---
author:
- |
    Alain Chenciner & Hugo Jiménez-Pérez\
    Observatoire de Paris, IMCCE (UMR 8028), ASD\
    77, avenue Denfert-Rochereau, 75014 Paris, France\
    `chenciner@imcce.fr, jimenez@imcce.fr`
title: 'Angular momentum and Horn’s problem'
---

We prove a conjecture made in [@C1]: given an $n$-body central configuration $X_0$ in the euclidean space $E$ of dimension $2p$, let $Im{\cal F}$ be the set of ordered real $p$-tuples $\{\nu_1,\nu_2,\cdots,\nu_p\}$ such that $\{\pm i\nu_1,\pm i\nu_2,\cdots,\pm i\nu_p\}$ is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of $X_0$ in $E$. Then $Im {\cal F}$ is a convex polytope. The proof consists in showing that there exist two, generically $(p-1)$-dimensional, convex polytopes ${\cal P}_1$ and ${\cal P}_2$ in ${\ensuremath{\mathbb{R}}}^{p}$ such that ${\cal P}_1\subset Im{\cal F}\subset {\cal P}_2$ and that these two polytopes coincide.

${\cal P}_1$, introduced in [@C1], is the set of spectra corresponding to the hermitian structures $J$ on $E$ which are “adapted" to the symmetries of the inertia matrix $S_0$; it is associated with Horn’s problem for the sum of $p\times p$ real symmetric matrices with spectra $\sigma_-$ and $\sigma_+$ whose union is the spectrum of $S_0$.

${\cal P}_2$ is the orthogonal projection onto the set of “hermitian spectra" of the polytope ${\cal P}$ associated with Horn’s problem for the sum of $2p\times 2p$ real symmetric matrices having each the same spectrum as $S_0$.

The equality ${\cal P}_1={\cal P}_2$ follows directly from a deep combinatorial lemma, proved in [@FFLP], which characterizes those of the sums $C=A+B$ of two $2p\times 2p$ real symmetric matrices $A$ and $B$ with the same spectrum, which are hermitian for some hermitian structure.

Origin of the problem: $N$-body relative equilibria and their angular momenta
=============================================================================

We recall here the results of [@AC; @C1; @C2] which are needed in order to understand the mechanical origin of the purely algebraic conjecture solved in the present paper: given a configuration $x_0=(\vec r_1,\cdots,\vec r_N)\in E^N$ of $N$ punctual positive masses in the euclidean space $E$, a [*rigid motion*]{} of the configuration under Newton’s attraction is a motion in which the mutual distances $||\vec r_i-\vec r_j||$ between the bodies stay constant. It is proved in [@AC] (see also [@C2]) that such a motion is necessarily a [*relative equilibrium*]{}. This implies that the motion takes place in a space of even dimension $2p$, which can be supposed to coincide with $E$, and that, in a galilean frame fixing the center of mass at the origin, it is of the form $x(t)=(e^{\Omega t}\vec r_1,e^{\Omega t}\vec r_2,\cdots,e^{\Omega t}\vec r_N)$, where $\Omega$ is a $2p\times 2p$-antisymmetric endomorphism of the euclidean space $E$ which is non degenerate. Choosing an orthonormal basis where $\Omega$ is normalized, this amounts to saying that there exists a hermitian structure on the space $E$ of motion and an orthogonal decomposition $E\equiv{\ensuremath{\mathbb{C}}}^p={\ensuremath{\mathbb{C}}}^{k_1}\times\cdots\times{\ensuremath{\mathbb{C}}}^{k_r}$ such that $$x(t)=(x_1(t),\cdots,x_r(t))=(e^{i\omega_1t}x_1,\cdots,e^{i\omega_rt}x_r),$$ where $x_m$ is the orthogonal projection on ${\ensuremath{\mathbb{C}}}^{k_m}$ of the $N$-body configuration $x$ and the action of $e^{i\omega_mt}$ on $x_m$ is the diagonal action on each body of the projected configuration. Such quasi-periodic motions exist only for very special configurations, called [*balanced configurations*]{} (see [@AC; @C2] for their characterization). The most degenerate balanced configurations are the [*central configurations*]{} for which all the frequencies $\omega_i$ are the same; this means that $\Omega=\omega J$, with $J$ a hermitian structure on $E$, and the motion is $$x(t)=(\vec r_1(t),\cdots,\vec r_N(t))=e^{i\omega t}x_0=(e^{i\omega t}\vec r_1,\cdots,e^{i\omega t}\vec r_N)$$ in the hermitian space $E\equiv{\ensuremath{\mathbb{C}}}^{p}$; in particular, it is periodic. In a space of dimension at most 3, $E$ is necessarily of dimension 2 and the configuration of any relative equilibrium is central.

Given a configuration $x=(\vec r_1,\cdots,\vec r_N)$ and a configuration of velocities $y=\dot x=(\vec v_1,\cdots, \vec v_N)$, both with center of mass at the origin: $\sum_{k=1}^Nm_k\vec r_k=\sum_{k=1}^Nm_k\vec v_k=0$, the [*angular momentum*]{} of $(x,y)$ is the bivector ${\mathcal C}=\sum_{k=1}^Nm_k\vec r_k\wedge\vec v_k$. If we represent $x$ and $y$ by the $2p\times N$ matrices $X$ and $Y$ whose $i$th column are respectively made of the components $(r_{1i},\cdots,r_{2pi})$ and $(v_{1i},\cdots,v_{2pi})$ of $\vec r_i$ and $\vec v_i$ in an orthonormal basis of $E$ and if $M=\hbox{diag}(m_1,\cdots,m_N)$, this bivector is represented by the antisymmetric matrix [*(we use the french convention $^{t\!}X$ for the transposed of $X$)*]{} $$C=-XM^{t\!}Y+YM^{t\!\!}X\;\;\hbox{with coefficients}\;\; c_{ij}
=\sum_{k=1}^Nm_k(-r_{ik}v_{jk}+r_{jk}v_{ik}).$$ The dynamics of a solid body is determined by its [*inertia tensor*]{} (with respect to its center of mass), represented in the case of a point masses configuration $X$ by the symmetric matrix $$S=XM^{t\!\!}X\;\;\hbox{with coefficients}\;\; 
s_{ij}=\sum_{k=1}^Nm_kr_{ik}r_{jk},$$ whose trace is the [*moment of inertia of the configuration $x$ with respect to its center of mass*]{}. In particular, the angular momentum of a relative equilibrium solution $X(t)=e^{t\Omega}X_0$ is represented by the antisymmetric matrix $C=S_0\Omega+\Omega S_0$, where $S_0=X_0M^{t\!\!}X_0$. Restricting to the case of central configurations, that is $\Omega=\omega J$, and making $\omega=1$, we consider in what follows the spectrum of $J$-skew-hermitian matrices of the form $S_0J+JS_0$ or, what amounts to the same, the spectrum of $J$-hermitian matrices[^1] of the form $J^{-1}S_0J+S_0$.

[*In the following, we identify $E$ with ${\ensuremath{\mathbb{R}}}^{2p}$ by the choice of some orthonormal basis. ${\ensuremath{\mathbb{R}}}^{2p}$ is endowed with its canonical basis $e_i=(0,\cdots,1,\cdots,0)$ and its canonical euclidean scalar product $x\cdot y=\sum_{i=1}^{2p}x_iy_i$; this allows identifying linear endomorphisms of $E={\ensuremath{\mathbb{R}}}^{2p}$ and $2p\times 2p$ matrices with real coefficients. When we say that $J$ is a hermitian structure, we mean that the standard euclidean structure is given and that $J$ is a complex structure which is orthogonal.*]{}

The frequency map
=================

We recall the definition, given in [@C1], of the [*frequency map*]{} ${\cal F}$ from the set of hermitian structures on ${\ensuremath{\mathbb{R}}}^{2p}$ to the positive Weyl chamber $W_p^+\subset {\ensuremath{\mathbb{R}}}^p$: given some $2p\times 2p$ real symmetric matrix $S_0$, we consider the mapping $J\mapsto J^{-1}S_0J+S_0$ from the space of hermitian structures on $E$ to the set of $2p\times 2p$ real symmetric matrices. We are only interested in the spectra of these matrices, hence choosing an orientation for $J$ is harmless and we shall consider only those of the form $J=R^{-1}J_0R$, where $J_0$ is the “standard" structure defined by $J_0=\begin{pmatrix}0&-Id\\ Id&0\end{pmatrix}$ and $R\in SO(2p)$. Two elements $R'$ and $R''$ of $SO(2p)$ defining the same $J$ if and only if there exists an element $U\in U(p)$ such that $R''=UR'$, it follows that the space of (oriented) hermitian structures is identified to the homogeneous space $U(p)\backslash SO(2p)$. The symmetric matrix $J^{-1}S_0J+S_0$ is $J$-hermitian, that is, it commutes with $J$. This implies that its spectrum is real, of the form $\{\nu_1,\nu_2,\cdots,\nu_p\}$ if considered as a $p\times p$ complex matrix (for the identification of ${\ensuremath{\mathbb{R}}}^{2p}$ to ${\ensuremath{\mathbb{C}}}^p$ defined by $J$) and of the form $\{(\nu_1, \nu_2, \cdots, \nu_p), (\nu_1, \nu_2, \cdots, \nu_p)\}$ if considered as a $2p\times 2p$ real matrix (see the next section for the trivial proof).

The [*frequency mapping*]{} $${\mathcal F}:U(p)\backslash SO(2p)\to W_p^+=\{(\nu_1,\cdots\nu_p)\in{\ensuremath{\mathbb{R}}}^p,   \nu_1\ge\cdots\ge\nu_p\}$$ associates to each hermitian structure $J$ the ordered spectrum $(\nu_1,\cdots,\nu_p)$ of the $J$-hermitian matrix $J^{-1}S_0J+S_0$.

Hermitian spectra
=================

Let $C:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be a symmetric endomorphism. The following assertions are equivalent:

1\) There exists a hermitian structure $J=R^{-1}J_0R$ such that $C$ be $J$-hermitian (i.e. $JC=CJ$);

2\) The spectrum $\sigma(C)$ of $C$ is of the form $$\sigma(C)=\{(\nu_1, \nu_2, \cdots, \nu_p), (\nu_1, \nu_2, \cdots, \nu_p)\}.$$

Let $J=R^{-1}J_0R$; the mapping $C$ is $J$-hermitian if and only if $RCR^{-1}$ is $J_0$-hermitian. This is equivalent to the existence of an isomorphism $U\in U(p)\subset SO(2p)$ such that $$U^{-1}RCR^{-1}U=\hbox{diag}\bigl((\nu_1,\cdots,\nu_p),(\nu_1,\cdots,\nu_p)\bigr).$$ Conversely, the identity $$R^{-1}CR=\hbox{diag}\bigl((\nu_1,\cdots,\nu_p),(\nu_1,\cdots,\nu_p)\bigr)$$ implies the commutation of $R^{-1}CR$ with $J_0$ and hence the commutation of $C$ with $J=R^{-1}J_0R$.

[**Notations.**]{} We shall call [*hermitian*]{} the spectra of this form and [*the diagonal*]{} the linear subspace $\Delta$ of $W_{2p}^+$ defined by $$\Delta=\{(\mu_1\ge\mu_2\ge\cdots\ge\mu_{2p}),\; \mu_1=\mu_2,\, \mu_3=\mu_4,\, \, \cdots,\mu_{2p-1}=\mu_{2p}\}.$$ Hence the ordered hermitian spectra are the ones belonging to $\Delta$.

Two convex polytopes
====================

Let $S_0:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be a symmetric endomorphism with spectrum $$\sigma(S_0)=\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}.$$ To $S_0$ we associate the subsets ${\cal P}_1$ and ${\cal P}_2$ of ${\ensuremath{\mathbb{R}}}^p$ (in fact of the positive Weyl chamber $W_p^+$ of ordered real $p$-tuples), defined as follows:

1\) ${\cal P}_1$ is the set of ordered spectra $$\sigma(c)=\{\nu_1\ge\nu_2\ge\cdots\ge\nu_p\}$$ of symmetric endomorphisms $c$ of ${\ensuremath{\mathbb{R}}}^p$ of the form $c=a+b$, where $a:{\ensuremath{\mathbb{R}}}^p\to{\ensuremath{\mathbb{R}}}^p$ and $b:{\ensuremath{\mathbb{R}}}^p\to{\ensuremath{\mathbb{R}}}^p$ are arbitrary symmetric endomorphisms with respective spectra $$\sigma(a)=\sigma_-:= \{\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}\},\quad \sigma(b)=\sigma_+:=\{\sigma_2,\sigma_4,\cdots,\sigma_{2p}\} ;$$

2\) ${\cal P}_2$ is the set of $p$-tuples $\{\nu_1\ge\nu_2\ge\cdots\ge\nu_p\}$ such that $$\{(\nu_1,\nu_2\cdots,\nu_{p}),(\nu_1,\nu_2\cdots,\nu_{p})\}$$ is the spectrum of some symmetric endomorphism $C$ of ${\ensuremath{\mathbb{R}}}^{2p}$ of the form $C=A+B$, where $A:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ and $B:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ are arbitrary symmetric endomorphisms with the same spectrum $$\sigma(A)=\sigma(B)=\sigma(S_0).$$ In other words, identifying canonically the diagonal $\Delta$ with ${\ensuremath{\mathbb{R}}}^p$, one can write $${\cal P}_2={\cal P}\cap\Delta,$$ where ${\cal P}$ is the $(2p-1)$-dimensional Horn polytope which describes the ordered spectra of sums $C=A+B$ of $2p\times 2p$ real symmetric matrices $A,B$ with the same spectrum as $S_0$.

${\cal P}_1$ and ${\cal P}_2$ are both contained in the hyperplane of ${\ensuremath{\mathbb{R}}}^p$ with equation $$\sum_{i=1}^p\nu_i=\sum_{j=1}^{2p}\sigma_j.$$ Moreover, ${\cal P}_1$ and ${\cal P}_2$ are both $(p-1)$-dimensional convex polytopes and $${\cal P}_1\subset Im {\cal F} \subset{\cal P}_2.$$

The first identity comes from the additivity of the trace function. The fact that both ${\cal P}_1$, ${\cal P}$ and hence ${\cal P}_2={\cal P}\cap\Delta$, are convex polytopes is a general fact coming from the interpretation of the Horn problem as a moment map problem. Finally, the second inclusion comes from the very definition of ${\cal F}$ and the first comes from Lemma 1 and the following identity, where $\sigma_-$ and $\sigma_+$ are considered as $p\times p$ diagonal matrices and $\rho\in SO(p)$: $$\begin{cases}
&\begin{pmatrix}
\sigma_-&0\\
0&\sigma_+
\end{pmatrix}+
\begin{pmatrix}
0&-\rho^{-1}\\
\rho&0
\end{pmatrix}^{-1}
\begin{pmatrix}
\sigma_-&0\\
0&\sigma_+
\end{pmatrix}
\begin{pmatrix}
0&-\rho^{-1}\\
\rho&0
\end{pmatrix}\\
=&
\begin{pmatrix}
\sigma_-+\rho^{-1}\sigma_+\rho&0\\
0&\rho\sigma_-\rho^{-1}+\sigma_+
\end{pmatrix}\cdot
\end{cases}$$

[**Remark.**]{} The choice of the partition $\sigma=\sigma_-\cup\sigma_+$ of $\sigma$ is dictated by the following theorem, which proves that any other partition of $\sigma$ into two subsets with $p$ elements will lead to a smaller polytope ${\cal P}_1$:

Let $A$ and $B$ be $p\times p$ Hermitian matrices. Let $\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}$ be the eigenvalues of $A$ and $B$ arranged in descending order. Then there exist Hermitian matrices $\tilde A$ and $\tilde B$ with eigenvalues $\sigma_1\ge\sigma_3\ge\cdots\ge\sigma_{2p-1}$ and $\sigma_2\ge\sigma_4\ge\cdots\ge\sigma_{2p}$ respectively, such that $\tilde A+\tilde B=A+B$.

This was used in [@C1] to prove that ${\cal P}_1= Im {\cal A}$ is the image under the frequency map ${\cal F}$ of the [*adapted*]{} hermitian structures, i.e. those $J$ which send some $p$-dimensional subspace of ${\ensuremath{\mathbb{R}}}^{2p}$ generated by eigenvectors of $S_0$ onto its orthogonal.

The projection property
=======================

In this section, we prove the

The two polytopes ${\cal P}_1$ and ${\cal P}_2$ coincide.

${Im {\cal F}}={\cal P}_1= Im {\cal A}$. In other words, the image by the frequency map ${\cal F}$ of the adapted structures is already the full image $Im {\cal F}$.

We need recall the inductive definition of the Horn inequalities which define the Horn polytopes (see [@F]). For a sum $a+b=c$ of symmetric $p\times p$ matrices with respective (ordered in decreasing order) spectra $$\alpha=(\alpha_1,\cdots,\alpha_p),\;  \beta=(\beta_1,\cdots,\beta_p),\; \gamma=(\gamma_1,\cdots,\gamma_p),$$ they read $$(^*IJK)\quad\quad\quad \forall r<p,\; \forall (I,J,K)\in T^p_r,\quad \sum_{k\in K}\gamma_k\le \sum_{i\in I}\alpha_i+\sum_{j\in J}\beta_j,$$ where $T^p_r$ (notation of [@F], noted $LR^p_r$ by reference to Littlewood-Richardson coefficients in [@FFLP]) is defined as follows: let $U^p_r$ be the set of triples $(I,J,K)$ of subsets of cardinal $r$ of $\{1,2,\cdots, p\}$ such that $$\sum_{i\in I}i+\sum_{j\in J}j=\sum_{k\in K}k+\frac{r(r+1)}{2}.$$ Then set $T^p_1=U^p_1$ and define recursively $T^p_r$ by $$T^p_r=
\begin{bmatrix}
(I,J,K)\in U^p_r, \forall s<r,\, \forall (F,G,H)\in T^r_s, \\
\sum_{f\in F}i_f+\sum_{g\in G}j_g\le \sum_{h\in H}k_h+\frac{s(s+1)}{2}
\end{bmatrix}$$

An immediate computation gives the following

Let $$\begin{cases}
I_2&=(2i_1-1,2i_2-1,\cdots,2i_r-1,2j_1,2j_2,\cdots,2j_r),\\
J_2&=(2i_1-1,2i_2-1,\cdots,2i_r-1,2j_1,2j_2,\cdots,2j_r),\\
K_2&=(2k_1-1,2k_1,2k_2-1,2k_2,\cdots,2k_r-1,2k_r),
\end{cases}$$

Then $(I_2,J_2,K_2)\in U^{2p}_{2r}$.

It suffices to check that $$2\left[\sum_{i\in I}(2i-1)+\sum_{j\in J}(2j)\right]=\sum_{k\in K}(2k-1)+\sum_{k\in K}2k+\frac{2r(2r+1)}{2}\cdot$$

Now, comes the key fact:

For any triple $(I,J,K)$ in $T^p_r$, the triple $(I_2,J_2,K_2)$ is in $T^{2p}_{2r}$

The proof of this theorem, which concerns the so-called “domino-decomposable Young diagrams", is based on a version of the Littlewood-Richardson rule due to Carré and Leclerc [@CL].

It implies that, for any a sum $A+B=C$ of real symmetric $2p\times 2p$ matrices with respective (ordered in decreasing order) spectra $$\hat\alpha=(\hat\alpha_1,\cdots,\hat\alpha_{2p}),\;  \hat\beta=(\hat\beta_1,\cdots,\hat\beta_{2p}),\; \hat\gamma=(\hat\gamma_1,\cdots,\hat\gamma_{2p}),$$ $(^*I_2,J_2,K_2)$ holds, that is $$\sum_{k\in K}(\hat\gamma_{2k-1}+\hat\gamma_{2k})\le \sum_{i\in I}(\hat\alpha_{2i-1}+\hat\beta_{2i-1})+\sum_{j\in J}(\hat\alpha_{2j}+\hat\beta_{2j}).$$ In particular, if $$\hat\alpha=\hat\beta=\sigma=(\sigma_1,\sigma_2,\cdots,\sigma_{2p}),$$ we get that $$\sum_{k\in K}\frac{\hat\gamma_{2k-1}+\hat\gamma_{2k}}{2}\le \sum_{i\in I}\sigma_{2i-1}+\sum_{j\in J}\sigma_{2j}.$$ Note that the mapping $$(\hat\gamma_1,\hat\gamma_2,\cdots,\hat\gamma_{2p-1},\hat\gamma_{2p})\mapsto (\frac{\hat\gamma_1+\hat\gamma_2}{2},\frac{\hat\gamma_1+\hat\gamma_2}{2},\cdots, \frac{\hat\gamma_{2p-1}+\hat\gamma_{2p}}{2},\frac{\hat\gamma_{2p-1}+\hat\gamma_{2p}}{2})$$ is the orthogonal projection of the ordered set $(\hat\gamma_1,\hat\gamma_2,\cdots,\hat\gamma_{2p-1},\hat\gamma_{2p})$ on the [*diagonal*]{} $\Delta$ of ${\ensuremath{\mathbb{R}}}^{2p}$ defined by the equations $\hat\gamma_1=\hat\gamma_2,\cdots,\hat\gamma_{2p-1}=\hat\gamma_{2p}$, that is on the subset of “hermitian" spectra. Hence a paraphrase of the above theorem is

Let $C=A+B$ be the sum of two $2p\times 2p$ real symmetric matrices with the same spectrum $\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}$.

If $\{\nu_1\ge\cdots\ge\nu_p\}$ is the orthogonal projection on the diagonal $\Delta\equiv{\ensuremath{\mathbb{R}}}^p$ of the spectrum $\{\hat\gamma_1\ge\hat\gamma_2\ge\cdots\ge\hat\gamma_{2p}\}$ of $C$, that is if $\nu_k=\frac{\hat\gamma_{2k-1}+\hat\gamma_{2k}}{2}$, the triple of ordered spectra $$\alpha=(\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}),\; \beta=(\sigma_2,\sigma_4,\cdots,\sigma_{2p}),\; \gamma=(\nu_1,\nu_2,\cdots,\nu_p)$$ satisfies the Horn inequality $(^*I,J,K)$.

This implies the following extremal property of the subset of “hermitian" spectra:

The orthogonal projection on the diagonal $\Delta$ of the $(2p-1)$-dimensional Horn polytope ${\cal P}\subset {\ensuremath{\mathbb{R}}}^{2p}$ associated with the spectra $$\sigma(A)=\sigma(B)=\{\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_{2p}\}$$ coincides with the $(p-1)$-dimensional Horn polytope ${\cal P}_1\in{\ensuremath{\mathbb{R}}}^p$ associated with the spectra $$\sigma(a)=(\sigma_1,\sigma_3,\cdots,\sigma_{2p-1}),\; \sigma(b)=(\sigma_2,\sigma_4,\cdots,\sigma_{2p}).$$

In particular, the intersection ${\cal P}_2={\cal P}\cap\Delta$ corresponding to the hermitian spectra, that is those such that $\hat\gamma_1=\hat\gamma_2,\cdots,\hat\gamma_{2p-1}=\hat\gamma_{2p}$, coincides with ${\cal P}_1$. Indeed, ${\cal P}_2$ coincides with the projection of ${\cal P}$, which itself coincides with ${\cal P}_1$.

[**Remark.**]{} The equality $Im {\cal F}={\cal P}_2$ implies the following

Let $C:{\ensuremath{\mathbb{R}}}^{2p}\to{\ensuremath{\mathbb{R}}}^{2p}$ be the sum $C=A+B$ of two symmetric endomorphisms $A$ and $B$ with the same spectrum $\sigma(A)=\sigma(B)=\sigma(S_0)$. Then $C$ is $J$-hermitian for some hermitian structure $J$ on ${\ensuremath{\mathbb{R}}}^{2p}$ if and only if it is conjuguate by an element of $SO(2p)$ to a matrix of the form $S_0+\tilde J^{-1}S_0\tilde J$, where $\tilde J$ is a hermitian structure on $R^{2p}$.

[**Note.**]{} The proof of the above results has been written by the first author after he was convinced by the numerical experiments made by the second author that the equality ${\cal P}_1={\cal P}_2$ was plausible when $p=3$ and more precisely that not only the intersection ${\cal P}_2={\cal P}\cap\Delta$ but also the orthogonal projection of the Horn polytope ${\cal P}$ on $\Delta$ was contained in ${\cal P}_1$ after the canonical identification of $\Delta$ with ${\ensuremath{\mathbb{R}}}^p$. This led first to a proof when $p=2$ or $3$, obtained by coping directly with Horn’s inequalities and then to the discovery that the general case followed from a lemma which turned out to be exactly the lemma 1.18 of [@FFLP]. The numerical experiments are described in the next section.

Numerical experiments
=====================

The numerical checking of the conjecture that $Im {\cal F}={\cal P}_1$, was made on the matrix $S_0= \frac{1}{32}{\rm diag}\left\{ 13,8,5,3,2,1 \right\}$, whose spectrum satisfies the inequalities in [@C1] (section 8). We wrote a program in TRIP [@GL11] producing different rotation matrices $R\in SO(2p)$ in a random way. Starting from the canonical basis $\xi=\{\xi_1,\dots,\xi_m\},$ $m=p(2p-1)$, of $\mathfrak{so}(2p)$, we created a list containing the $m$ one-parameter subgroups $G_i(t)= e^{t\xi_i}\subset SO(2p)$. We created a second list of $m$ random values $[t_i]_{i=1}^m$ and a random permutation $[1,2,\dots,m]\to [i_1,i_2,\dots ,i_m]$. The random rotation matrix was defined as $$\begin{aligned}
    R = \prod_{j=1}^m G_{i_j}(t_j),\quad 0\leq t_i \leq 2\pi.
    \label{eqn:R}\end{aligned}$$ The program which plots $\mathcal P_1$ is very simple (the fact that we replaced the conjugation of $J_0$ by the conjugation of $S_0$ is immaterial and comes from the formulation of the conjecture in [@C1]):\

  --------------------------------------------
  `create` $S_0$ `and` $J_0$
  `for` $i=1$ `to` $N_{max}$ `do`
  $\quad$ `create` $R$ `and` $R^{-1}$;
  $\quad$ $S = RS_0R^{-1}$;
  $\quad$ $C = S - J_0 S J_0$;
  $\quad$ `lst = eigenvalues(`$C$`)`;
  $\quad$ `plot ( lst[5], lst[3], lst[1] )`;
  `end for`.
  --------------------------------------------

We have assigned the value $N_{max}=25000$ obtaining the results shown in Figure \[fig:AA\]. The figure shows also the simplex $\gamma_1+\gamma_2+\gamma_3=1$ and its intersection with $W_3^+$.

![${\cal P}_1=Im {\cal A}$: 25000 random adapted hermitian structures[]{data-label="fig:AA"}](fig1)

The modified algorithm to estimate the shape of $\mathcal P_2=\mathcal 
P\cap\Delta$ in $W^+_3$ is similar. For a random $R$ in $SO(6)$, the ordered spectrum $spec(C)=\left( \gamma_1,\dots,\gamma_6 \right)$ of $C=S_0+R^{-1}S_0R$ is projected orthogonally onto the diagonal $\Delta$ by the map $\pi_\Delta:\mathbb R^6 \to \Delta$: $$\begin{aligned}
 (\gamma_1,\gamma_2,\dots,\gamma_6)&\mapsto& \left( \frac{\gamma_1+\gamma_2}{2},
 \frac{\gamma_3+\gamma_4}{2},
 \frac{\gamma_5+\gamma_6}{2}
    \right).\end{aligned}$$ At first, when $spec(C)$ was $\varepsilon$-close to $\Delta$ *i.e.*, if $\sum_{k=1}^3 |\gamma_{2k-1}-\gamma_{2k}|^2 < 2\varepsilon^2$ for $\varepsilon$ small, the projected point was plotted in green; otherwise it was plotted in red. No particular pattern was found for the green points meanwhile the red ones were all contained in $\mathcal P_1$. The algorithm to plot $\pi_\Delta(\mathcal P)$ is

  ------------------------------------------
  `create` $S_0$

  `for` $i=1$ `to` $N_{max}$ `do`

  $\quad$ `create` $R$ `and` $R^{-1}$;

  $\quad$ $C = S_0 + R^{-1} S_0 R$;

  $\quad$ `lst = eigenvalues(`$C$`)`;

  $\quad$ `sort( lst )`;

  $\quad$ `plot` $\left( 
        \frac{lst[6]+lst[5]}{2}, 
        \frac{lst[4]+lst[3]}{2}, 
        \frac{lst[2]+lst[1]}{2}\right)$`;`

  `end for`.
  ------------------------------------------

The results of the projection $\pi_\Delta(spec(C))$ for $50000$ random rotation matrices are shown in Figure \[fig:BB\].

![Projection of ${\cal P}$: 50000 random rotations[]{data-label="fig:BB"}](fig2)

The matrix $S_0$ and hence the polytope $\mathcal P_1$ are the same as in Figure \[fig:AA\] (the interior lines correspond to the polytopes associated to different partitions of the spectrum of $S_0$, as depicted in the corresponding figure in [@C1]). Recall that the polytope ${\cal P}$ has dimension 5; this explains that in order to get a better filling one should have taken many more points. This was not done because the evidence was sufficiently convincing to ask for a proof.

Acknowledgements
================

Warm thanks to Sun Shanzhong and Zhao Lei for their interest in this work and numerous discussions.

[99]{}

A. Albouy, A. Chenciner [*Le Problème des $N$ corps et les distances mutuelles*]{}, Inventiones mathematicae 131 (1998), 151-184.

A. Chenciner [*The angular momentum of a relative equilibrium*]{}, arXiv:1102.0025v1, final version to appear in D.C.D.S.

A. Chenciner [*The Lagrange reduction of the N-body problem: a survey*]{}, to appear in Acta Mathematica Vietnamica

C. Carré, B. Leclerc [*Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts*]{}, Journal of Algebraic Combinatorics 4 (1995), 201-231

W. Fulton [*Eigenvalues, invariant factors, highest weights, and Schubert calculus*]{}, Bull. Amer. Math. Soc. (N.S.) [**37**]{} no. 3, 209-249 (2000)

S. Fomin, W. Fulton, C.K. Li, Y.T. Poon, [*Eigenvalues, singular values, and Littlewood-Richardson coefficients*]{}, Amer. J. Math. [**127**]{}, no. 1, 101–127 (2005)

M. Gastineau and J. Laskar, 2011. [TRIP]{} 1.1.18, *TRIP Reference manual*. IMCCE, Paris Observatory. [http://www.imcce.fr/trip/]{}.

[^1]: Notice that this is the same as the spectrum of the $J_0$-hermitian matrix $\Sigma=J_0^{-1}SJ_0+S$, where $S=RS_0R^{-1}$, which was considered in [@C1].
