D_j^{\ddagger}$ (for $e_j$ odd). Else it depends on $d_1$. If $d_1 \ge 2$ then $I_{11}$ suits. If $d_1 = 1, n \ge 2$ and $d_2 = 1$ then $I_{12}$ suits. Otherwise, no polynomial in $\mathcal{E}$ is flipped and $\mathbb{C}(\hat{S})^{\hat{\mathbb{N}}} = \mathbb{C}(\hat{S})^{\hat{\mathbb{N}}^{\ddagger}}$. $\square$

Almost free $\mathcal{O}_3(\mathbb{C})$ actions have a stably rational invariant field [CS07, Proposition 4.10]. The above theorem provides the following stronger result, yet with a different hypothesis as already discussed after Theorem 3.9.
• If $n^\sharp \ge 1$, choose $P := T_j^\sharp$ (for some $e_j$ even) or $P := D_j^\sharp$ (for some $e_j$ odd).

• Else, if $d_1 \ge 2$, choose $P := I_{d_1 1}$.

• Else, if $n \ge 2$ and $d_2 = 1$, choose $P := I_{12}$.

• Else, choose $P := 1$.

Then, the following set of algebraically independent polynomials generates both $\mathbb{R}(S)^{\mathbb{N}^\sharp}$ or $\mathbb{C}(\hat{S})^{\mathbb{N}^\sharp}$:

$$ \mathcal{E}^{\dagger} = \left\{ \begin{array}{l} D_0^{\dagger} = D_0 \\ T_j^{\dagger} = T_j, \qquad d_j \text{ even} \\ D_j^{\dagger} = D_j, \qquad d_j \text{ odd} \\ R_{mj}^{\dagger} = \left\{ \begin{array}{l} R_{mj}, \qquad d_1 m + d_j \text{ even} \\ PR_{mj}, \qquad d_1 m + d_j \text{ odd} \end{array} \right. \\ I_{mj}^{\dagger} = \left\{ \begin{array}{l} PI_{mj}, \qquad d_1 m + d_j \text{ even} \\ I_{mj}, \qquad d_1 m + d_j \text{ odd} \end{array} \right. \end{array} \right\} \cup \left\{ \begin{array}{l} T_j^{\ddagger} = P T_j^{\ddagger}, \qquad e_j \text{ even} \\ D_j^{\ddagger} = P D_j^{\ddagger}, \qquad e_j \text{ odd} \\ R_{mj}^{\ddagger} = \left\{ \begin{array}{l} PR_{mj}^{\ddagger}, \qquad d_1 m + e_j \text{ even} \\ R_{mj}^{\ddagger}, \qquad d_1 m + e_j \text{ odd} \end{array} \right. \\ I_{mj}^{\ddagger} = \left\{ \begin{array}{l} I_{mj}^{\ddagger}, \qquad d_1 m + e_j \text{ even} \\ PI_{mj}^{\ddagger}, \qquad d_1 m + e_j \text{ odd} \end{array} \right. \end{array} \right\} $$

*Proof.* We notice the direct product $\hat{N}^\sharp = \hat{N} \times C_2$. This gives $\mathbb{C}(\hat{S})^{\mathbb{N}^\sharp} = \mathbb{C}(\mathbb{C}(\hat{S})^{\mathbb{N}})^{C_2}$. Theorem 3.7 gives the same minimal set of generators for $\mathbb{C}(\hat{S})^{\mathbb{N}}$, independantly with the sharp:

$$ \mathcal{E} = \left\{ \begin{array}{l} T_j(y_0, h) = \begin{array}{l} y_{0,j} \\ y_{0,j}^{\sharp} \\ y_0^2 \end{array}, \qquad d_j \text{ even} \\ T_j^{\sharp}(y_0, h) = \begin{array}{l} y_{0,j} \\ y_0^2 \end{array}, \qquad e_j \text{ even} \\ D_0(y_0, h) = \begin{array}{l} y_0 \\ y_0 \end{array}, \qquad d_j \text{ odd} \\ D_j(y_0, h) = \begin{array}{l} y_0 \cdot y_{0,j}, \\ y_0 y_{0,j}^{\sharp} \end{array}, \qquad e_j \text{ odd} \\ D_j^{\sharp}(y_0, h) = \begin{array}{l} y_0 \cdot y_{0,j}^{\sharp}, \\ y_0 y_{0,j}^{\sharp} \end{array}, \qquad 1 \le j \le n, m \le d_j \\ R_{mj}(y_0, h) = \begin{array}{l} y_{m,j} y_{-1,1}^{m} + y_{-m,j} y_{1,1}^{m}, \\ y_{m,j}^{\sharp} y_{-1,1}^{\sharp} + y_{-m,j}^{\sharp} y_{1,1}^{\sharp} \end{array}, \qquad 1 \le j \le n^\sharp, m \le e_j \\ R_{mj}^{\sharp}(y_0, h) = \begin{array}{l} y_{m,j}^{\sharp} y_{-1,1}^{\sharp} + y_{-m,j}^{\sharp} y_{1,1}^{\sharp}, \\ y_{m,j}^{\sharp} y_{-1,1}^{\sharp} - y_{-m,j}^{\sharp} y_{1,1}^{\sharp} \end{array}, \qquad 1 \le j \le n^\sharp, m \le d_j, (j, m) \ne (1, 1) \\ I_{mj}(y_0, h) = \begin{array}{l} iy_0 \left(y_{m,j} y_{-1,1}^{m} - y_{-m,j} y_{1,1}^{m}\right), \\ iy_0 \left(y_{m,j}^{\sharp} y_{-1,1}^{\sharp} - y_{-m,j}^{\sharp} y_{1,1}^{\sharp}\right) \end{array}, \qquad 1 \le j \le n^\sharp, m \le e_j \\ I_{mj}^{\sharp}(y_0, h) = \begin{array}{l} iy_0 \left(y_{m,j}^{\sharp} y_{-1,1}^{\sharp} - y_{-m,j}^{\sharp} y_{1,1}^{\sharp}\right), \\ iy_0 \left(y_{m,j}^{\sharp} y_{-1,1}^{\sharp} - y_{-m,j}^{\sharp} y_{1,1}^{\sharp}\right) \end{array}, \qquad 1 \le j \le n^\sharp, m \le e_j \end{array} \right\} $$

Take then the symmetry $\sigma = -I_3 \in \hat{N}^\sharp$. In spherical coordinates it acts as $\forall (r, \theta, \varphi), \sigma((r, \theta, \varphi)) = (r, \pi - \theta, \varphi + \pi)$. By composition, for a vector $Y_{d_j}^m \in \mathcal{H}_{d_j}$, $\sigma$ acts as

$$ \begin{aligned} \rho_{d_j}(\sigma)(Y_{d_j}^m)((r, \theta, \varphi)) &= Y_{d_j}^m(\sigma((r, \theta, \varphi))) \\ &= Y_{d_j}^m(r, \pi - \theta, \varphi + \pi) \\ &= r^d c_d^m e^{im(\varphi+\pi)} P_{d_j}^m(-\cos(\theta)) \\ &= (-1)^{d_j} Y_{d_j}^m((r, \theta, \varphi)) \end{aligned} $$

and in a sharped representation $\mathcal{H}_{d_j}^\sharp$,

$$ \begin{aligned} \rho_{d_j}^{\sharp}(\sigma)(Y_{d_j}^m)((r, \theta, \varphi)) &= -Y_{d_j}^m(\sigma((r, \theta, \varphi))) \\ &= -Y_{d_j}^m(r, \pi - \theta, \varphi + \pi) \\ &= -r^d c_d^m e^{im(\varphi+\pi)} P_{d_j}^m(-\cos(\theta)) \\ &= (-1)^{d_j+1} Y_{d_j}^m((r, \theta, \varphi)) \end{aligned} $$
**Corollary 4.2.** Let $\rho: \mathrm{O}_3(\mathbb{R}) \to \mathrm{GL}(\mathcal{H})$ a linear representation including at least once the irreducible representation $(\rho_1, \mathcal{H}_1)$. Then, both $\mathrm{R}(\mathcal{V})^{\mathrm{O}_3(\mathbb{R})}$ and $\mathrm{C}(\hat{\mathcal{V}})^{\mathrm{O}_3(\mathbb{C})}$ are rational.

**Example 4.3.** We follow up on Example 3.11 where the representation space was $\mathcal{T} = \mathcal{H}_1 \oplus \mathcal{H}_3$. We examined the $\mathrm{SO}_3(\mathbb{R})$-invariants, giving their restrictions to the slice. The restrictions to the slice $\mathcal{S}_\mathcal{T}$ of a generating set of rational $\mathrm{O}_3(\mathbb{R})$-invariant is given by the following algebraically independent generating set for $\mathrm{R}(\mathcal{S}_\mathcal{T})^{\mathbb{N}^2}$:

$$ \mathcal{E}_{\mathcal{T}}^{\dagger} = \left\{ \begin{array}{l} D_{1}^{\dagger} = y_{0,0}^{2} \\ D_{2}^{\dagger} = y_{0,0}y_{0,1} \\ R_{1}^{\dagger} = 2y_{1,1}y_{-1,1} \\ R_{2}^{\dagger} = (y_{1,1}^{2}y_{-2,1} + y_{-1,1}^{2}y_{2,1})^{2} \\ R_{3}^{\dagger} = y_{1,1}^{3}y_{-3,1} + y_{-1,1}^{3}y_{3,1} \\ I_{2}^{\dagger} = iy_{0,0}(y_{1,1}^{2}y_{-2,1} - y_{-1,1}^{2}y_{2,1}) \\ I_{3}^{\dagger} = iy_{0,0}(y_{1,1}^{3}y_{-3,1} - y_{-1,1}^{3}y_{3,1})(y_{1,1}^{2}y_{-2,1} + y_{-1,1}^{2}y_{2,1}) \end{array} \right\} $$

# 5 Applications

We have exhibited invariants for representations of $\mathrm{SO}_3(\mathbb{R})$ and $\mathrm{O}_3(\mathbb{R})$ on spaces $\mathbb{R}^3 \oplus \mathcal{H}$. Among those is the well studied case of $\mathbb{R}^3 \times \dots \times \mathbb{R}^3$ treated to some extent in Example 3.12. This representation and its invariants are relevant in several areas, in particular to design invariant neural networks [BV23; DCP21; DG24; GD04]. In this section we apply our method for three other cases of relevance.

## 5.1 Odd degree ternary forms

Among the cases of interest to which our construction applies is the representation of $\mathrm{SO}_3(\mathbb{R})$ or $\mathrm{O}_3(\mathbb{R})$ on the vector space of homogeneous polynomials $\mathrm{R}[U, V, W]_d$ of odd degree $d$. The case of even degrees was treated in [GHP19]. The starting point for both odd and even degrees is the harmonic decomposition.

**Lemma 5.1.** [AH12, Theorem 2.18] $\mathrm{R}[U, V, W]_d = \bigoplus_{j=0}^{[d/2]} Q^j \mathcal{H}_{d-2j}$ where $Q = U^2 + V^2 + W^2$.

Whenever $d$ is odd, the last term of this decomposition is $\mathcal{H}_1$ and thus we can apply the construction provided in previous sections.

**Corollary 5.2.** For any $d \in \mathbb{N}$, the invariant field of the $\mathrm{SO}_3(\mathbb{R})$ or $\mathrm{O}_3(\mathbb{R})$ action on $\mathrm{R}[U, V, W]_d$ is rational.

**Example 5.3.** The vector space $\mathcal{T} = \mathcal{H}_1^3 \oplus \mathcal{H}_3^3$ introduced in Example 3.11 is isomorphic to the space of ternary cubics $\mathrm{R}[U, V, W]_3$. The basis of $\mathrm{C}[U, V, W]_3$ we use is:

$$ \{Q Y_{-1,1}, Q Y_{0,1}, Q Y_{1,1}; Y_{-3,3}, Y_{-2,3}, Y_{-1,3}, Y_{0,3}, Y_{1,3}, Y_{2,3}, Y_{3,3}\}. $$

In this basis an element of $\mathrm{C}[U, V, W]_3$ is given by its vector of coefficients

$$ (y_{-1,0}, y_{0,0}, y_{1,0}; y_{-3,1}, y_{-2,1}, y_{-1,1}, y_{0,1}, y_{1,1}, y_{2,1}, y_{3,1})^T. $$

The Seshadri slice for $\mathrm{R}[U, V, W]_3$ is $\mathcal{S} = \mathrm{RQW} \oplus \mathcal{H}_3^3$ with stabilizer $\mathbb{N} \cong \mathrm{O}_2(\mathbb{R})$. The set $\mathcal{E}_{\mathcal{T}}$ in Example 3.11 is an algebraically independent generating set of $\mathrm{R}(\mathcal{S})^N$. By Theorem 3.6, $\mathcal{E}_{\mathcal{T}}$ is an algebraically generating set of $\mathrm{R}(\mathcal{T})^{\mathrm{SO}_3(\mathbb{R})}$. Wigner matrices give the expressions of the pulled back invariants with respect to spherical
It thus has an induced action on each elements in $\mathcal{E}$:

$$ \left\{ \begin{array}{lcl} \sigma(T_j) & = & T_j, \quad d_j \text{ even} \\ \sigma(T_j^\sharp) & = & -T_j^\sharp, \quad e_j \text{ even} \\ \sigma(D_0) & = & D_0 \\ \sigma(D_j) & = & D_j, \quad d_j \text{ odd} \\ \sigma(D_j^\sharp) & = & -D_j^\sharp, \quad e_j \text{ odd} \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{lcl} \sigma(R_{mj}) & = & (-1)^{d_j+d_1m} R_{mj}, \\ \sigma(R_{mj}^\sharp) & = & (-1)^{1+e_j+d_1m} R_{mj}^\sharp \\ \sigma(I_{mj}) & = & (-1)^{1+d_j+d_1m} I_{mj} \\ \sigma(I_{mj}^\sharp) & = & (-1)^{e_j+d_1m} I_{mj}^\sharp, \end{array} \right. $$

We next apply lemma 2.5. We need a polynomial $P \in \mathcal{E}$ flipped by $\sigma$. Whenever $n^\sharp \ge 1$, one can choose $P := T_j^\sharp$ (for $e_j$ even) or $P := D_j^\sharp$ (for $e_j$ odd). Else it depends on $d_1$. If $d_1 \ge 2$ then $I_{11}$ suits. If $d_1 = 1, n \ge 2$ and $d_2 = 1$ then $I_{12}$ suits. Otherwise, no polynomial in $\mathcal{E}$ is flipped and $C(\hat{S})^{\mathcal{N}} = C(\hat{S})^{\mathcal{N}^\sharp}$. $\square$

<figure><img src="image_4.png" /></figure>

Almost free $O_3(\mathbb{C})$ actions have a stably rational invariant field [CS05, Proposition 4.10]. The above theorem provides the following stronger result, yet with a different hypothesis as already discussed after Theorem 3.7.

**Corollary 4.2.** Let $\rho: O_3(\mathbb{R}) \to GL(\mathcal{H})$ a linear representation including at least once the irreducible representation $(\rho_1, \mathcal{H}_1)$. Then, both $\mathbb{R}(\mathcal{H})^{O_3(\mathbb{R})}$ and $\mathbb{C}(\mathcal{H})^{O_3(\mathbb{C})}$ are rational.

**Example 4.3.** The ternary cubics $O_3(\mathbb{R})$-representation $\mathbb{R}[U, V, W]_3$ is isomorphic to $\mathcal{T} = \mathcal{H}_1 \oplus \mathcal{H}_3$. With the basis introduced in Example 3.2, an algebraically independent generating set for the slice invariants $\mathbb{R}(S_{\mathcal{T}})^{\mathcal{N}^\sharp}$ is thus

$$ \mathcal{E}_{\mathcal{T}}^{\dagger} = \left\{ \begin{array}{lcl} D_1^{\dagger} & = & y_0^2 \\ D_3^{\dagger} & = & y_0 y_{0,3} \\ R_1^{\dagger} & = & 2y_{1,3}y_{-1,3} \\ R_2^{\dagger} & = & (y_{1,3}^2 y_{-2,3} + y_{-1,3}^2 y_{2,3})^2 \\ R_3^{\dagger} & = & y_{1,3}^3 y_{-3,3} + y_{-1,3}^3 y_{3,3} \\ I_2^{\dagger} & = & i y_0 (y_{1,3}^2 y_{-2,3} - y_{-1,3}^2 y_{2,3}) \\ I_3^{\dagger} & = & i y_0 (y_{1,3}^3 y_{-3,3} - y_{-1,3}^3 y_{3,3}) (y_{1,3}^2 y_{-2,3} + y_{-1,3}^2 y_{2,3}) \end{array} \right\} $$

# 5 Applications

We have exhibited invariants for representations of $SO_3(\mathbb{R})$ and $O_3(\mathbb{R})$ on spaces $\mathbb{R}^3 \oplus \mathbb{H}$. Among those is the well studied case of $\mathbb{R}^3 \times \dots \times \mathbb{R}^3$ treated to some extent in Example 3.15. This representation and its invariants are relevant in several areas [BV23; DCP21; DG22; GD04; Wey46]. In [HJ24a] we treat the general case, that of $SO_n(\mathbb{R})$ and $O_n(\mathbb{R})$ acting on $\mathbb{R}^n \times \dots \mathbb{R}^n$ and provide algebraically independent sets of generators for the invariant field, relating them with the classical Weyl invariants [Wey46]. In this section we expand on two other cases of relevance for which we give explicitly the restrictions of the invariants to the slice. As explained in Section 3, from these expressions we can evaluate the invariants at any point outside of an hypersurface, as well as solve the inverse problem.

## 5.1 Odd degree ternary forms

Among the cases of interest to which our construction applies is the representation of $SO_3(\mathbb{R})$ or $O_3(\mathbb{R})$ on the vector space of homogeneous polynomials $\mathbb{R}[U, V, W]_d$ of odd degree $d$. The case of even degrees was treated in [GHP19]. The starting point for both odd and even degree is the harmonic decomposition.

**Lemma 5.1.** [AH12, Theorem 2.18] The usual representation of $SO_3(\mathbb{R})$ on $\mathbb{R}[U, V, W]_d$ is isomorphic to $\bigoplus_{j=0}^{[d/2]} \mathcal{H}_{d-2j}$. $$
coordinates. Symbolic computations, that we implemented in Maple, provide their expressions in terms of Cartesian coordinates w.r.t. bases of spherical harmonics. We give the three first ones:

$$ \tilde{D}_0 = y_{0,0}^2 + 2y_{-1,0}y_{1,0} $$

$$ \tilde{D}_1 = \frac{\sqrt{3y_{-1,0}y_{1,0}(y_{-1,0}y_{1,3}+y_{1,0}y_{-1,3})+\sqrt{5}(y_{1,0}^3y_{-3,3}+y_{-1,0}^3y_{3,3})}}{\frac{\sqrt{2}(y_{0,0}^2+2y_{1,0}y_{-1,0})}{2\sqrt{3}y_{0,0}^2(y_{-1,0}y_{1,3}+y_{1,0}y_{-1,3})+y_{0,0}y_{0,3}(y_{0,0}^2-3\sqrt{5}y_{-1,0}y_{1,0})}} - \frac{i\sqrt{3y_{0,0}(y_{1,0}^2y_{-2,3}-y_{-1,0}^2y_{2,3})}}{2(y_{0,0}^2+2y_{1,0}y_{-1,0})} $$

$$
\begin{aligned}
\tilde{R}_{11} = & \frac{1}{4D_0^3} (-12i\sqrt{2}\sqrt{3}\sqrt{5}y_{-2,1}y_{-1,0}y_{0,0}y_{1,0}y_{1,0}^3 + 4i\sqrt{2}\sqrt{3}\sqrt{5}y_{-2,1}y_{-1,0}y_{0,1}y_{1,0}y_{1,0}^4 + 8i\sqrt{2}\sqrt{3}\sqrt{5}y_{-2,1}y_{-1,0}y_{0,1}y_{1,0}y_{1,0}^2 \\
& +20i\sqrt{5}y_{-1,0}y_{-1,1}y_{0,0}y_{1,0}y_{1,0}^3 + 30i\sqrt{3}y_{-3,1}y_{-2,1}y_{0,0}y_{1,0}^5 + 8i\sqrt{5}y_{-2,1}y_{0,0}y_{1,0}y_{1,1} + 32i\sqrt{5}y_{-1,0}y_{0,0}y_{1,1}y_{1,2} \\
& +10i\sqrt{3}y_{-3,1}y_{-1,0}y_{0,0}y_{1,0}^3y_{1,0}^2y_{1,2} + 20i\sqrt{3}y_{-3,1}y_{-1,0}y_{0,0}y_{1,0}^3y_{1,0}^2y_{1,2} + 18i\sqrt{5}y_{-2,1}y_{-1,0}y_{0,0}y_{1,0}^3y_{1,1} \\
& +26i\sqrt{5}y_{-2,1}y_{-1,0}y_{-1,1}y_{0,0}y_{1,0}y_{1,0}^4 - y_{-1,0}^4y_{1,0}^2y_{1,1}^2 - y_{-1,0}^2y_{-1,1}^2y_{1,0}^4 + 30y_{-3,1}y_{-1,0}y_{1,0}y_{1,0}^3y_{3,1} + 20y_{-2,1}y_{-1,0}y_{1,0}^3y_{1,0}^2y_{2,1} \\
& +64y_{-1,0}^3y_{0,0}y_{1,0}y_{1,1}^2 - 48y_{-1,0}^2y_{0,0}y_{1,0}^2y_{0,1}y_{1,0}^2 + 64y_{-1,0}y_{-1,1}^2y_{0,0}y_{1,0}^3y_{1,0}^2 + 48y_{-1,0}y_{0,0}y_{1,0}^4y_{0,1}y_{1,0} + 2y_{-1,0}y_{-1,1}y_{1,0}^3y_{1,0}y_{1,1} \\
& +8\sqrt{3}\sqrt{5}y_{-1,0}^3y_{-1,1}y_{0,0}y_{1,0}y_{1,0}y_{3,1} + 22\sqrt{3}\sqrt{2}y_{-1,0}y_{0,0}y_{0,1}y_{1,0}y_{1,1} + 22\sqrt{3}\sqrt{2}y_{-1,0}^2y_{-1,1}y_{0,0}y_{0,1}y_{1,0}^3 \\
& -48\sqrt{3}\sqrt{2}y_{-1,0}^2y_{0,0}y_{0,1}y_{1,0}y_{1,1} - 48\sqrt{3}\sqrt{2}y_{-1,0}y_{-1,1}y_{0,0}y_{0,1}y_{1,0}^2 - 24y_{-1,0}^2y_{0,0}y_{1,0}^4y_{1,1}^2 \\
& +6\sqrt{5}\sqrt{2}y_{-3,1}y_{-1,0}y_{0,0}y_{0,1}y_{1,0}y_{1,0}y_{3,1} - 10i\sqrt{3}y_{-2,1}y_{-1,0}y_{0,0}y_{1,0}y_{1,0}y_{1,0}^2 \\
& -20i\sqrt{3}y_{-2,1}y_{-1,0}y_{0,0}y_{1,0}y_{1,0}^2y_{2,1} + 8\sqrt{3}\sqrt{5}y_{-3,1}y_{-1,0}y_{0,0}y_{1,0}y_{1,0}y_{1,1} + 12y_{-1,0}^3y_{0,1}^2y_{1,0}^2 - 24y_{-1,1}^2y_{0,0}y_{1,0}^4y_{1,0}^2 \\
& -18i\sqrt{5}y_{-1,0}y_{-1,1}y_{0,0}y_{1,0}y_{1,0}^2y_{2,1} + 8\sqrt{3}\sqrt{5}y_{-3,1}y_{-1,0}y_{0,0}y_{1,0}y_{1,0}y_{1,1} + 12y_{-1,0}^3y_{0,1}^2y_{1,0}^2y_{1,1}^2 - 24y_{-1,1}^2y_{0,0}y_{1,0}^4y_{1,0}^2 \\
& -10y_{-2,1}^2y_{-1,0}y_{1,0}y_{1,0}^5 + 40y_{-2,1}^2y_{0,0}y_{1,0}^4 - 10y_{-1,0}^5y_{1,0}y_{1,0}y_{2,1}^2 + 40y_{-1,0}^4y_{0,0}y_{2,1}^2 + 8y_{-1,1}y_{0,0}^6y_{1,1} \\
& -4i\sqrt{2}\sqrt{3}\sqrt{5}y_{-1,0}y_{0,1}y_{1,0}y_{1,0}^2y_{2,1} - 8i\sqrt{2}\sqrt{3}\sqrt{5}y_{-1,0}y_{0,0}y_{1,0}y_{2,1} - 15y_{-2,1}^2y_{1,0}^6 - 15y_{-1,0}^6y_{3,1}^2 \\
& +2\sqrt{3}\sqrt{5}y_{-3,1}y_{-1,0}y_{1,0}y_{1,1}^2 - 2\sqrt{3}\sqrt{5}y_{-3,1}y_{-1,0}y_{-1,1}y_{1,0}^5 + 14\sqrt{3}\sqrt{5}y_{-3,1}y_{-1,1}y_{0,0}y_{1,0}^2y_{1,0}^4 \\
& -2\sqrt{3}\sqrt{5}y_{-1,0}^5y_{1,0}y_{1,1}y_{3,1} + 2\sqrt{3}\sqrt{5}y_{-1,0}^4y_{-1,1}^2y_{1,0}^2y_{3,1} + 14\sqrt{3}\sqrt{5}y_{-1,0}^4y_{0,0}y_{1,0}y_{1,1}y_{3,1} - 4\sqrt{3}\sqrt{5}y_{-1,0}^2y_{-1,1}y_{0,0}y_{1,0}^4y_{3,1} \\
& +8\sqrt{3}\sqrt{2}y_{-1,1}y_{0,0}y_{0,1}y_{1,0} - 12\sqrt{5}\sqrt{2}y_{-3,1}y_{0,0}y_{0,1}y_{1,0}^3 - 12\sqrt{5}\sqrt{2}y_{-1,0}y_{0,0}y_{0,1}y_{3,1} \\
& -30i\sqrt{3}y_{-1,0}y_{0,0}y_{2,1}y_{3,1} - 32i\sqrt{5}y_{-2,1}y_{-1,1}y_{0,0}y_{1,0}^3 - 8i\sqrt{5}y_{-1,0}y_{-1,1}y_{0,0}y_{2,1} - 4\sqrt{3}\sqrt{5}y_{-3,1}y_{0,0}y_{1,0}y_{1,1}^2 \\
& +12i\sqrt{2}\sqrt{3}\sqrt{5}y_{-1,0}^3y_{0,0}y_{1,0}y_{1,0}y_{2,1} + 30y_{-3,1}y_{-1,0}y_{2,1}^2y_{0,0}y_{1,0}y_{1,0}^2y_{2,1} + 40y_{-2,1}y_{-1,0}y_{0,0}y_{1,0}y_{1,0}y_{2,1} \\
& +114y_{-1,0}^2y_{-1,1}y_{0,0}y_{1,0}y_{1,0}^2y_{2,1} - 40y_{-1,0}y_{-1,1}y_{0,0}y_{1,0}y_{1,1} + 8\sqrt{3}\sqrt{2}y_{-1,0}y_{0,0}y_{0,1}y_{1,1})
\end{aligned}
$$

The expressions of $\tilde{R}_{12}$, $\tilde{R}_{13}$, $\tilde{I}_{12}$ and $\tilde{I}_{13}$ are too large to be exposed here. One can nevertheless find them, and the codes to compute them, at https://sites.google.com/view/martin-jalard/accueil.

## 5.2 Piezoelectricity tensors

For some material of industrial interest(single crystals, ceramics, thin films,...), an electric displacement is induced in response to an applied mechanical stress. This property is called the piezoelectricity effect and a third-order tensor relates the induced electric displacement vector to the second-order stress tensor [AG20]. The orthogonal group $SO_3(\mathbb{R})$ acts naturally on the 18-dimensional space of piezoelectricity tensors

$$ \text{Piez} := \{e \in \otimes^3 \mathbb{R}^3, e_{ijk} = e_{ikj}\}. $$

As such it decomposes into $SO_3(\mathbb{R})$-irreducible subspaces as [Azz+22, Section 7.1]

$$ \text{Piez} \cong \mathcal{H}_1 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \mathcal{H}_3. $$

A tensor in $\text{Piez}$ is accordingly given by a coordinate vector

$$ (y_{-1,0}, y_{0,0}, y_{1,0}; y_{-1,1}, y_{0,1}, y_{1,1}; y_{-2,2}, y_{-1,2}, y_{0,2}, y_{1,2}y_{2,2}; y_{-3,3}, y_{-2,3}, y_{-1,3}, y_{0,3}, y_{1,3}, y_{2,3}, y_{3,3}), $$

where $y_{-m,j} = \bar{y}_{m,j}$, when the components $\mathcal{H}_d$ are endowed the bases of Theorem 3.2. A Seshadri slice for $SO_3(\mathbb{R})$ acting on $\text{Piez}$ is given by the subspace $\mathcal{S}$ defined by the equations $y_{-1,0} = 0$ and $y_{1,0} = 0$ and which is stabilized by $\mathbb{N} \cong \mathbb{O}_2(\mathbb{R})$. Theorem 2.6 gives us an algebraically independent set of generators for $\mathbb{R}(\mathcal{S})^N$:
Whenever $d$ is odd, the last term of this decomposition is $\mathcal{H}_1$ and thus we can apply the construction provided in previous sections.

**Corollary 5.2.** For any $d \in \mathbb{N}$, the invariant field of the $\text{SO}_3(\mathbb{R})$ or $\text{O}_3(\mathbb{R})$ action on $\mathbb{R}[U, V, W]_d$ is rational.

**Example 5.3.** The vector space $T = \mathcal{H}_1 \oplus \mathcal{H}_3$ introduced in Example 3.9 is isomorphic to the space of ternary cubics $\mathbb{R}[U, V, W]_3$. Completing the basis obtained in Example 3.2, one endows $\mathcal{T}$ with the real harmonic basis

$$ \mathcal{B}_{\mathcal{T}} = \left\{ \begin{array}{lll} X_{-1} = QV, & X_0 = QU, & X_1 = -QW \\ X_3^3 = 3W^2V - V^3 & X_3^2 = UVW & X_3^1 = W(4U^2 - V^2 - W^2) \\ X_3^{-3} = 3WV^2 - W^3 & X_3^0 = U(2U^2 - 3V^2 - 3W^2) & X_3^{-1} = -V(4U^2 - V^2 - W^2) \end{array} \right\} $$

The set $\mathcal{E}_{\mathcal{T}}$ in Example 3.9 generates then $\mathbb{R}(S_{\mathcal{T}})^N$ (with respect to the associated complex basis). Theorem 3.14 gives a set of rational generators for $\mathbb{R}(\mathcal{T})^{\text{SO}_3(\mathbb{R})}$. It separates orbits everywhere but in $\{\tilde{D}_0 = 0\} = \{x_{-1}^2 + x_0^2 + x_1^2 = 0\}$ and $\{\tilde{R}_{11} = 0\}$.

## 5.2 Piezoelectricity tensors

In Continuum Mechanics, the orthogonal group $\text{SO}_3(\mathbb{R})$ acts on the piezoelectricity tensors space $\text{Piez}$ as presented in [Che+19, Section 2.1] and in [Azz+22]:

$$ \text{Piez} := \{e \in \otimes^3 \mathbb{R}^3, e_{ijk} = e_{ikj}\} $$

For physical reasons this action is mostly studied over the reals. The 18-dimensional space $\text{Piez}$ decomposes into $\text{SO}_3(\mathbb{R})$-irreducible subspaces as [Azz+22, Section 7.1]

$$ \text{Piez} \cong \mathcal{H}_1 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \mathcal{H}_3. $$

One can choose $(\mathcal{S}, \mathbb{N})$ as in Theorem 2.8 and get:

**Proposition 5.4.** The following set is an algebraically independent and generates $\mathbb{R}(\mathcal{S})^N$.

$$ \mathcal{E}_{\text{Piez}} = \left\{ \begin{array}{lll} T_2 = y_{0,2} & R_{11} = 2y_{-1,1}y_{1,1} & I_{12} = iy_0 (y_{1,2}y_{-1,1} - y_{-1,2}y_{1,1}) \\ D_0 = y_0^2 & R_{12} = y_{1,2}y_{-1,1} + y_{-1,2}y_{1,1} & I_{22} = iy_0 (y_{2,2}y_{-1,1}^2 - y_{-2,2}y_{1,1}^2) \\ D_1 = y_0y_{0,1} & R_{22} = y_{2,2}y_{-1,1}^2 + y_{-2,2}y_{1,1}^2 & I_{13} = iy_0 (y_{1,3}y_{-1,1} - y_{-1,3}y_{1,1}) \\ D_3 = y_0y_{0,3} & R_{13} = y_{1,3}y_{-1,1} + y_{-1,3}y_{1,1} & I_{23} = iy_0 (y_{2,3}y_{-1,1}^2 - y_{-2,3}y_{1,1}^2) \\ & R_{23} = y_{2,3}y_{-1,1}^2 + y_{-2,3}y_{1,1}^2 & I_{33} = iy_0 (y_{3,3}y_{-1,1}^3 - y_{-3,3}y_{1,1}^3) \\ & R_{33} = y_{3,3}y_{-1,1}^3 + y_{-3,3}y_{1,1}^3 & \end{array} \right\} $$

Here is its expression in real coordinates (see Section 3.3):

$$ \mathcal{E}_{\text{Piez}} = \left\{ \begin{array}{lll} T_2 = x_{0,2} & R_{11} = 2(x_{1,1}^2 + x_{-1,1}^2) & \\ D_0 = x_0^2 & R_{12} = 4x_{-2}x_{-1}x_1 - 2x_{-1}^2x_2 + 2x_1^2x_2 & \\ D_1 = x_0x_{0,1} & R_{22} = 4x_{-2}x_{-1}x_1 - 2x_{-1}^2x_2 + 2x_1^2x_2 & \\ D_3 = x_0x_{0,3} & R_{13} = -2x_{-3}x_{-1}^3 + 6x_{-3}x_{-1}x_1^2 - 6x_{-1}^2x_1x_3 + 2x_1^3x_3 & \\ & R_{23} = -2x_{-3}x_{-1}^3 + 6x_{-3}x_{-1}x_1^2 - 6x_{-1}^2x_1x_3 + 2x_1^3x_3 & \\ & R_{33} = -2x_{-3}x_{-1}^3 + 6x_{-3}x_{-1}x_1^2 - 6x_{-1}^2x_1x_3 + 2x_1^3x_3 & \\ & I_{12} = 2x_0(x_{-2}x_{-1}^2 - x_{-2}x_1^2 + 2x_{-1}x_1x_2) & \\ & I_{22} = 2x_0(x_{-2}x_{-1}^2 - x_{-2}x_1^2 + 2x_{-1}x_1x_2) & \\ & I_{13} = 2x_0(3x_{-3}x_{-1}^2x_1 - x_{-3}x_1^3 - x_{-1}^3x_3 + 3x_{-1}x_1^2x_3) & \\ & I_{23} = 2x_0(3x_{-3}x_{-1}^2x_1 - x_{-3}x_1^3 - x_{-1}^3x_3 + 3x_{-1}x_1^2x_3) & \\ & I_{33} = 2x_0(3x_{-3}x_{-1}^2x_1 - x_{-3}x_1^3 - x_{-1}^3x_3 + 3x_{-1}x_1^2x_3) & \end{array} \right\} $$
**Proposition 5.4.** The following set is an algebraically independent and generates $\mathbb{R}(S)^N$.

$$ \mathcal{E}_{\text{Piez}} = \left\{ \begin{array}{l} T_2 = y_{0,2} \qquad R_{11} = 2y_{-1,1}y_{1,1} \qquad I_{12} = iy_{0,0}(y_{1,2}y_{-1,1} - y_{-1,2}y_{1,1}) \\ D_0 = y_{0,0}^2 \qquad R_{12} = y_{1,2}y_{-1,1} + y_{-1,2}y_{1,1} \qquad I_{22} = iy_{0,0}(y_{2,2}y_{-1,1}^2 - y_{-2,2}y_{1,1}^2) \\ D_1 = y_{0,0}y_{0,1} \qquad R_{22} = y_{2,2}y_{-1,1}^2 + y_{-2,2}y_{1,1}^2 \qquad I_{13} = iy_{0,0}(y_{1,3}y_{-1,1} - y_{-1,3}y_{1,1}) \\ D_3 = y_{0,0}y_{0,3} \qquad R_{13} = y_{1,3}y_{-1,1} + y_{-1,3}y_{1,1} \qquad I_{23} = iy_{0,0}(y_{2,3}y_{-1,1}^2 - y_{-2,3}y_{1,1}^3) \\ \qquad \qquad \qquad R_{23} = y_{2,3}y_{-1,1}^2 + y_{-2,3}y_{1,1}^2 \qquad I_{33} = iy_{0,0}(y_{3,3}y_{-1,1}^3 - y_{-3,3}y_{1,1}^3) \\ \qquad \qquad \qquad R_{33} = y_{3,3}y_{-1,1}^3 + y_{-3,3}y_{1,1}^3 \end{array} \right\} $$

Here is its expression in real coordinates (see Section 3.3):

$$ \mathcal{E}_{\text{Piez}} = \left\{ \begin{array}{l} T_2 = x_{0,2} \qquad R_{1,1} = x_{-1,1}^2 + x_{1,1}^2 \\ D_0 = x_{0,0}^2 \qquad R_{1,2} = x_{-1,1}x_{-1,2} + x_{1,1}x_{1,2} \\ D_1 = x_{0,0}x_{0,1} \qquad R_{2,2} = \frac{\sqrt{2}}{2}(2x_{-2,2}x_{-1,1}x_{1,1} - x_{-1,1}^2x_{2,2} + x_{1,1}^2x_{2,2}) \\ D_3 = x_{0,0}x_{0,3} \qquad R_{1,3} = x_{-1,1}x_{-1,3} + x_{1,1}x_{1,3} \\ \qquad \qquad \qquad R_{2,3} = \frac{\sqrt{2}}{2}(2x_{-2,3}x_{-1,1}x_{1,1} - x_{-1,1}^2x_{2,3} + x_{1,1}^2x_{2,3}) \\ \qquad \qquad \qquad R_{3,3} = -\frac{1}{2}x_{-3,3}x_{-1,1}^3 + \frac{3}{2}x_{-3,3}x_{-1,1}x_{1,1}^2 - \frac{3}{2}x_{-1,1}^2x_{1,1}x_{3,3} + \frac{1}{2}x_{1,1}^3x_{3,3} \\ \qquad \qquad \qquad I_{1,2} = -x_{0,0}(x_{-1,1}x_{1,2} - x_{-1,2}x_{1,1}) \\ \qquad \qquad \qquad I_{2,2} = -\frac{\sqrt{2}}{2}x_{0,0}(x_{-2,2}x_{-1,1}^2 - x_{-2,2}x_{1,1}^2 + 2x_{-1,1}x_{1,1}x_{2,2}) \\ \qquad \qquad \qquad I_{1,3} = -x_{0,0}(x_{-1,1}x_{1,3} - x_{-1,3}x_{1,1}) \\ \qquad \qquad \qquad I_{2,3} = -\frac{\sqrt{2}}{2}x_{0,0}(x_{-2,3}x_{-1,1}^2 - x_{-2,3}x_{1,1}^2 + 2x_{-1,1}x_{1,1}x_{2,3}) \\ \qquad \qquad \qquad I_{3,3} = -\frac{1}{2}x_{0,0}(3x_{-3,3}x_{-1,1}^2x_{1,1} - x_{-3,3}x_{1,1}^3 - x_{-1,1}^3x_{3,3} + 3x_{-1,1}x_{1,1}^2x_{3,3}) \end{array} \right\} $$

Theorem 3.16 allows us to deduce an algebraically independent set $\tilde{\mathcal{E}}_{\text{Piez}}$ generating $\mathbb{R}(\text{Piez})^{SO_3(\mathbb{R})}$. In [Azz+22, Section 7.1], the group acting on the space of piezoelectricity tensors is actually $O_3(\mathbb{R})$. The $O_3(\mathbb{R})$ representation is then $\text{Piez} \cong \mathcal{H}_1 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2^\sharp \oplus \mathcal{H}_3$. Here the first term $d_1 = 1$ is odd, and $T_2^\sharp = y_{0,2}$ of Theorem 4.1 can be chosen as the flipped polynomial $P$. Applying Theorem 4.1:

**Proposition 5.5.** The following set of cardinal 15 is an algebraically independent generating set for both $\mathbb{C}(\hat{S})^{\mathbb{N}^\sharp}$ or $\mathbb{R}(S)^{\mathbb{N}^\sharp}$.

$$ \mathcal{E}_{\text{Piez}}^{\sharp} = \left\{ \begin{array}{l} T_2^{\sharp} = y_{0,2}^2 \qquad R_{11} = y_{1,1}y_{-1,1} + y_{-1,1}y_{1,1} \qquad I_{12}^{\sharp} = iy_{0,0}(y_{1,2}y_{-1,1} - y_{-1,2}y_{1,1}) \\ D_0 = y_{0,0}^2 \qquad R_{12}^{\sharp} = y_{0,2}(y_{1,2}y_{-1,1} + y_{-1,2}y_{1,1}) \qquad I_{22}^{\sharp} = iy_{0,0}y_{0,2}(y_{2,2}y_{-1,1}^2 - y_{-2,2}y_{1,1}^2) \\ D_1 = y_{0,0}y_{0,1} \qquad R_{22}^{\sharp} = y_{2,2}y_{-1,1}^2 + y_{-2,2}y_{1,1}^2 \qquad I_{13} = iy_{0,0}y_{0,2}(y_{1,3}y_{-1,1} - y_{-1,3}y_{1,1}) \\ D_3 = y_{0,0}y_{0,3} \qquad R_{13} = y_{1,3}y_{-1,1} + y_{-1,3}y_{1,1} \qquad I_{23} = iy_{0,0}(y_{2,3}y_{-1,1}^2 - y_{-2,3}y_{1,1}^3) \\ \qquad \qquad \qquad R_{23} = y_{0,2}(y_{2,3}y_{-1,1}^2 + y_{-2,3}y_{1,1}^2) \qquad I_{33} = iy_{0,0}y_{0,2}(y_{3,3}y_{-1,1}^3 - y_{-3,3}y_{1,1}^3) \\ \qquad \qquad \qquad R_{33} = y_{3,3}y_{-1,1}^3 + y_{-3,3}y_{1,1}^3 \end{array} \right\} $$

Here again Theorem 3.16 allows us to deduce an algebraically independent set $\tilde{\mathcal{E}}_{\text{Piez}}$ generating $\mathbb{R}(\text{Piez})^{O_3(\mathbb{R})}$.

## 5.3 3D Moment Invariants

Several field of engineering are confronted with differentiating or classifying 3D shapes up to a group of transformations, and in particular the group $SO_3(\mathbb{R})$ of rotations. Moments are descriptors of shapes that are obtained by acquisition devices or by computation. Moment invariants are then the descriptors of the shape independently of rotation. For two dimensional images, many approaches to the computation and application of moment invariants were developed. The books [FSZ09; FSZ17] are entry points to the intricate literature for mathematical and methodological advances together with their concrete applications. The seminal papers in the 3D case seem to be [Guo93; Hu62; LD89]. A sample of applications in medicine and biology can be found in [Gon+13; Man+04; Som+07].
This set provides an algebraically independent set $\mathcal{E}_{\tilde{\text{Piez}}}$ generating $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$, which separates orbits whenever $\tilde{D}_0$ and the denominator of $\tilde{R}_{11}$ do not vanish. The denominator of $\tilde{R}_{11}$ is the same as in Example 3.15:

$$-y_{-1}^2 y_{1,1}^2 - 2y_{-1} y_0 y_{0,1} y_{1,1} + 2y_{-1} y_1 y_{-1,1} y_{1,1} + 2y_{-1} y_1 y_{0,1}^2 + 2y_0^2 y_{-1,1} y_{1,1} - 2y_0 y_1 y_{-1,1} y_{0,1} - y_1^2 y_{-1,1}^2$$

In [Azz+22, Section 7.1], the group acting on the Piezoelectricity tensors is actually $O_3(\mathbb{R})$. The $O_3(\mathbb{R})$ representation is then $\tilde{\text{Piez}} \cong \mathcal{H}_1 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2^\sharp \oplus \mathcal{H}_3$. Here the first term $d_1 = 1$ is odd, and $T_2^\sharp = y_{0,2}$ of Theorem 4.1 can be chosen as the flipped polynomial $P$. Applying Theorem 4.1:

**Proposition 5.5.** The following set of cardinal 15 is an algebraically independent generating set for both $\mathbb{C}(\hat{\mathcal{S}})^{\mathbb{N}\sharp}$ or $\mathbb{R}(\mathcal{S})^{\mathbb{N}\sharp}$.

$$\mathcal{E}_{\text{Piez}} = \left\{ \begin{array}{llll} T_2^\sharp = y_{0,2}^2 & R_{11} = y_{1,1}y_{-1,1} + y_{-1,1}y_{1,1} & I_{12}^\sharp = iy_0(y_{1,2}y_{-1,1} - y_{-1,2}y_{1,1}) \\ D_0 = y_0^2 & R_{12}^\sharp = y_{0,2}(y_{1,2}y_{-1,1} + y_{-1,2}y_{1,1}) & I_{22}^\sharp = iy_0y_{0,2}(y_{2,2}y_{-1,1}^2 - y_{-2,2}y_{1,1}^2) \\ D_1 = y_0y_{0,1} & R_{22}^\sharp = y_{2,2}y_{-1,1}^2 + y_{-2,2}y_{1,1}^2 & I_{13} = iy_0y_{0,2}(y_{1,3}y_{-1,1} - y_{-1,3}y_{1,1}) \\ D_3 = y_0y_{0,3} & R_{13} = y_{1,3}y_{-1,1} + y_{-1,3}y_{1,1} & I_{23} = iy_0(y_{2,3}y_{-1,1}^2 - y_{-2,3}y_{1,1}^3) \\ & R_{23} = y_{0,2}(y_{2,3}y_{-1,1}^2 + y_{-2,3}y_{1,1}^2) & I_{33} = iy_0y_{0,2}(y_{3,3}y_{-1,1}^3 - y_{-3,3}y_{1,1}^3) \\ & R_{33} = y_{3,3}y_{-1,1}^3 + y_{-3,3}y_{1,1}^3 & \end{array} \right\}$$

## A Rewriting an invariant in terms of the generators

In this article we exhibited algebraically independent generators of the invariant field $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$ emphasizing their separation properties. One can also use a generating set of invariants as new variables of an invariant problem, thus factoring out the symmetry; see for instance [HL12; HL13; HL16] where are examined the use of invariants of Abelian groups to solve polynomial systems or reduce the number of parameters in dynamical systems. In this appendix we show how to rewrite any invariants in $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$ in terms of the generators exhibited.

Consider $\tilde{R} \in \mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$. Then the restriction of $\tilde{R}$ to the slice $\mathcal{S}$ is an $O_2(\mathbb{R})$-invariant; in other words $R = \tilde{R}_{|\mathcal{S}} \in \mathbb{R}(\mathcal{S})^{O_2(\mathbb{R})}$. If we can identify a rational function $F$ in $\mathbb{R}(T_j, D_j, R_{mj}, I_{mj})$ such that $R = F(\mathcal{E})$ then $\tilde{R} = F(\tilde{\mathcal{E}})$ by Theorem 3.4. One thus can rewrite any element of $\mathbb{R}(\mathcal{H})^{\text{SO}_3(\mathbb{R})}$ as a rational function of $\tilde{\mathcal{E}}$ thanks to the following rewrite rules.

**Theorem A.1.** Assume $P, Q \in \mathbb{R}[\mathcal{S}]$ and take $p$ and $q$ in

$$\mathbb{R}(T_j, D_j, R_{mj}, I_{mj})[y_0; y_{1,1}; y_{-1,1}; y_{-d_j, j}, \dots, y_{-d_j, j}, 1 \le j \le n]$$

the reductions of $P$ and $Q$ by the following confluent system of rewrite rules:

$$\begin{array}{lclcl} y_{0,j} & \rightarrow & T_j & & 1 \le j \le p \\ y_{0,j} & \rightarrow & \frac{D_j}{D_0} y_0 & & p+1 \le j \le n \\ y_{m,j} & \rightarrow & 2^{m-1} \left( \frac{R_{mj}}{R_{11}^m} - \frac{i I_{mj}}{y_0 R_{11}^m} \right) y_{1,1}^m & & 1 \le j \le n, 1 \le m \le d_j \\ y_{-m,j} & \rightarrow & 2^{m-1} \left( \frac{R_{mj}}{R_{1,1}^m} + \frac{i I_{mj}}{y_0 R_{1,1}^m} \right) y_{-1,1}^m & & 1 \le j \le n, 1 \le m \le d_j \\ y_{1,1} y_{-1,1} & \rightarrow & R_{11} & & \\ y_0^2 & \rightarrow & D_0 & & \end{array}$$

Then $\frac{P}{Q}$ belongs to $\mathbb{R}(\mathcal{S})^N$ iff $p = \tilde{p}r$ and $q = \tilde{q}r$ with $\tilde{p}, \tilde{q} \in \mathbb{R}(T_j, D_j, R_{mj}, I_{mj})$ and $r \in \mathbb{R}[y]$. In which case

$$\frac{P}{Q} = \frac{\tilde{p}(T_j(y_0, h), D_j(y_0, h), R_{mj}(y_0, h), I_{mj}(y_0, h))}{\tilde{q}(T_j(y_0, h), D_j(y_0, h), R_{mj}(y_0, h), I_{mj}(y_0, h))}$$
Finding analytical expressions of moment invariants is the one facet that we wish to expand on here. Several methods have been designed to produce 3D moment invariants under SO₃(R) with recent contributions [BB23; Li+17; SFB15; Sun24]. Explicit expressions for some moment invariants up to tenth order were exhibited. The main result of this article, Theorem 3.14, applies straightforwardly and provide explicit expressions of an algebraically independent set of moment invariants, up to any order, that is generating.

Consider a measure μ with compact support in R³. The associated moment functional is the linear form

$$ \mathfrak{m}_{\mu}: \mathbb{R}[U, V, W] \to \mathbb{R} \\ p \mapsto \int_{\mathbb{R}^3} p \, d\mu $$

The measure can for instance be given by a density function ω : R³ → R≥0 with compact support so that the moments are

$$ \mathfrak{m}_{\mu}(p) = \int_{\mathbb{R}^3} p(U, V, W) \omega(U, V, W) \, du \, dv \, dw. $$

For instance ω can be the characteristic function of a 3D shape. Another important case is when μ is an atomic measure, i.e., a linear combination of a finite number Diracs centered at ξ₁, ..., ξₚ ∈ R³ with positive coefficients ωᵢ. Then

$$ \mathfrak{m}_{\mu}(p) = \sum_{i=1}^{r} \omega_i p(\xi_i). $$

Assume ω is the characteristic function of a shape and that this shape is transformed by an invertible linear map a : R³ → R³. Then the characteristic function of the transformed shape is ω ◦ a⁻¹. More generally, for a measure μ and its transform μª by a we have mμª(p) = mμ(p ◦ a).

For a group representation G → GL(R³), if Aₚ⁽'d⁾: G → GLₚ₉(R) is the matrix of the induced representation on R[U, V, W]₉ in the basis {p₁, ..., pₚ₉}, i.e. [p₁ ◦ g⁻¹ ... pₚ₉ ◦ g⁻¹] = [p₁ ... pₚ₉] Aₚ⁽'d⁾(g), then, in the dual basis, the matrix of the induced representation on the dual space (R[U, V, W]₉)ª is Mₚª⁽'d⁾: G → GLₚ₉(R) with Mₚª⁽'d⁾(g) = (Aₚ⁽'d⁾(g))⁻¹. Concomitantly

$$ \begin{bmatrix} \mathfrak{m}_{\mu^g}(p_1) \\ \vdots \\ \mathfrak{m}_{\mu^g}(p_{r_d}) \end{bmatrix} = (A^{(d)}(g^{-1}))^T \begin{bmatrix} \mathfrak{m}_{\mu}(p_1) \\ \vdots \\ \mathfrak{m}_{\mu}(p_{r_d}) \end{bmatrix}. $$

In the case of a compact group we can always find bases where the matrices Aₚ⁽'d⁾(g) are orthogonal so that the induced representations on R[U, V, W]₉ and its dual (R[U, V, W]₉)ª are equivalent. For SO₃(R), the real spherical harmonics defined in Section 3.1 provide such bases.

The action of SO₃(R) on moments of order 0 up to order d is thus equivalent to the action of SO₃(R) on M₉ = R ⊕ R[U, V, W]₁ ⊕ ... ⊕ R[U, V, W]₉. Given that

$$ \mathbb{R}[U, V, W]_d = \bigoplus_{k=0}^{\lfloor \frac{d}{2} \rfloor} Q^k \mathcal{H}_{d-2k} $$

we can apply Theorem 3.14 to provide the moment invariants up to any order. For instance for d = 4 the basis of M₄ is, with Q = U² + V² + W²,

$$ \{1\} \cup \{X_1^{-1}, X_1^0, X_1^0\} \cup \{Q; X_2^{-2}, \dots, X_2^2\} \cup \{QX_1^{-1}, QX_1^0, QX_1^0; X_3^{-3}, \dots, X_3^3\} \cup \{Q^2; QX_2^{-2}, \dots, QX_2^2; X_4^{-4}, \dots, X_4^4\}. $$

The moments m(1), m(Q), m(Q²) are already invariants. To obtain a generating set of moment invariants one has to add the invariants obtained from Theorem 3.14 with the following coordinates that are arranged according to parity of the degrees (p = 3, n = 5 in the notation of Theorem 3.14).
Proof. The proof relies on [MS99]. A Gröbner basis of the ideal in $\mathbb{R}(y)[Y]$ generated by

$$ \begin{aligned} & \{Y_{0,j} - y_{0,j}, 1 \le j \le p\} \cup \{Y_0 Y_{0,j} - y_0 y_{0,j}, p+1 \le j \le n\} \\ & \cup \{ (Y_{-1,1}^m Y_{m,j} + Y_{1,1}^m Y_{-m,j}) - (y_{-1,1}^m y_{m,j} + y_{1,1}^m y_{-m,j}), 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1) \} \\ & \cup \{ Y_0 (Y_{-1,1}^m Y_{m,j} - Y_{1,1}^m Y_{-m,j}), -y_0 (y_{-1,1}^m y_{m,j} - y_{1,1}^m y_{-m,j}), 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1) \} \\ & \cup \{ Y_{-1,1} Y_{1,1} - y_{-1,1} Y_{1,1}, Y_0^2 - y_0^2 \} \end{aligned} $$

w.r.t. to a term order that eliminates $Y_{0,j}$, $Y_{m,j}$ and $Y_{-m,j}$ is given by

$$ \begin{aligned} & \{\mathbf{Y}_0^2 - y_0^2, \mathbf{Y}_{1,1}\mathbf{Y}_{-1,1} - y_{1,1}y_{-1,1}\} \\ & \cup \{\mathbf{Y}_{0,j} - y_{0,j} | 1 \le j \le p\} \cup \{\mathbf{Y}_{0,j} - \frac{y_{0,j}}{y_0} Y_0 | p+1 \le j \le n\} \\ & \cup \left\{ \mathbf{Y}_{m,j} - \left( \frac{(y_{m,j}y_{-1,1}^m + y_{-m,j}y_{1,1}^m)}{2(y_{-1,1}y_{1,1})^m} + \frac{(y_{m,j}y_{-1,1}^m - y_{-m,j}y_{1,1}^m)}{2y_0(y_{-1,1}y_{1,1})^m} \right) Y_0 \right\} Y_{1,1}^m | 1 \le j \le n, 1 \le m \le d_j, (m,j) \neq (1,1) \}, \\ & \cup \left\{ \mathbf{Y}_{-m,j} - \left( \frac{(y_{m,j}y_{-1,1}^m + y_{-m,j}y_{1,1}^m)}{2y_{-1,1}^m y_{1,1}^m} - \frac{(y_{m,j}y_{-1,1}^m - y_{-m,j}y_{1,1}^m)}{2y_0(y_{-1,1}y_{1,1})^m} \right) Y_0 \right\} Y_{-1,1}^m | 1 \le j \le n, 1 \le m \le d_j, (m,j) \neq (1,1) \end{aligned} $$

where the leading terms are in bold font. This latter can be written as :

$$ \begin{aligned} & \{\mathbf{Y}_0^2 - D_0, \mathbf{Y}_{1,1}\mathbf{Y}_{-1,1} - \frac{1}{2}R_{11}\} \\ & \cup \{\mathbf{Y}_{0,j} - T_j | 1 \le j \le p\} \cup \{\mathbf{Y}_{0,j} - \frac{D_j}{D_0}Y_0 | p+1 \le j \le n\} \\ & \cup \left\{ \mathbf{Y}_{m,j} - 2^{m-1} \left( \frac{R_{mj}}{R_{11}^m} - \frac{iI_{mj}}{D_0 R_{11}^m} Y_0 \right) Y_{1,1}^m | 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1) \right\}, \\ & \cup \left\{ \mathbf{Y}_{-m,j} - 2^{m-1} \left( \frac{R_{mj}}{R_{11}^m} + \frac{iI_{mj}}{D_0 R_{11}^m} Y_0 \right) Y_{-1,1}^m | 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1) \right\}. \end{aligned} $$

By [MS99, Lemma 1.7 and Algorithm 1.10], we obtain the announced result. □

**Example A.2.** In the space of piezoelectricity tensor, the invariant $\tilde{P}$ whose restriction to the slice equals $P = y_{-2,1}^3 y_{3,1}^2 + y_{2,1}^3 y_{-3,1}^2$ can be rewritten as

$$ \tilde{P} = \frac{256}{\tilde{R}_{1,1}^6} \left( \tilde{R}_{21}^3 \tilde{R}_{31}^2 - \frac{\tilde{R}_{21}^3 \tilde{I}_{31}^2}{\tilde{D}_0} + \frac{6 \tilde{R}_{21}^2 \tilde{I}_{21} \tilde{R}_{31} \tilde{I}_{31}}{\tilde{D}_0} - \frac{3 \tilde{R}_{21} \tilde{I}_{21}^2 \tilde{R}_{31}^2}{\tilde{D}_0} + \frac{3 \tilde{R}_{21} \tilde{I}_{21}^2 \tilde{I}_{31}^2}{\tilde{D}_0^2} - \frac{2 \tilde{I}_{21}^3 \tilde{R}_{31} \tilde{I}_{31}}{\tilde{D}_0^2} \right). $$

As predicted by Corollary 3.18, the denominator is a power of $\tilde{D}_0 = Q$ though the rewriting system of Theorem A.1 could lead us to believe that $\tilde{R}_1$ could appear as denominator as well.

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[AH12] K. Atkinson and W. Han. Spherical harmonics and approximations on the unit sphere: an introduction. Vol. 2044. Springer Science & Business Media, 2012.

[Azz+22] P. Azzi, R. Desmorat, B. Kolev, and F. Priziac. “Distance to a constitutive tensor isotropy stratum by Lasserre polynomial optimization method”. In: arXiv preprint arXiv:2207.04729 (2022).

[BV23] B Blum-Smith and S. Villar. “Machine learning and invariant theory”. In: Notices of the American Mathematical Society (2023).

[Bür+21] P. Bürgisser, M. Doğan, V. Makam, M. Walter, and A. Wigderson. “Polynomial time algorithms in invariant theory for torus actions”. In: arXiv preprint arXiv:2102.07727 (2021).

[Che+19] Y. Chen, Z. Ming, L. Qi, and W. Zou. “A Polynomially Irreducible Functional Basis of Hemitropic Invariants of Piezoelectric Tensors”. In: arXiv preprint arXiv:1901.01701 (2019).
$$y_{-1,0} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_1^{-1}) + i \mathfrak{m}(X_1^1)), y_{0,0} = \mathfrak{m}(X_1^0), y_{1,0} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_1^{-1}) - i \mathfrak{m}(X_1^1));$$

$$y_{-2,1} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_2^{-2}) + i \mathfrak{m}(X_2^2)), \dots, y_{2,1} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_2^{-2}) - i \mathfrak{m}(X_2^2));$$

$$y_{-2,2} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_2^{-2}) + i \mathfrak{m}(QX_2^2)), \dots, y_{-2,2} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_2^{-2}) - i \mathfrak{m}(QX_2^2));$$

$$y_{-4,3} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_2^{-4}) + i \mathfrak{m}(X_2^4)), \dots, y_{-4,3} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_2^{-4}) - i \mathfrak{m}(X_2^4));$$

$$y_{-1,4} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_1^{-1}) + i \mathfrak{m}(QX_1^1)), \dots, y_{-1,4} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_1^{-1}) - i \mathfrak{m}(QX_1^1));$$

$$y_{-3,5} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_3^{-3}) + i \mathfrak{m}(X_3^3)), \dots, y_{-3,5} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_3^{-3}) - i \mathfrak{m}(X_3^3));$$

If is nonetheless often assumed that $\mathfrak{m}(\mathbb{R}[U, V, W]_1) = \{0\}$. This is obtained by applying an appropriate translation to the measure. One thus has to deal with the action of $SO_3(\mathbb{R})$ on $\mathfrak{M}_d^o = \mathbb{R} \oplus \mathbb{R}[U, V, W]_2 \oplus \mathbb{R}[U, V, W]_3 \oplus \dots \oplus \mathbb{R}[U, V, W]_d$. One can still apply Theorem 3.14 since $\mathbb{R}[U, V, W]_3 = Q\mathcal{H}_1 \oplus \mathcal{H}_3$. We reorganize the components of $\mathfrak{M}_d^o$ as

$$\mathfrak{M}_d^o = Q\mathcal{H}_1 \oplus \mathbb{R} \oplus (Q\mathbb{R} \oplus \mathcal{H}_2) \oplus (Q^2\mathbb{R} \oplus Q\mathcal{H}_2 \oplus \mathcal{H}_4) \oplus \mathcal{H}_3.$$

Again, $\mathfrak{m}(1)$, $\mathfrak{m}(Q)$, $\mathfrak{m}(Q^2)$ being already invariants, we can apply Theorem 3.14 on the following coordinates $(p=3)$:

$$y_{-1,0} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_1^{-1}) + i \mathfrak{m}(QX_1^{-1})), y_{0,0} = \mathfrak{m}(QX_1^0), y_{-1,0} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_1^{-1}) - i \mathfrak{m}(QX_1^{-1})), ;$$

$$y_{-2,1} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_2^{-2}) + i \mathfrak{m}(X_2^2)), \dots, y_{2,1} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_2^{-2}) - i \mathfrak{m}(X_2^2));$$

$$y_{-2,2} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_2^{-2}) + i \mathfrak{m}(QX_2^2)), \dots, y_{2,2} = \frac{1}{\sqrt{2}} (\mathfrak{m}(QX_2^{-2}) - i \mathfrak{m}(QX_2^2));$$

$$y_{-4,3} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_4^{-4}) + i \mathfrak{m}(X_4^4)), \dots, y_{4,3} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_4^{-4}) - i \mathfrak{m}(X_4^4)).$$

$$y_{-3,4} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_3^{-3}) + i \mathfrak{m}(X_3^3)), \dots, y_{3,4} = \frac{1}{\sqrt{2}} (\mathfrak{m}(X_3^{-3}) - i \mathfrak{m}(X_3^3));$$

The expressions of the invariants can be obtained with symbolic computation. Nevertheless, they become quickly unwieldy. To evaluate these invariants numerically, the better way would probably be to apply a rotation that brings

$$\left(-\frac{1}{\sqrt{2}}(y_{1,0} + y_{-1,0}), y_{0,0}, \frac{i}{\sqrt{2}}(y_{1,0} - y_{-1,0})\right)^T \quad \text{to} \quad \left(0, 0, \sqrt{y_{0,0}^2 + 2y_{-1,0}y_{1,0}}\right)^T$$

and then evaluate the restriction of the invariants to the slice as given in Theorem 3.7.

An alternative to provide explicit expressions of moment invariants is to extend the method of [GHP19] that is based on a natural Seshadri slice for $\mathbb{R}[U, V, W]_2$. The natural bases for the polynomial rings are then the cubic harmonics. Note that if the measure is centrally symmetric then all moments of odd order are zero and therefore the rational invariants obtained by applying Theorem 3.14 are zero or undefined. The method based on [GHP19] would provide generically separating moment invariants then though.
[CS05] J-L. Colliot-Thélène and J-J. Sansuc. "The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)". In: *arXiv preprint math/0507154* (2005).

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[Dai13] F. Dai. *Approximation theory and harmonic analysis on spheres and balls*. Springer, 2013.

[DK15] H. Derksen and G. Kemper. *Computational invariant theory*. 2nd. Springer-Verlag, 2015.

[Des+19] R. Desmorat, N. Auffray, B. Desmorat, B. Kolev, and M. Olive. "Generic separating sets for three-dimensional elasticity tensors". In: *Proceedings of the Royal Society A* 475.2226 (2019), p. 20190056.

[DCP21] G. Dhont, P. Cassam-Chenaï, and F. Patras. "Molien generating functions and integrity bases for the action of the SO(3) and O(3) groups on a set of vectors". In: *Journal of Mathematical Chemistry* 59.10 (2021), pp. 2294-2326.

[DG22] N. Dym and S-J Gortler. "Low dimensional invariant embeddings for universal geometric learning". In: *arXiv preprint arXiv:2205.02956* (2022).

[FH13] W. Fulton and J. Harris. *Representation theory: a first course*. Vol. 129. Springer Science & Business Media, 2013.

[GSS88] M. Golubitsky, I. Stewart, and D. Schaeffer. *Singularities and groups in bifurcation theory*. Vol. II. Vol. 69. Applied Mathematical Sciences. New York: Springer-Verlag, 1988.

[GHP19] Paul Görlach, Evelyne Hubert, and Théo Papadopoulo. "Rational invariants of even ternary forms under the orthogonal group". In: *Foundations of Computational Mathematics* 19.6 (2019), pp. 1315-1361.

[GD04] John C. Gower and Garmt B. Dijksterhuis. *Procrustes problems*. Vol. 30. Oxf. Stat. Sci. Ser. Oxford: Oxford University Press, 2004.

[HJ24a] E. Hubert and M. Jalard. "Rationality of the invariant field of $O_n(\mathbb{R})$ and $SO_n(\mathbb{R})$ representations". In: (2024).

[HJ24b] E. Hubert and M. Jalard. "Stratification and separation of the orbit space of piezoelectricity tensors". In: (2024).

[Hub12] E. Hubert. "Algebraic and Differential Invariants". In: *Foundations of computational mathematics, Budapest 2011*. Ed. by F. Cucker, T. Krick, A. Pinkus, and A. Szanto. London Mathematical Society Lecture Note Series 403. Cambridge University Press, 2012, pp. 168-190.

[Hub19] E. Hubert. "Invariant algebraic sets and symmetrization of polynomial systems". In: *Journal of Symbolic Computation* 95 (2019), pp. 53-67.

[HK07a] E. Hubert and I. Kogan. "Rational invariants of a group action. Construction and rewriting". In: *Journal of Symbolic Computation special issue: Effective Methods in Algebraic Geometry (MEGA 05)* 42.1-2 (2007), pp. 203-217.

[HK07b] E. Hubert and I. Kogan. "Smooth and algebraic invariants of a group action: local and global constructions". In: *Foundations of Computational Mathematics* 7.4 (2007), pp. 455-493.

[HL12] E. Hubert and G. Labahn. "Rational invariants of scalings from Hermite normal forms". In: *ISSAC 2012—Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation*. ACM, New York, 2012, pp. 219-226.

[HL13] E. Hubert and G. Labahn. "Scaling invariants and symmetry reduction of dynamical systems". In: *Found. Comput. Math.* 13.4 (2013), pp. 479-516.

[HL16] E. Hubert and G. Labahn. "Computation of the Invariants of Finite Abelian Groups". In: *Mathematics of Computations* 85.302 (2016), pp. 3029-3050.
A Rewriting an invariant in terms of the generators

We exhibited algebraically independent generators of the invariant field $\mathbb{R}(V)^{SO_3(\mathbb{R})}$ emphasizing their separation properties. One can also use a generating set of invariants as new variables of an invariant problem, thus factoring out the symmetry; see for instance [HL12; HL13; HL16] where the invariants are used to solve polynomial systems with symmetry or reduce the number of parameters in dynamical systems. In this appendix we show how to rewrite any invariants in $\mathbb{R}(V)^{SO_3(\mathbb{R})}$ in terms of the generators exhibited.

Consider $\tilde{R} \in \mathbb{R}(V)^{SO_3(\mathbb{R})}$. Then the restriction of $\tilde{R}$ to the slice $S$ is an $O_2(\mathbb{R})$-invariant; in other words $R = \tilde{R}|_S \in \mathbb{R}(S)^{O_2(\mathbb{R})}$. If we can identify a rational function $F$ such that $R = F(T_j, D_j, R_{mj}, I_{mj})$ then $\tilde{R} = F(\tilde{T}_j, \tilde{D}_j, \tilde{R}_{mj}, \tilde{I}_{mj})$ by Theorem 3.6. One thus can rewrite any element of $\mathbb{R}(V)^{SO_3(\mathbb{R})}$ as a rational function of $\tilde{\mathcal{E}}$ thanks to the following rewriting rules.

**Theorem A.1.** Assume $P, Q \in \mathbb{R}[S]$ and take $p$ and $q$ in

$$ \mathbb{R}(T_j, D_j, R_{mj}, I_{mj})[y_{0,0}; y_{1,1}; y_{-1,1}; y_{-d_j, j}, \dots, y_{-d_j, j}, 1 \le j \le n] $$

the reductions of $P$ and $Q$ by the following confluent system of rewrite rules:

$$
\begin{align*}
y_{0,j} &\rightarrow \frac{D_j}{D_0} y_{0,0} & 1 \le j \le p \\
y_{0,j} &\rightarrow T_j & p+1 \le j \le n \\
y_{m,j} &\rightarrow 2^{m-1} \left( \frac{R_{mj}}{R_{11}^m} - \frac{i I_{mj}}{y_{0,0} R_{11}^m} \right) y_{1,1}^m & 1 \le j \le n, 1 \le m \le d_j \\
y_{-m,j} &\rightarrow 2^{m-1} \left( \frac{R_{mj}}{R_{1,1}^m} + \frac{i I_{mj}}{y_{0,0} R_{1,1}^m} \right) y_{-1,1}^m & 1 \le j \le n, 1 \le m \le d_j \\
y_{1,1} y_{-1,1} &\rightarrow R_{11} \\
y_{0,0}^2 &\rightarrow D_0
\end{align*}
$$

Then $\frac{P}{Q}$ belongs to $\mathbb{R}(S)^N$ iff $p = \tilde{p}r$ and $q = \tilde{q}r$ with $\tilde{p}, \tilde{q} \in \mathbb{R}(T_j, D_j, R_{mj}, I_{mj})$ and $r \in \mathbb{R}[y]$. In which case

$$ \frac{P}{Q} = \frac{\tilde{p}(T_j(H_1), D_j(H_1), R_{mj}(H_1), I_{mj}(H_1))}{\tilde{q}(T_j(H_1), D_j(H_1), R_{mj}(H_1), I_{mj}(H_1))} $$

*Proof.* The proof relies on [MS99]. A Gröbner basis of the ideal in $\mathbb{R}(y)[Y]$ generated by

$$
\begin{align*}
& \{Y_0 Y_{0,j} - y_{0,0} y_{0,j}, 1 \le j \le p\} \cup \{Y_{0,j} - y_{0,j}, p+1 \le j \le n\} \\
\cup & \{(Y_{-1,1} Y_{m,j} + Y_{1,1}^m Y_{-m,j}) - (y_{-1,1}^m y_{m,j} + y_{1,1}^m y_{-m,j}), 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1)\} \\
\cup & \{Y_0 (Y_{-1,1}^m Y_{m,j} - Y_{1,1}^m Y_{-m,j}), -y_{0,0} (y_{-1,1}^m y_{m,j} - y_{1,1}^m y_{-m,j}), 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1)\} \\
& \cup \{Y_{-1,1} Y_{1,1} - y_{-1,1} Y_{1,1}, Y_0^2 - y_{0,0}^2\}
\end{align*}
$$

w.r.t. to a term order that eliminates $Y_{0,j}$, $Y_{m,j}$ and $Y_{-m,j}$ is given by

$$
\begin{align*}
& \{\mathbf{Y}_0^2 - y_0^2, \mathbf{Y}_{1,1}\mathbf{Y}_{-1,1} - y_{1,1}y_{-1,1}\} \\
& \cup \{\mathbf{Y}_{0,j} - \frac{y_{0,j}}{y_0} Y_0 | 1 \le j \le p\} \cup \{\mathbf{Y}_{0,j} - y_{0,j} | p+1 \le j \le n\} \\
& \cup \left\{ \mathbf{Y}_{m,j} - \left( \frac{(y_{m,j}y_{-1,1}^m + y_{-m,j}y_{1,1}^m)}{2(y_{-1,1}y_{1,1})^m} + \frac{(y_{m,j}y_{-1,1}^m - y_{-m,j}y_{1,1}^m)}{2y_0(y_{-1,1}y_{1,1})^m} \right) Y_0 \right\}_{1,1} | 1 \le j \le n, 1 \le m \le d_j, (m,j) \neq (1,1) \}, \\
& \cup \left\{ \mathbf{Y}_{-m,j} - \left( \frac{(y_{m,j}y_{1,1}^m + y_{-m,j}y_{1,1}^m)}{2y_{-1,1}^m y_{1,1}^m} - \frac{(y_{m,j}y_{-1,1}^m - y_{-m,j}y_{1,1}^m)}{2y_0(y_{-1,1}y_{1,1})^m} \right) Y_0 \right\}_{-1,1} | 1 \le j \le n, 1 \le m \le d_j, (m,j) \neq (1,1) \}
\end{align*}
$$

where the leading terms are in bold font. This latter can be written as :

$$
\begin{align*}
& \{\mathbf{Y}_0^2 - D_0, \mathbf{Y}_{1,1}\mathbf{Y}_{-1,1} - \frac{1}{2}R_{11}\} \\
& \cup \{\mathbf{Y}_{0,j} - \frac{y_{0,j}}{y_0} Y_0 | 1 \le j \le p\} \cup \{\mathbf{Y}_{0,j} - y_{0,j} | p+1 \le j \le n\} \\
& \cup \left\{ \mathbf{Y}_{m,j} - 2^{m-1} \left( \frac{R_{mj}}{R_{11}^m} - \frac{i I_{mj}}{D_0 R_{11}^m} Y_0 \right) Y_{1,1}^m | 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1) \right\}, \\
& \cup \left\{ \mathbf{Y}_{-m,j} - 2^{m-1} \left( \frac{R_{mj}}{R_{11}^m} + \frac{i I_{mj}}{D_0 R_{11}^m} Y_0 \right) Y_{-1,1}^m | 1 \le j \le n, 1 \le m \le d_j, (m,j) \ne (1,1) \right\}.
\end{align*}
$$
[MB99] J. Müller-Quade and T. Beth. "Calculating generators for invariant fields of linear algebraic groups". In: *Applied algebra, algebraic algorithms and error-correcting codes* (Honolulu, HI, 1999). Vol. 1719. Lecture Notes in Computer Science. Springer, 1999.

[MS99] J. Müller-Quade and R. Steinwandt. "Basic algorithms for rational function fields". In: *Journal of Symbolic Computation* 27.2 (1999), pp. 143–170. DOI: 10.1006/jsco.1998.0246.

[Oli14] M. Olive. "Géométrie des espaces de tenseurs Une approche effective appliquée à la mécanique des milieux continus". PhD thesis. AMU, 2014.

[PV94] V. Popov and E. Vinberg. "Invariant theory". In: *Algebraic geometry. IV*. Springer, 1994, pp. 122–278.

[Ros56] M. Rosenlicht. "Some basic theorems on algebraic groups". In: *American Journal of Mathematics* 78 (1956), pp. 401–443.

[Stu93] B. Sturmfels. *Algorithms in invariant theory*. Texts and Monographs in Symbolic Computation. Vienna: Springer-Verlag, 1993.

[Wey46] H Weyl. *The classical groups: their invariants and representations*. Vol. 45. Princeton University Press, 1946.

[Wig59] E. Wigner. *Group theory and its application to the quantum mechanics of atomic spectra*. Academic Press Inc. (London), 1959.
By [MS99, Lemma 1.7 and Algorithm 1.10], we obtain the announced result.

<figure><img src="image_6.png" /></figure>

**Example A.2.** In the space of piezoelectricity tensors, the invariant $\tilde{P}$ whose restriction to the slice equals $P = y_{-2,1}^3 y_{3,1}^2 + y_{2,1}^3 y_{-3,1}^2$ can be rewritten as

$$ \tilde{P} = \frac{256}{\tilde{R}_{1,1}^6} \left( \tilde{R}_{21}^3 \tilde{R}_{31}^2 - \frac{\tilde{R}_{21}^3 \tilde{I}_{31}^2}{\tilde{D}_0} + \frac{6 \tilde{R}_{21}^2 \tilde{I}_{21} \tilde{R}_{31} \tilde{I}_{31}}{\tilde{D}_0} - \frac{3 \tilde{R}_{21} \tilde{I}_{21}^2 \tilde{R}_{31}^2}{\tilde{D}_0} + \frac{3 \tilde{R}_{21} \tilde{I}_{21}^2 \tilde{I}_{31}^2}{\tilde{D}_0^2} - \frac{2 \tilde{I}_{21}^3 \tilde{R}_{31} \tilde{I}_{31}}{\tilde{D}_0^2} \right). $$

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