structure: $e_{1},\ldots,e_{n+3}$ are the elements of the canonical basis,
that is $e_{i}=(\delta_{ij})_{j=1}^{n+3}$ for $i=1,\ldots,n+3$. One denotes by
$x_{1},\ldots,x_{n+3}$ the corresponding standard coordinates, and one has

$\big{(}x,y\big{)}=\sum_{k=1}^{n+3}x_{k}y_{k}\qquad\mbox{ and }\qquad\lvert
x\lvert=\sqrt{(x,x)}=\sqrt{\,(x_{1})^{2}+\cdots+(x_{n+3})^{2}}$

for any two elements $x=(x_{k})_{k=1}^{n+3}$ and $y=(y_{k})_{k=1}^{n+3}$ of
$\mathbf{R}^{n+3}$. Finally, we will denote by $\mathbf{R}^{n+2}_{i}$ the
coordinate hyperplane in $\mathbf{R}^{n+3}$ cut out by the equation $x_{i}=0$.

Let $\mathcal{T}^{or}$ be the tautological bundle over
$G_{2}^{or}(\mathbf{R}^{n+3})$. Its Euler's class ${\rm E}_{n}^{or}={\rm
E}(\mathcal{T}^{or})$ is a non trivial element in
$\boldsymbol{H}^{2}(G_{2}^{or}(\mathbf{R}^{n+3}),\mathbf{R})$. Since the
oriented grassmannian is a symmetric space, there exists a unique ${\rm
SO}_{n+2}(\mathbf{R})$-invariant global differential 2-form $E_{n}^{or}$ on it
such that $[E_{n}^{or}]={\rm E}(\mathcal{T}^{or})$. To simplify, we will no
longer write the superscript or and just write $E_{n}$ in what follows.

We consider the Stiefel manifold
$\boldsymbol{\mathbb{S}}_{2}^{or}(\mathbf{R}^{n+3})$ of oriented 2-frame in
$\mathbf{R}^{n+3}$: it is the manifold of dimension $2n+3$ formed by
orthogonal 2-frames in $\mathbf{R}^{n+3}$:

$\boldsymbol{\mathbb{S}}_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}=\left\\{\,\big{(}e_{1},e_{2}\big{)}\in\big{(}\mathbf{R}^{n+3}\big{)}^{2}\hskip
2.84544pt\Big{\lvert}\hskip 2.84544pt\lvert e_{1}\lvert=\lvert
e_{2}\lvert=1\mbox{ and }\,\langle e_{1},e_{2}\rangle=0\,\right\\}\,.$

It comes with a natural map
$\mathcal{S}:\boldsymbol{\mathbb{S}}_{2}^{or}(\mathbf{R}^{n+3})\rightarrow
G_{2}^{or}(\mathbf{R}^{n+3})$, $(e_{1},e_{2})\mapsto e_{1}\wedge e_{2}$ which
makes of the Stiefel variety a ${\rm SO}_{2}(\mathbf{R})$-bundle over the
oriented grassmannian. The interest of considering the Stiefel manifold is
that there is a simple explicit formula for the pull-back of $E_{n}$ under
$\mathcal{S}$.

Indeed, for $i=1,2$, let $de_{i}$ be the $\mathbf{R}^{n+3}$-valued 1-form
whose components are the exterior derivatives of the scalar components of
$e_{i}$: if $e_{i}=(e_{i}^{k})_{k=1}^{n+3}$, then
$de_{i}=(de_{i}^{k})_{k=1}^{n+3}$. Then the componentwise wedge product of
$de_{1}$ with $de_{2}$ is a multiple of the pull-back of $E_{n}$ under
$\mathcal{S}$ since

(66) $\mathscr{E}_{n}=de_{1}\wedge de_{2}=\sum_{k=1}^{n+3}de_{1}^{k}\wedge
de_{2}^{k}=\frac{-1\,}{2\pi}\,\mathcal{S}^{*}\Big{(}E_{n}\Big{)}$

(cf. [Da1, §6.2] or [Da2, §6] for more details and references). Since we are
only interested in $E_{n}$ up to a non zero multiple, it will be more
convenient to work with
$\mathscr{E}_{n}\in\Gamma\Big{(}\Omega^{2}(\boldsymbol{\mathbb{S}}_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}\big{)}\Big{)}$
in what follows.

Formula (66) above gives us an effective way to compute $\mathscr{E}_{n}$.
Indeed, let $\gamma:U\rightarrow G_{2}^{or}(\mathbf{R}^{n+3})$ be a smooth map
where $U$ stands for an open domain in an affine space of dimension $N$. Given
affine coordinates $u_{1},\ldots,u_{N}$ on $U$, the question is to give an
explicit formula for (a multiple of) $\gamma^{*}(E_{n}^{or})$ in the
$u_{i}$'s. We assume that $\gamma$ is given by
$u\mapsto\gamma_{1}(u)\wedge\gamma_{2}(u)$ where both $\gamma_{1}$ and
$\gamma_{2}$ are smooth maps from $U$ to $\mathbf{R}^{n+3}$ such that
$\gamma_{1}(u)$ and $\gamma_{2}(u)$ are not colinear for any $u\in U$. Denote
by $GS$ the Gram-Schmidt orthogonalization process, defined here as the map

$\displaystyle GS:\,\Omega^{n+3}$
$\displaystyle\longrightarrow\boldsymbol{\mathbb{S}}_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}$
$\displaystyle\big{(}x,y\big{)}$
$\displaystyle\longmapsto\left(\frac{x}{\lvert
x\lvert}\,,\,\frac{\tilde{y}}{\lvert\tilde{y}\lvert}\right)$

where $\Omega^{n+3}$ stands for the set of pairs of two linearly independent
vectors of $\mathbf{R}^{n+3}$ and where for such a pair $(x,y)$, we have set
$\tilde{y}=y-\big{(}(x,y)/\lvert x\lvert^{2}\big{)}\,x$.

Then $\Gamma=GS\circ\gamma$ is a lift of $\gamma$ to the Stiefel variety, i.e.
the following diagram is commutative

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\boldsymbol{\mathbb{S}}_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}}$$\scriptstyle{\mathcal{S}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma}$$\scriptstyle{\gamma}$$\textstyle{G_{2}^{or}(\mathbf{R}^{n+3})}$

hence we obtain that $\Gamma^{*}(\mathscr{E}_{n})$ coincides with $-1/(2\pi)$
times the pull-back of $E_{n}$ under $\gamma$: one has

(67)
$\Gamma^{*}(\mathscr{E}_{n})=\frac{-1\,}{2\pi}\,\gamma^{*}\Big{(}E_{n}\Big{)}\,.$

We use the material of the previous paragraph in the somehow `most general map
$\gamma$' in order to deduce some formulas for (pull-backs of) Euler's form
which will be used later on. The map we are referring to here, denoted by
$\xi$, is just the map associating to a $2\times(n+3)$ matrix $M$ the oriented
2-plane $\xi(M)$ spanned by the first and the second rows of $M$, denoted by
$x(M)$ and $y(M)$ respectively: one has $\xi(M)=x(M)\wedge y(M)$. For
$x,y\in\mathbf{R}^{n+3}$ we denote by $M_{x,y}$ the matrix such that
$x(M_{x,y})=x$ and $y(M_{x,y})=y$ and we set $\xi(x,y)=\xi(M_{x,y})\in
G_{2}^{or}(\mathbf{R}^{n+3})$. The map $\xi$ under scrutiny here is defined on
the subset $\Omega_{2}\subset{\rm M}_{2\times(n+3)}(\mathbf{R})$ of rank 2
matrices:

(68) $\displaystyle\xi:\,\Omega_{2}$ $\displaystyle\longrightarrow
G_{2}^{or}(\mathbf{R}^{n+3})$ (71) $\displaystyle
M=\Big{[}\begin{tabular}[]{c}$x$\vspace{-0.15cm}\\\ $y$\end{tabular}\Big{]}$
$\displaystyle\longmapsto\xi(M)=\xi(x,y)=x\wedge y\,.$

From $\gamma:\Omega_{2}\rightarrow G_{2}^{or}(\mathbf{R}^{n+3})$, we build
other such maps by setting for any $M_{x,y}\in\Omega_{2}$

$\check{\gamma}(M_{x,y})={\gamma}(M_{x,y})\qquad\mbox{ and
}\qquad\gamma_{G}(M_{x,y})=\gamma(M_{x,y}\cdot G)\,$

where $G$ stands for a previously given constant element of the linear group
${\rm GL}_{n+3}(\mathbf{R})$. For
$\epsilon=(\epsilon_{k})_{k=1}^{n+3}\in\\{\pm 1\\}$, we denote by
$D_{\epsilon}$ the diagonal matrix with $\epsilon_{k}$ for its $k$-th
coefficient, that is $D_{\epsilon}={\rm
diag}(\epsilon_{1},\ldots,\epsilon_{n+3})\in{\rm GL}_{n+3}(\mathbf{R})$ and we
set $\gamma_{\epsilon}=\gamma_{D_{\epsilon}}$. With the preceding notation,
one can state some transformation formulas for (the pull-backs of) Euler's
form that will prove to be important in the sequel.

######  Lemma 5.1.

The 2-form $E_{n}$ satisfies the following transformation formulas:

  1. 1.

$\check{\gamma}^{*}\big{(}E_{n}\big{)}=-{\gamma}^{*}\big{(}E_{n}\big{)}$;

  2. 2.

$\gamma_{\epsilon}^{*}\big{(}E_{n}\big{)}={\gamma}^{*}\big{(}E_{n}\big{)}$ for
any $\epsilon\in\\{\pm 1\\}^{n+3}$;

  3. 3.

$\gamma_{G}^{*}\big{(}E_{n}\big{)}={\gamma}^{*}\big{(}E_{n}\big{)}$ for any
$G\in{\rm SO}_{n+3}(\mathbf{R})$.

###### Proof.

As the notation suggests, one has $\check{\gamma}=D\circ\gamma$ where $D$ is
the change of orientation. Since $D$ has for $\mathcal{S}$-equivariant lift
the exchange map $(e_{1},e_{2})\rightarrow(e_{2},e_{1})$ on the Stiefel
manifold, 1. follows from $de_{2}\wedge de_{1}=-de_{1}\wedge de_{2}$ together
with (66). The second point is a direct consequence of this formula as well.
As for 3., it follows immediatly from the ${\rm
SO}_{n+3}(\mathbf{R})$-invariance of $E_{n}$. ∎

For $i\in\\{1,\ldots,n+3\\}$, the natural inclusion
$\mathbf{R}^{n+2}_{i}\subset\mathbf{R}^{n+3}$ induces a natural embedding
$\iota_{i}:G^{or}_{2}(\mathbf{R}^{n+2}_{i})\hookrightarrow
G^{or}_{2}(\mathbf{R}^{n+3})$ which is such that the following lemma holds
true:

######  Lemma 5.2.

For any $k\geq 1$, denote by $E_{n}^{k}$ the $k$-th wedge product of $E_{n}$.

  1. 1.

The pull-back $\iota_{i}^{*}\big{(}E_{n})$ coincides with the Euler form of
the tautological bundle of $G^{or}_{2}(\mathbf{R}^{n+2}_{i})$.

  2. 2.

The pull-back $\iota_{i}^{*}\big{(}E_{n}^{n})$ is a ${\rm SO}_{n+2}$-invariant
volume form on $G^{or}_{2}(\mathbf{R}^{n+2}_{i})$.

###### Proof.

Since $\iota_{i}^{*}(\mathcal{T}^{or})$ coincides with the tautological bundle
on $G^{or}_{2}(\mathbf{R}^{n+2}_{i})$, the first point follows from the
naturality of the Euler's class and of the ${\rm SO}_{n+3}$-invariance of the
representative $E_{n}$.

The second point follows from the first combined with all the following facts:
$i.$ $G^{or}_{2}(\mathbf{R}^{n+2})$ is orientable and compact hence admits a
necessarily non exact volume form; and $ii.$ the top degree cohomology space
$\boldsymbol{H}^{2n}\big{(}G^{or}_{2}(\mathbf{R}^{n+2}),\mathbf{R}\big{)}\simeq\mathbf{R}$
is generated over $\mathbf{R}$ by (the class of) the ${\rm
SO}_{n+2}$-invariant form $E_{n-1}^{n}$. ∎

With the constructions and results discussed above at hand, one can now follow
Damiano's construction of an AR from a power of the Euler class $E_{n}$. It is
the subject of the next subsection.

####  5.1.3. The positive Eulerian abelian relation

Here, we review Damiano's construction (which itself heavily relies on
Gelfand-MacPherson theory [GM]) of the Euler abelian relation for
$\boldsymbol{\mathcal{W}}_{0,n+3}$ on the positive part
$\mathcal{M}_{0,n+3}^{>0}$, which is the privileged component of the moduli
space $\mathcal{M}_{0,n+3}$ we will work on. We will discuss later the Euler's
abelian relations on the other components $\mathcal{M}(\boldsymbol{\sigma})$.

Let $U\subset\mathbf{R}^{n}$ be the set of $n$-tuples
$u=(u_{i})_{i=1}^{n}\in\mathbf{R}^{n}$ such that $1<u_{1}<u_{2}<\cdots<u_{n}$.
The map

(72) $\displaystyle\varphi:U\,$
$\displaystyle\longrightarrow\mathcal{M}_{0,n+3}^{\,or,\,>0}$ $\displaystyle
u$
$\displaystyle\longmapsto\Big{[}\,\infty\,,\,0\,,\,-1\,,\,-u_{1}\,,\,-u_{2}\,,\,\ldots\,,\,-u_{n}\,\Big{]}$

can be seen as a global isomorphism hence the $u_{i}$'s form a global
coordinate system on $\mathcal{M}_{0,n+3}^{>0}$, which is the one we will work
with.

Recall that a matrix $M\in{\rm M}_{2,n+3}(\mathbf{R})$ is said `positive' if
all its $2\times 2$ minors are positive. We then define the `positive oriented
grassmannian' $G^{or}_{2}(\mathbf{R}^{n+2})^{>0}$ as the image by the map (68)
of the open subset $\Omega_{2}^{>0}\subset\Omega_{2}$ of positive matrices:
one has

(73) $G^{or}_{2}(\mathbf{R}^{n+2})^{>0}=\xi\Big{(}\Omega_{2}^{>0}\Big{)}\,.$

One verifies that the positive grassmannian
$G^{or}_{2}(\mathbf{R}^{n+2})^{>0}$ is

  * $-$

included in $\widehat{G}^{or}_{2}(\mathbf{R}^{n+2})$;

  * $-$

stable under the action of the `positive part' $H_{0}$ of the Cartan torus
$H\subset{\rm SL}_{n+3}(\mathbf{R})$;

  * $-$

such that the quotient of it by $H$ is $\mathcal{M}_{0,n+3}^{>0}$, i.e. one
has (cf. diagram (65)):

$\pi_{H}\Big{(}G^{or}_{2}\big{(}\mathbf{R}^{n+2})^{>0}\Big{)}=\mathcal{M}_{0,n+3}^{>0}\,.$

We consider the following map

(74) $\displaystyle M\hskip 4.26773pt:\hskip 4.26773ptU\hskip 4.26773pt$
$\displaystyle\longrightarrow\hskip 4.26773pt\Omega_{2}^{>0}$ $\displaystyle
u$ $\displaystyle\longmapsto
M_{u}=\begin{bmatrix}\,1&0&-1&-u_{1}&-u_{2}&\cdots&-u_{n}\,\\\
\,0&1&\,1&1&1&\cdots&1\,\end{bmatrix}$

which is such that (1) the composition $\gamma=\xi\circ M:U\rightarrow
G^{or}_{2}(\mathbf{R}^{n+2})$ takes values into the positive grassmannian; and
(2) makes the following diagram commutative:

(75) $\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}}$$\scriptstyle{\pi_{H}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ | $\gamma=\xi\circ M$
---

$\scriptstyle{\varphi}$$\textstyle{\mathcal{M}_{0,n+3}^{>0}\,.}$

It is not difficult to give an explicit formula for
$\pi=\varphi^{-1}\circ\pi_{H}:G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\rightarrow
U$. Given $\xi^{or}\in G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$, one has
$\xi^{or}=\xi_{x,y}=\xi\big{(}\big{[}{\begin{tabular}[]{c}$x$\vspace{-0.15cm}\\\
$y$\end{tabular}}\big{]}\big{)}$ for $x,y\in\mathbf{R}^{n+3}$ such that
$x_{i}y_{j}-y_{i}x_{j}>0$ for any $i,j$ such that $1\leq i<j\leq n+3$. In
particular, the submatrix $\big{[}{\begin{tabular}[]{c}$x_{1}$
$x_{2}$\vspace{-0.15cm}\\\ $y_{1}$ $y_{2}$\end{tabular}}\big{]}$ is
invertible, hence one has $\big{[}{\begin{tabular}[]{c}$x_{1}$
$x_{2}$\vspace{-0.15cm}\\\ $y_{1}$
$y_{2}$\end{tabular}}\big{]}\cdot\big{[}{\begin{tabular}[]{c}$x$\vspace{-0.15cm}\\\
$$ $y$\end{tabular}}\big{]}=\big{[}{\begin{tabular}[]{c}$1$ $0$
$\tilde{x}_{3}$ $\cdots$ $\tilde{x}_{n+3}$\vspace{-0.1cm}\\\ $$ $0$ $1$
$\tilde{y}_{3}$ $\cdots$ $\tilde{y}_{n+3}$\end{tabular}}\big{]}$ for some
$\tilde{x}_{k}$'s and $\tilde{y}_{k}$'s with $k=3,\ldots,n+3$, which are
explicit rational expressions in the original $x_{i}$'s and $y_{i}$'s. Notice
that, since all $2\times 2$ minors are positive, one necessarily has
$\tilde{x}_{k}<0$ and $\tilde{y}_{k}>0$ for any $k=3,\ldots,n+3$. Then setting
$h_{1}=-\tilde{x}_{3}$, $h_{2}=\tilde{y}_{3}$ and
$h_{k}=\tilde{y}_{3}/\tilde{y}_{k}$ for all such $k$, one has

$\begin{bmatrix}h_{1}^{-1}&0\\\
0&h_{2}^{-1}\end{bmatrix}\cdot\begin{bmatrix}1&0&\tilde{x}_{3}&\cdots&\tilde{x}_{n+3}\\\
0&1&\tilde{y}_{3}&\cdots&\tilde{y}_{n+3}\end{bmatrix}\cdot\begin{bmatrix}{h_{1}}&&0\\\
&\ddots&\\\
0&&{h_{n+3}}\end{bmatrix}=\begin{bmatrix}1&0&-1&-\frac{\tilde{x}_{4}\tilde{y}_{3}}{\tilde{y}_{4}\tilde{x}_{3}}&\cdots&-\frac{\tilde{x}_{n+3}\tilde{y}_{3}}{\tilde{y}_{n+3}\tilde{x}_{3}}\\\
0&1&1&1&\cdots&1\end{bmatrix}$

from which it follows that the map $\pi$ under scrutiny is given in
coordinates by

(76) $\displaystyle\pi\,:\,G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$
$\displaystyle\longrightarrow U$ $\displaystyle\xi_{x,y}$
$\displaystyle\longmapsto\left(\,\frac{\tilde{x}_{4}\tilde{y}_{3}}{\tilde{y}_{4}\tilde{x}_{3}}\,,\,\ldots\,,\,\frac{\tilde{x}_{n+3}\tilde{y}_{3}}{\tilde{y}_{n+3}\tilde{x}_{3}}\right)$

(it is easy and left to the reader to show that this is well-defined and is
the right formula).

$\star$

Recall the way of parametrizing the $H_{0}$-orbit of any generic oriented
2-plane $\xi^{or}$ by means of the hypersimplex
$\Delta_{2}^{n+3}=\big{\\{}\,(t_{i})_{i=1}^{n+3}\in]0,1[^{n+3}\,\big{\lvert}\,\sum_{i=1}^{n+3}t_{i}=2\,\big{\\}}$
(see [GM]): the map

$h:\Delta_{2}^{n+3}\longrightarrow(\mathbf{R}_{>0})^{n+3},\,t=\big{(}t_{i}\big{)}_{i=1}^{n+3}\longmapsto
h(t)=\Big{(}t_{1}/(1-t_{i})\Big{)}_{i=1}^{n+3}$

is such that the image of

$\displaystyle Dh={\rm Diag}\circ h\,:\hskip 4.26773pt\Delta_{2}^{n+3}$
$\displaystyle\longrightarrow{\bf P}{\rm SL}_{n+3}(\mathbf{R})$ $\displaystyle
t$ $\displaystyle\longmapsto{\rm
Diag}(h(t)\big{)}={$\begin{bmatrix}\frac{t_{1}}{1-t_{1}}&&0\\\ &\ddots&\\\
0&&\frac{t_{n+3}}{1-t_{n+3}}\end{bmatrix}$}$

is equal to the positive component $H_{0}$ of the Cartan torus $H\subset{\bf
P}{\rm SL}_{n+3}(\mathbf{R})$.

Using $Dh$ and the maps $\xi$ and $M$ defined in 68 and (74), one constructs

(77) $\textstyle{\Phi_{U}=\xi\circ\big{(}M\cdot Dh\big{)}\hskip
4.26773pt:\hskip 5.69046ptU\times\Delta_{2}^{n+3}\hskip
4.26773pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
4.26773ptG_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\hskip
56.9055pt}$$\textstyle{\hskip 125.19194pt\big{(}u,t\big{)}\hskip
1.42271pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
4.26773pt\xi\Big{(}M_{u}\cdot Dh(t)\Big{)}\hskip 56.9055pt}$

which can easily be proven to be an isomorphism (from the direct product
$U\times\Delta_{2}^{n+3}$ onto the whole positive oriented grassmannian
$G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$). Let

(78) $\delta:\,U\times\Delta_{2}^{n+3}\longrightarrow U\,$

be the projection onto the first factor. Since $E_{n}$ is a global smooth
2-form on the oriented grassmannian, for any $k\geq 1$, its $k$-th wedge
product $E_{n}^{k}$ is integrable along the fibers of the $H_{0}$-action.
Consequently, for any such $k$ the corresponding $2k$-form
$\widetilde{E}_{n}^{k}=\big{(}\Phi_{U}\big{)}^{*}(E_{n}^{k})$ on
$U\times\Delta_{2}^{n+3}$ admits a push-forward by $\delta$ which we will
denote by

$\mathscr{E}_{n}^{k}=\delta_{*}\big{(}\,\widetilde{E}_{n}^{k}\,\big{)}\,.$

It is a smooth $(2k-{n+2})$-form on $U$, where we use the convention that a
$\ell$-differential form is 0 when $\ell$ is negative. Two important
properties of these forms are given in the following:

######  Proposition 5.3 ([Da2]).

1\. The $(n-2)$-form $\mathscr{E}_{n}^{n}$ is trivial: one has
$\mathscr{E}_{n}^{n}=0$;

2\. The $n$-form $\mathscr{E}_{n}^{n+1}$ is non zero and can be written

(79) $\mathscr{E}_{n}^{n+1}(u)=e_{n}(u)\,du_{1}\wedge\ldots\wedge du_{n}$

for a non vanishing global analytic function $e_{n}$ on $U$.

###### Proof.

The first point is proved through a direct computation by Damiano (see [Da1,
§6.3]). The proof of 2. relies on the very same arguments used to establish
the second point of Lemma 5.2. ∎

In order to build the Eulerian AR discussed here, it is necessary to extend
$\Phi_{U}$ to the product $U\times\overline{\Delta}_{2}^{n+3}$ where
$\overline{\Delta}_{2}^{n+3}$ stands for the closure of the hypersimplex:

$\overline{\Delta}_{2}^{n+3}=\Big{\\{}\,\big{(}t_{1},\ldots,t_{n+3}\big{)}\in[0,1]^{n+3}\hskip
4.26773pt\Big{\lvert}\hskip
4.26773pt\begin{tabular}[]{l}$t_{1}+\cdots+t_{n+3}=2$\end{tabular}\Big{\\}}\,.$

In the classical case, that is when the target is the usual grassmannian, the
extension property of the corresponding map
$\nu\circ\Phi_{U}:U\times\Delta_{2}^{n+3}\rightarrow
G_{2}\big{(}\mathbf{R}^{n+3}\big{)}$ is well-known (cf. [GM, Prop. 2.3.2]) but
since the map takes values into the oriented grassmannian in the case under
scrutiny, we believe that it has to be justified, which is easy to do.

######  Proposition 5.4.

  1. 1.

The map $\Phi_{U}$ in (77) admits a unique continuous extension

(80) $\textstyle{\overline{\Phi}_{U}\hskip 4.26773pt:\hskip
5.69046ptU\times\overline{\Delta}_{2}^{n+3}\hskip
4.26773pt\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
4.26773ptG_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}\,.}$

  2. 2.

Moreover, this extension $\overline{\Phi}_{U}$ takes values into the totally
non negative oriented grassmannian
$G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{\geq 0}$ and is actually smooth as
a map between manifolds with corners.

This proposition is rather direct consequence of the following lemma:

######  Lemma 5.5.

  1. 1.

The preimage of $G_{2}(\mathbf{R}^{n+3})^{>0}$ by the covering
$\nu:G_{2}^{or}(\mathbf{R}^{n+3})\rightarrow G_{2}(\mathbf{R}^{n+3})$ is the
disjoint union of the positive and the negative oriented grassmannians: one
has

$\nu^{-1}\left(G_{2}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\right)=G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\sqcup
G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{<0}\,.$

  2. 2.

The map (77) is a smooth diffeomorphism (in particular, it is surjective);

  3. 3.

The isomorphism between $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$ and
its non oriented counterpart $G_{2}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$
induced by the restriction of $\nu$ (again denoted by $\nu$ to simplify)
extends to an isomorphism of smooth varieties with corner

$\nu:G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{\geq 0}\longrightarrow
G_{2}\big{(}\mathbf{R}^{n+3}\big{)}^{\geq 0}\,.$

###### Proof.

The first point is obvious and the second is hardly more difficult to prove.
The third point follows from the first two together with [GM, Prop. 2.3.2]. ∎

######  Remark 5.6.

That $\Phi_{U}$ is an isomorphism has the following interesting corollary:
since $U\simeq\mathcal{M}_{0,n+3}^{>0}$ is known to be isomorphic to (the
interior of) a polyhedron, the so-called $n$-th associahedron ${\rm Ass}_{n}$,
(77) provides a description of the positive grassmannian of 2-planes in
$\mathbf{R}^{n+3}$ as the product of (the interiors of) two polyhedra: one has

$G_{2}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\simeq{\rm
Ass}_{n}\times\Delta_{2}^{n+3}\,.$

As far as we are aware of, such a polyhedral description of
$G_{2}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$ is not mentioned in the literature
yet, although it is quite likely that it is known by specialists.

From the second point of Proposition 5.4, one deduces that
$\widetilde{E}_{n}^{k}$ admits a unique smooth extension
$\overline{E}_{n}^{k}$ to $U\times\overline{\Delta}_{2}^{n+3}$, which is such
that

$\overline{E}_{n}^{k}=\big{(}\overline{\Phi}_{U}\big{)}^{*}\big{(}E_{n}^{k}\big{)}\,.$

Considering the boundary
$\partial{\Delta}_{2}^{n+3}=\overline{\Delta}_{2}^{n+3}\setminus{\Delta}_{2}^{n+3}$,
the natural inclusion
$\partial{\Delta}_{2}^{n+3}\subset\overline{\Delta}_{2}^{n+3}$ gives rise to
an injective morphism of bundles over $U$ denoted by
$\tau:U\times\partial{\Delta}_{2}^{n+3}\hookrightarrow
U\times\overline{\Delta}_{2}^{n+3}$. The map
$\delta^{\partial}=\delta\circ\tau$ is nothing else than the projection map
$U\times\partial{\Delta}_{2}^{n+3}\rightarrow U$. Then, from the Stokes-type
formula for integration along fibers with boundary, we get that the following
identity (which is an equality between differential $(n-1)$-forms on $U$) is
satisfied:

(81)
$(-1)^{n+2}d\mathscr{E}_{n}^{n}=\big{(}\delta^{\partial}\big{)}_{*}\Big{(}\tau^{*}(\mathscr{E}_{n}^{n}\big{)}\Big{)}\,.$

Let us now explain how this relation gives rise to an AR for the web
$\boldsymbol{\mathcal{W}}_{0,n+3}$ on $\mathcal{M}_{0,n+3}^{>0}\simeq U$.
First, it follows from Proposition 5.3.1 that the RHS in (81) is 0. Secondly,
for $i=1,\ldots,n+3$ and $\upsilon=0,1$, let
$d_{i}^{\upsilon}:\mathbf{R}^{n+2}\hookrightarrow\mathbf{R}^{n+3}$ be the
affine map associating $(t_{1},\ldots,t_{i-1},,\upsilon,t_{i},\ldots,t_{n+2})$
to any $(n+2)$-tuple $(t_{s})_{s=1}^{n+2}$ (`$\upsilon$ is inserted at the
$i$-th position'). Then the restriction of this map to

$\overline{\Delta}_{2-\upsilon}^{n+2}=\left\\{\,(s_{k})_{k=1}^{n+2}\in[0,1]^{n+2}\,\big{\lvert}\,\sum_{k}s_{k}=2-\upsilon\,\right\\}$

induces a linear inclusion whose image, denoted by
$\overline{\Delta}_{2-\upsilon}^{n+2}(i)$, is the facet of
$\overline{\Delta}_{2}^{n+3}$ obtained by intersecting it with the affine
hyperplane cut out by $x_{i}=\upsilon$: one has

$\overline{\Delta}_{2-\upsilon}^{n+2}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\overline{\Delta}_{2-\upsilon}^{n+2}(i)=d_{i}^{\upsilon}\Big{(}\overline{\Delta}_{2-\upsilon}^{n+2}\Big{)}=\overline{\Delta}_{2}^{n+3}\cap\big{\\{}\,x_{i}=\upsilon\,\big{\\}}\subset\overline{\Delta}_{2}^{n+3}\subset\mathbf{R}^{n+3}\,.$

The codimension 1 boundary of $\overline{\Delta}_{2}^{n+3}$ is exactly the
disjoint union of the interiors ${\Delta}_{2-\upsilon}^{n+2}(i)$ of the
(hyper)simplices $\overline{\Delta}_{2-\upsilon}^{n+2}(i)$ for
$i=1,\ldots,n+3$ and $\upsilon=0,1$.

An important fact from which a crucial property satisfied by
$\mathcal{E}_{n}^{n}$ will be deduced below is the following homological
relation

$\partial\bigg{[}\,\overline{\Delta}_{2}^{n+3}\bigg{]}=\sum_{i=1}^{n+3}(-1)^{i}\,\big{(}d_{i}^{1}\big{)}_{*}\bigg{[}\,\overline{\Delta}_{1}^{n+2}\bigg{]}-\sum_{i=1}^{n+3}(-1)^{i}\,\big{(}d_{i}^{0}\big{)}_{*}\bigg{[}\,\overline{\Delta}_{2}^{n+2}\bigg{]}$

which holds true in the relative homology group
$\boldsymbol{H}_{n+3}\big{(}\overline{\Delta}_{2}^{n+3},\partial\Delta_{2}^{n+3},\mathbf{Z}\big{)}$
(cf. [GM, Prop. 2.1.3]).

Since the facets $\overline{\Delta}_{1}^{n+2}(i)$ for $i=1,\ldots,n+3$ (all
isomorphic to the $(n+1)$-dimensional simplex $\Delta_{1}^{n+2}$) do not play
any role in building Euler's abelian relation, we will no longer consider them
in what follows and we will write $d_{i}=d_{i}^{0}$ from now on.

Combining the above homological relation with (81) and Proposition 5.3.1, we
obtain that

(82)
$0=\sum_{i=1}^{n+3}(-1)^{i}\big{(}\delta_{i}\big{)}_{*}\Big{(}\tau_{i}^{*}\big{(}\mathscr{E}_{n}^{n}\big{)}\Big{)}$

where $\tau_{i}:U\times{\Delta}_{2}^{n+2}(i)\subset
U\times\partial{\Delta}_{2}^{n+3}$ is the morphism of trivial bundles over $U$
induced by the inclusion
${\Delta}_{2}^{n+2}(i)\subset\partial{\Delta}_{2}^{n+3}$ and where
$\delta_{i}=\delta\circ\tau_{i}:U\times{\Delta}_{2}^{n+2}(i)\rightarrow U$ is
the natural projection.

In order of interpreting (82) as an abelian relation, one has now to express
each term
$\big{(}\delta_{i}\big{)}_{*}\big{(}\tau_{i}^{*}\big{(}\mathscr{E}_{n}^{n}\big{)}\big{)}$
in a different way. The key point to do this is Lemma 5.2.1, which says that
up to the natural identification of
$G_{2}^{or}\big{(}\mathbf{R}^{n+2}_{i}\big{)}$ with
$G_{2}^{or}\big{(}\mathbf{R}^{n+2}\big{)}$, the restriction of $E_{n}$ along
$G_{2}^{or}\big{(}\mathbf{R}^{n+2}_{i}\big{)}$ coincides with the invariant
representative $E_{n-1}$ of the Euler class of the tautological bundle on
$G_{2}^{or}\big{(}\mathbf{R}^{n+2}\big{)}$.

To build Euler's AR, we then consider the following diagram

(83)
$\textstyle{U\times\overline{\Delta}_{2}^{n+3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\Phi}_{U}}$$\textstyle{G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{\geq
0}}$$\textstyle{U\times\overline{\Delta}_{2}^{n+2}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{i}}$$\scriptstyle{\delta_{i}}$$\scriptstyle{\overline{\Phi}_{U}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{2}^{or}\big{(}\mathbf{R}^{n+2}_{i}\big{)}^{\geq
0}}$$\textstyle{G_{2}^{or}\big{(}\mathbf{R}^{n+2}\big{)}^{\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\mu_{i}}$$\scriptstyle{\sim}$$\textstyle{U^{\prime}\times\overline{\Delta}_{2}^{n+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{\hskip
11.38092pt\overline{\Phi}_{U^{\prime}}}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{i}}$$\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{\prime}}$
---

where

  * •

$U^{\prime}$ stands for the set of $(n-1)$-tuples
$u^{\prime}=(u^{\prime}_{k})_{k=1}^{n}\in\mathbf{R}^{n-1}$ such that
$1<u^{\prime}_{1}<\cdots<u^{\prime}_{n-1}$;

  * •

$\gamma^{\prime}$, $\pi^{\prime}$, $\overline{\Phi}_{U^{\prime}}$ and
$\delta^{\prime}$ correspond respectively to the maps (75), (76), (80) and
(78) above but in the case when the dimension has been taken to be $n-1$
instead of $n$;

  * •

the map $\mu_{i}$ is the restriction of the natural identification between
$G_{2}^{or}\big{(}\mathbf{R}^{n+2}\big{)}$ and
$G_{2}^{or}\big{(}\mathbf{R}^{n+2}_{i}\big{)}$ induced by the linear map
$d_{i}:\mathbf{R}^{n+2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{\rm
Im}(d_{i})=\mathbf{R}^{n+2}_{i}\subset\mathbf{R}^{n+3}$;

  * •

the maps $\psi_{i}:U\rightarrow U^{\prime}$ are given by

$\psi_{1}(u)=\left(\,\frac{u_{1}(u_{j}-1)}{u_{j}(u_{1}-1)}\,\right)_{j=2}^{n}\,,\qquad\psi_{2}(u)=\left(\,\frac{u_{j}-1}{u_{1}-1}\,\right)_{j=2}^{n}\,,\qquad\psi_{3}(u)=\left(\,\frac{u_{j}}{u_{1}}\,\right)_{j=2}^{n}$

and $\psi_{3+i}(u)=\big{(}u_{1},\ldots,\widehat{u_{i}},\ldots,u_{n}\big{)}$
for $i=1,\ldots,n$.

  * •

for $i=1,\ldots,n+3$ and $u^{\prime}\in U^{\prime}$, $\gamma_{i}(u^{\prime})$
is the oriented 2-plane in $\mathbf{R}^{n+2}_{i}\subset\mathbf{R}^{n+3}$
associated to the $2\times(n+3)$ matrix obtained from $M_{u^{\prime}}\in{\rm
Mat}_{2,n+2}(\mathbf{R})$ (see (74)) by inserting in it the zero column at the
$i$-th place.

By straightforward computations (left to the reader), it can be verified that
(83) is commutative. Combining this with the fact recalled in the paragraph
just before this diagram, one deduces the

######  Proposition 5.7.

  1. 1. One has $\big{(}\delta_{i}\big{)}_{*}\big{(}\tau_{i}^{*}\big{(}\mathscr{E}_{n}^{n}\big{)}\big{)}=\psi_{i}^{*}\big{(}\mathscr{E}_{n-1}^{n}\big{)}$ for $i=1,\ldots,n+3$.

  2. 2.

Consequently (82) can be written
$0=\sum_{i=1}^{n+3}(-1)^{i}\psi_{i}^{*}\big{(}\mathscr{E}_{n-1}^{n}\big{)}$.

  3. 3.

Since $\mathscr{E}_{n-1}^{n}$ never vanishes (according to Proposition 5.3.2)
it follows that

(84)
$\boldsymbol{\mathcal{E}}_{n}^{>0}=\bigg{(}\,(-1)^{i-1}\,\psi_{i}^{*}\left(\mathscr{E}_{n-1}^{n}\right)\,\bigg{)}_{i=1}^{n+3}$

is a complete hence in particular non-trivial abelian relation for
$\boldsymbol{\mathcal{W}}^{\,>0}_{0,n+3}$.

The construction above gives indeed an AR on
$U\simeq\mathcal{M}_{0,n+3}^{\,>0}$ but several questions about it remain to
be answered:

(85) $i.$ Does the construction above of $\boldsymbol{\mathcal{E}}_{n}^{>0}$
depend on some choices? If yes, on which ones
---
and to what extent?
$ii.$ Following a completely similar approach, one can construct an Eulerian
abelian re-
lation $\boldsymbol{\mathcal{E}}_{n}(\boldsymbol{\sigma})$ for
$\boldsymbol{\mathcal{W}}_{0,n+3}$ on the component
$\mathcal{M}(\boldsymbol{\sigma})$ of $\mathcal{M}_{0,n+3}(\mathbf{R})$ for
any $\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}$ (cf.
§2.1.5). How are the $\boldsymbol{\mathcal{E}}_{n}(\boldsymbol{\sigma})$'s
related, especially with respect to the birational action
of $\mathfrak{S}_{n+3}$ on the real moduli space
$\mathcal{M}_{0,n+3}(\mathbf{R})$?

We will come back to these important points further on but we will first
discuss how to make $\boldsymbol{\mathcal{E}}_{n}^{>0}$ explicit by giving
formulas for its components. Since all are pull-backs of the form
$\mathcal{E}_{n-1}^{n}$ on $U^{\prime}\simeq\mathcal{M}_{0,n+2}^{\,>0}$, this
amounts to studying the function $e_{n-1}$ defined in Proposition 5.3.2.

####  5.1.4. An integral formula for $e_{n}$

A multivariable integration scheme for computing $e_{n-1}$ has already been
described by Damiano in [Da2, §6] (more details are given in [Da1, §6]). But
it is not quite explicit and the author was able to use it only in the case
when $n=2$, recovering the well-known Rogers' dilogarithm. Our goal below is
to go a bit further by describing a slight simplification of Damiano's
integration scheme, which gives rise to a more explicit integral formula for
$e_{n-1}$ that we will use to get effective informations about $e_{n-1}$ for
higher but still small $n$ (namely $n=2,3$ and $4$).

In what follows, $m$ is an integer bigger than or equal to $1$. We do not make
any assumption about the parity of $m$. The integer $m$ has to be thought of
as $n$ shifted by 1, namely $m=n-1$, this just in order to simplify the
writing. We use below the same notation as above, but in which $n$ has been
replaced by $m$. For instance, $U$ here stands for the set of $m$-tuples
$u=(u_{1},\ldots,u_{m})\in\mathbf{R}^{m}$ such that $1<u_{1}<\cdots
u_{m-1}<u_{m}$, etc.

Our main goal here is to give a nice closed formula for the function
$e_{m}(u)$ defined by equality (79). The main facts we use for this purpose
are first that $E_{m}^{m+1}$ coincides (up to sign) with an invariant volume
form on $G_{2}^{or}(\mathbf{R}^{m+3})$ (cf. Lemma 5.2.2) and secondly that
there is a simple formula for such a volume form when working with a natural
parametrization.

We use the notation introduced circa (68). For $M\in\Omega_{2}$ (that is, $M$
is a $2\times(m+3)$-matrix of rank 2), we write $M=[M_{12},N]$ where $M_{12}$
is the square $2\times 2$ matrix obtained by considering the first 2 columns
of $M$ while $N$ is the one formed by the $n+1$ others. Then
$\Omega_{2}^{\prime}=\\{\,M\,\lvert\,{\rm det}(M_{12})>0\\}$ is an open subset
of $\Omega_{2}$ and the map $A:\Omega_{2}^{\prime}\rightarrow
M_{2,m+1}(\mathbf{R}),\,M\mapsto M_{12}^{-1}\cdot N$ is a surjective ${\rm
GL}_{2}^{>0}(\mathbf{R})$-fibration. The injection
$B:{M}_{2,m+1}(\mathbf{R})\hookrightarrow\Omega_{2}^{\prime},\,N\mapsto\big{[}{\rm
Id}_{2},N\big{]}$ is such that $A\circ B$ is the identity. Denoting again by
$\xi$ the restriction of the map (68) to $\Omega_{2}^{\prime}$, we define a
map $\Xi:{M}_{2,m+1}(\mathbf{R})\rightarrow G_{2}^{or}(\mathbf{R}^{m+3})$ by
requiring that the following diagram commutes:

(86)
$\textstyle{\Omega_{2}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\scriptstyle{A}$$\textstyle{G_{2}^{or}\big{(}\mathbf{R}^{m+3}\big{)}}$$\textstyle{{M}_{2,m+1}(\mathbf{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\Xi}$$\scriptstyle{B}$
---

More prosaically, $\Xi$ is noting else but the map that associates to the
$2\times(m+1)$ matrix whose two lines correspond to the vectors
$(x_{s})_{s=0}^{m},(y_{s})_{s=0}^{m}\in\mathbf{R}^{m+1}$, the oriented 2-plane
of $\mathbf{R}^{m+3}$ directly spanned by $(1,0,x_{0},\ldots,x_{m})$ and
$(0,1,y_{0},\ldots,y_{m})$, taken in this order. It is well-known that $\Xi$
induces a coordinates chart on $G_{2}^{or}\big{(}\mathbf{R}^{m+3}\big{)}$, the
corresponding coordinates being the $x_{s}$ and $y_{s}$ for $s=0,\ldots,m$. We
will denote by ${\rm dVol}(N)$ the standard Euclidean volume in these
coordinates, i.e.

${\rm dVol}(N)=\big{(}dx_{0}\wedge dy_{0}\big{)}\wedge\big{(}dx_{1}\wedge
dy_{1}\big{)}\cdots\wedge\big{(}dx_{m}\wedge dy_{m}\big{)}\,.$

The formula in the coordinates $x_{s},y_{s}$ for the pull-back under $\Xi$ of
an invariant volume form on $G_{2}^{or}\big{(}\mathbf{R}^{m+3}\big{)}$ is well
known:363636This formula is a particular case of a more general one holding
true for some affine parametrizations (in terms of Jordan structures) of a
vast class of homogeneous varieties, see Proposition X.6.3 in [Ber]. We have
not been able to locate a classical reference for this formula in the case of
real grassmannians.

######  Lemma 5.8.

There exists a non zero constant $C_{m}$ such that373737One has $C_{m}=(m+1)!$
but knowing the exact value of this constant is actually not relevant for our
purpose hence we will not give a proof of this.

(87) $\Xi^{*}\Big{(}E_{m}^{m+1}\Big{)}=C_{m}\cdot\frac{{\rm dVol}(N)}{{\rm
det}\Big{(}\,{\rm Id}_{2}+N\,{}^{t}N\,\Big{)}^{\frac{m+3}{2}}}\,.$

It is easy to get a nice integral formula for $e_{m}$ from (87). We list below
the facts and notation we need to establish the sought-after formula:

  * •

one denotes by $k=(k_{1},k_{2})$ (resp. by $h=(h_{1},\ldots,h_{m})$) an
element of $\big{(}\mathbf{R}_{>0}\big{)}^{2}$ (resp. of
$\big{(}\mathbf{R}_{>0}\big{)}^{m}$). By means of the map
$(k,h)\mapsto(k_{1},k_{2},1,h_{1},\ldots,h_{m})$, we naturally identify
$\big{(}\mathbf{R}_{>0}\big{)}^{2}\times\big{(}\mathbf{R}_{>0}\big{)}^{m}=\big{(}\mathbf{R}_{>0}\big{)}^{2+m}$
with the open subset of $(m+3)$-tuples in $\mathbf{R}^{m+3}$ with positive
coordinates and such that the third coordinate is 1;383838The fact that we
have chosen to normalize the third coordinate is not important at all. We
could have decided to normalize any other coordinate with essentially no
change.

  * •

we will consider the following map which is an isomorphism:

(88)
$\Delta_{2}^{m+3}\longrightarrow\big{(}\mathbf{R}_{>0}\big{)}^{2+m},\,\big{(}t_{i}\big{)}_{i=1}^{m+3}\longmapsto\left(\frac{t_{s}(t_{3}-1)}{t_{3}(t_{s}-1)}\right)_{s=1}^{m+3}\,;$

  * •

we set $dk=dk_{1}\wedge dk_{2}$, $dh=dh_{1}\wedge\cdots\wedge dh_{m}$ and
$du=du_{1}\wedge\cdots\wedge du_{m}$;

  * •

the notation $k,h>0$ will be used to mean that the pair $(k,h)$ varies in
$\big{(}\mathbf{R}_{>0}\big{)}^{2+m}$;

  * •

the following map is an isomorphism onto its image which is an open subset in
$\Omega_{2}$:

(89) $\displaystyle F:\,U\times\big{(}\mathbf{R}_{>0}\big{)}^{2+m}$
$\displaystyle\longrightarrow M_{2,n+3}(\mathbf{R})$
$\displaystyle\big{(}u,k,{h})$
$\displaystyle\longmapsto\begin{bmatrix}\sqrt{k_{1}}&0&-1&-u_{1}\sqrt{h_{1}}&\cdots&-u_{m}\sqrt{h_{m}}\,\\\
0&\sqrt{k_{2}}&1&\sqrt{h_{1}}&\cdots&\sqrt{h_{m}}\end{bmatrix}$

  * •

$X$ and $Y$ stand for the line vectors of the matrix $F(u,k,h)$; one has:

$\displaystyle X=$
$\displaystyle\,\Big{(}\sqrt{k_{1}},0,-1,-u_{1}\sqrt{h_{1}},\ldots,-u_{m}\sqrt{h_{m}}\,\Big{)}$
$\displaystyle\mbox{ and }\quad Y=$
$\displaystyle\,\Big{(}\,0,\sqrt{k_{2}},1,\sqrt{h_{1}},\ldots,\sqrt{h_{m}}\,\Big{)}\,;$

  * •

for $(u,k,h)\in U\times\big{(}\mathbf{R}_{>0}\big{)}^{2+m}$, we denote by
$S_{u,k,h}$ the matrix $F(u,k,h)\cdot F(u,k,h)^{t}$;

  * •

one has

$S_{u,k,h}=\begin{bmatrix}X^{2}&XY\\\ XY&Y^{2}\end{bmatrix}$

with

(90) $X^{2}=1+k_{1}+\sum_{i=1}^{m}h_{i}\,{u_{i}}^{2},\qquad
XY=-1-\sum_{i=1}^{m}h_{i}\,u_{i}\qquad\mbox{ and }\qquad
Y^{2}=1+k_{2}+\sum_{i=1}^{m}h_{i}\,.$

######  Proposition 5.9.

  1. 1.

Up to the isomorphism (88) between $\Delta_{2}^{m+3}$ and
$(\mathbf{R}_{>0})^{2+m}$, the map

$\xi\circ F:U\times(\mathbf{R}_{>0})^{2+m}\rightarrow
G_{2}^{or}(\mathbf{R}^{m+3})$

identifies with the parametrization (77) of
$G_{2}^{or}(\mathbf{R}^{m+3})^{>0}$.

  2. 2.

In particular, the projection $(u,k,h)\rightarrow u$ corresponds to
quotienting by the torus action.

  3. 3.

There exists a non zero scalar $\lambda_{m}$ such that the pull-back of
$E_{m}^{m+1}$ under $\xi\circ F$ is given by

$\big{(}\xi\circ F\big{)}^{*}\Big{(}E_{m}^{m+1}\Big{)}=\lambda_{m}\,{\rm
det}\left(S_{u,k,h}\right)^{-\frac{m+3}{2}}\,{dk\wedge dh\wedge du}\,.$

  4. 4.

Consequently, up to multiplication by a non zero constant, one has

(91) $e_{m}(u)=\int_{k,h>0}{\Big{(}\,X^{2}\cdot
Y^{2}-(XY)^{2}\,\Big{)}^{-\frac{m+3}{2}}}{dk\wedge dh}$

for any $u\in U$, where $X^{2},Y^{2}$ and $XY$ are the expressions given in
(90) above.

###### Proof.

Both the two first points are easy to establish (this is left to the reader).
Since 4. follows immediatly from 3., we will only discuss the latter.

For $(u,k,h)\in U\times(\mathbf{R}_{>0})^{2+m}$, we denote by
$\widetilde{F}(u,k,h)$ the $2\times(m+1)$ submatrix of $F(u,k,h)$ given by its
last $m+1$ columns and we consider the map

$\displaystyle f:U\times(\mathbf{R}_{>0})^{2+m}$ $\displaystyle\rightarrow
M_{2,m+1}(\mathbf{R})$ $\displaystyle(u,k,h)$
$\displaystyle\longmapsto\Big{[}{\begin{tabular}[]{cc}$1/\sqrt{k_{1}}$&\\\
0&$1/\sqrt{k_{2}}$\end{tabular}}\Big{]}\cdot\widetilde{F}(u,k,h)=\begin{bmatrix}{-1}/{\sqrt{k_{1}}}&-u_{1}\sqrt{h_{1}/k_{1}}&\cdots&-u_{m}\sqrt{h_{m}/k_{1}}\\\
\hskip 8.5359pt{1}/{\sqrt{k_{2}}}&\hskip
8.5359pt\sqrt{h_{1}/k_{2}}&\cdots&\hskip
8.5359pt\sqrt{h_{m}/k_{2}}\end{bmatrix}\,.$

The two following identities hold true:

  * •

$f^{*}\Big{(}{\rm det}\big{(}{I}_{2}+N\,{}^{t}N\big{)}\Big{)}={\rm
det}\big{(}S_{u,k,h}\big{)}/(k_{1}k_{2})$;

  * •

$f^{*}\big{(}{\rm
dVol}(N)\big{)}=\lambda^{\prime}_{m}\cdot(k_{1}k_{2})^{-(m+3)/2}dk\wedge
dh\wedge du$ for a certain non zero constant $\lambda^{\prime}_{m}$.

(The first identity is elementary and the second can be obtained by an easy
recurrence on $m$).

Since $\xi\circ F=\Xi\circ f$, point 3. follows from (87) combined with the
two preceding identities. ∎

######  Remark 5.10.

There is no point in justifying the convergence of the integral in (91): it
follows immediately from the fact that $E_{m}^{m+1}$ is a global smooth form
on $G_{2}^{or}(\mathbf{R}^{m+3})$ which is compact.

The interest of (91) lies in the fact that it is an explicit and closed
formula. This offers a way to study Euler's abelian relation more concretely.
We use this formula below to describe explicitly Gelfand-MacPherson's
computation leading to Rogers dilogarithm in the case $n=2$. Then we discuss
the case when $n=3$ and give the first terms of the Taylor series of $e_{2}$
that can be computed from (91).

##### Computation of $e_{1}$: Rogers dilogarithm

In this case, the preceding proposition gives us that for any
$u\in]1,+\infty[$, one has

$\displaystyle e_{1}(u)=$
$\displaystyle\,\int\int\int_{k_{1},k_{2},h_{1}=0}^{+\infty}\frac{dk_{1}dk_{2}dh_{1}}{\Big{(}\alpha_{0}-2\alpha_{1}u+\alpha_{2}u^{2}\Big{)}^{2}}$

with $\alpha_{0}=k_{1}k_{2}+h_{1}(k_{1}+1)+k_{2}$, $\alpha_{1}=h_{1}$ and
$\alpha_{2}=h_{1}(1+k_{2})$.

By successive direct computations, we get that

$\displaystyle e_{1}(u)=$
$\displaystyle\,\int\int_{k_{1},k_{2}=0}^{+\infty}\frac{dk_{1}dk_{2}}{\Big{(}k_{1}k_{2}+k_{1}+k_{2}\Big{)}\Big{(}\big{(}k_{2}+1\big{)}u^{2}-2u+(k_{1}+1)\Big{)}}$
$\displaystyle=$
$\displaystyle\,\int\int_{k_{1},k_{2}=1}^{+\infty}\frac{dk_{1}dk_{2}}{\Big{(}k_{1}k_{2}-1\Big{)}\Big{(}k_{2}u^{2}-2u+k_{1}\Big{)}}$
$\displaystyle=$
$\displaystyle\,\int_{k_{2}=1}^{+\infty}\frac{\log\Big{(}k_{2}/(k_{2}-1)\Big{)}+\log\Big{(}k_{2}u^{2}-2u+1\Big{)}}{\big{(}u\,k_{2}-1\big{)}^{2}}{dk_{2}}$
$\displaystyle=$ $\displaystyle\hskip
7.11317pt2\left(\frac{\log(u)}{u-1}-\frac{\log(u-1)}{u}\right)=4\,{\mathcal{R}}^{\prime}(u)$

with ${\mathcal{R}}(u)=R\left(\frac{u-1}{u}\right)$ for any $u>1$, where $R$
is Rogers dilogarithm (3).

It would be interesting to study (91) further and in particular to see whether
the explicit computation above can be generalized for $n>2$. Our few
preliminary attempts in this direction have not been successful and it might
be the case that this is something difficult. However, again by means of
explicit computations (performed on a computer algebra system), we have been
able to extract interesting information about $e_{n-1}$ for $n=3,4$. We
present some of the results we have obtained in the following two paragraphs.

##### On the Taylor expansion of $e_{2}$

We consider here the case $n=3$ which corresponds to the function $e_{2}$.
This function takes as arguments elements of $U\subset]1,+\infty[^{2}$, that
we will denote here by $(u_{2},u_{3})$ (hence $u_{2}$ and $u_{3}$ are such
that $1<u_{2}<u_{3}$).

For such a pair $(u_{2},u_{3})$, modulo an elementary and unimportant change
of the integration scheme, (91) reads in explicit form

(92) $e_{2}(u_{1},u_{2})=\int_{h_{2},\ldots,h_{5}>0}B(u,h)^{-5/2}dh$

with $dh=dh_{2}\wedge\ldots\wedge dh_{5}$ and where $B(u,h)$ is the following
polynomial of degree 2 in $u_{1}$ and $u_{2}$

$B(u,h)=\Big{(}\beta_{11}\,u_{1}^{2}-2h_{4}h_{5}u_{1}u_{2}+\beta_{22}\,u_{2}^{2}\Big{)}-2h_{3}\Big{(}h_{4}\,u_{1}+h_{5}\,u_{2}\Big{)}+\beta_{0}$

with $\beta_{11}=h_{4}(h_{2}+h_{3}+h_{5})$,
$\beta_{22}=h_{5}(h_{2}+h_{3}+h_{4})$ and
$\beta_{0}=(1+h_{3})(1+h_{2}+h_{4}+h_{5})-1$.

Even with the help of a symbolic integration software, we have not been able
to compute the quadruple multiple integral above for $u_{1},u_{2})$ arbitrary
in $U$. However, it is not the same if working with jets of finite order at a
suitable base-point in $U$, such as

$u^{*}=(2,3)\,.$

Let $\varepsilon_{2}(u,h)$ be the integrand in (92), which is obviously
analytic at $u^{*}$. We denote by
$\varepsilon_{2}(u,h)=\sum_{w=0}^{+\infty}\sum_{\lvert\ell\lvert=w}\varepsilon_{2}^{\ell}(h)(u-u^{*})^{\ell}$
its Taylor expansion at $u^{*}$, where we use the notation
$(u-u^{*})^{\ell}=(u_{1}-2)^{\ell_{1}}(u_{2}-3)^{\ell_{2}}$ for any
$\ell=(\ell_{1},\ell_{2})\in\mathbf{N}^{2}$. Setting $B(h)=B(u^{*},h)$, one
can verify that there are polynomials $P^{\ell}(h)$ for any $\ell$393939One
can give an explicit formula for $P^{\ell}(k,h)$ but since it is a bit
involved and not relevant for the discussion here we will not elaborate on
this. such that
$\varepsilon_{2}^{\ell}(h)={P^{\ell}(h)}/{B(h)^{\frac{5}{2}+w}}$ for any
$w\geq 0$ and any $\ell$ such that $\lvert\ell\lvert=\ell_{1}+\ell_{2}=w$.
With some work (left to the reader), one can justify the exchange of the
integration and the summation below

$e_{2}(u_{1},u_{2})=\int_{h>0}\Bigg{[}\sum_{\begin{subarray}{c}w\geq 0\\\
\lvert\ell\lvert=w\end{subarray}}\varepsilon_{2}^{\ell}(h)(u-u^{*})^{\ell}\Bigg{]}\,dh=\sum_{\begin{subarray}{c}w\geq
0\\\
\lvert\ell\lvert=w\end{subarray}}\Bigg{[}\int_{h>0}\varepsilon_{2}^{\ell}(h)dh\Bigg{]}\,(u-u^{*})^{\ell}\,.$

The interest of doing this lies in the fact that we have been able to compute
in closed form the integrals $\int_{h>0}\varepsilon_{2}^{\ell}(h)\,dh$, at
least for the pairs $\ell$ of weight $\lvert\ell\lvert$ small enough and we
have been able to get that way the first terms of the Taylor expansion of
$e_{2}$ at $u^{*}$:

$\displaystyle\frac{4}{\pi}\,e_{2}(u)=\frac{2}{9}-$
$\displaystyle\left(\frac{\mathit{(u_{1}-2)}}{9}+\frac{5\,(\mathit{u_{2}}-3)}{27}\right)+\frac{(\mathit{u_{1}}-2)^{2}}{18}+\frac{5\,(\mathit{u_{1}}-2)(\mathit{u_{2}}-3)}{54}+\frac{19\,(\mathit{u_{2}}-3)^{2}}{162}$
$\displaystyle-$
$\displaystyle\left(\frac{(\mathit{u_{1}}-2)^{3}}{36}+\frac{5\,(\mathit{u_{1}}-2)^{2}(\mathit{u_{2}}-3)}{108}+\frac{19\,(\mathit{u_{1}}-2)(\mathit{u_{2}}-3)^{2}}{324}+\frac{65\,(\mathit{u_{2}}-3)^{3}}{972}\right)$
$\displaystyle+$
$\displaystyle\,\frac{(\mathit{u_{1}}-2)^{4}}{72}+\frac{5\,(\mathit{u_{1}}-2)^{3}(\mathit{u_{2}}-3)}{216}+\frac{19\,(\mathit{u_{1}}-2)^{2}(\mathit{u_{2}}-3)^{2}}{648}+\frac{65\,(\mathit{u_{1}}-2)(\mathit{u_{2}}-3)^{3}}{1944}$
$\displaystyle+$
$\displaystyle\,\frac{211\,(\mathit{u_{2}}-3)^{4}}{5832}+O\Big{(}\big{(}u-u^{*}\big{)}^{5}\Big{)}\,.$

We observe that, up to the fourth order, the RHS of this equality coincides
with the Taylor series at $u^{*}$ of

$\frac{8}{3\,u_{1}u_{2}(u_{2}-1)}=\frac{2}{9}\sum_{w=0}^{+\infty}\sum_{l=0}^{w}(-1)^{w}\frac{\big{(}3^{l+1}-2^{l+1}\big{)}}{2^{w}\,3^{l}}\big{(}u_{1}-2\big{)}^{w-l}\big{(}u_{2}-3\big{)}^{l}\,.$

This coincidence could leads us to think that we have identically

(93) $e_{2}(u_{1},u_{2})=\frac{(2\pi/3)}{u_{1}u_{2}(u_{2}-1)}$

and we will see a bit further that this is indeed the case.

####  5.1.5. Invariance properties of the function $e_{n}$: some
transformation formulas

The positive grassmannian $G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$ admits dihedral
symmetries. Our goal here is to explain how one can deduce from them some
quite strong invariance properties for the function $e_{n-1}$, this for any
$n\geq 2$.

Euler's abelian relation $\boldsymbol{\mathcal{E}}_{n}^{>0}$ (see (84)) is
defined on the positive part $\mathcal{M}_{0,n+3}^{\,>0}$. As explained in
§2.1.5, this space admits a dihedral group of symmetries and a natural
question is about the behavior of $\boldsymbol{\mathcal{E}}_{n}^{>0}$ with
respect to them and what can be deduced from that for $e_{n-1}$.

The subgroup $D_{0,n+3}$ of $\mathfrak{S}_{n+3}$ letting
$\mathcal{M}_{0,n+3}^{\,>0}$ invariant is generated by the maps $C$ and $R$,
respectively induced by

  * •

the shift
$\tilde{C}:(\mathbf{P}^{1})^{n+3}\rightarrow(\mathbf{P}^{1})^{n+3},\,(z_{i})_{i=1}^{n+3}\mapsto(z_{n+3},z_{1},\ldots,z_{n+2})$
which is of order $n+3$;

  * •

the involutive transformation
$\tilde{R}:(\mathbf{P}^{1})^{n+3}\rightarrow(\mathbf{P}^{1})^{n+3},\,(z_{i})_{i=1}^{n+3}\mapsto(z_{1},z_{n+3},z_{n+2},\ldots,z_{3},z_{2})$
which, as an automorphism of a regular $(n+3)$-gon, corresponds to the
reflection with respect to the dihedral axis joining the center on the
$(n+3)$-gon to that of its vertices labeled by 1.

We set $n^{\prime}=\lceil(n+3)/2\rceil$. The two permutations of
$\\{1,\dots,n+3\\}$ corresponding to $C$ and $R$ are the $(n+3)$-cycle
$c=(1\ldots n+3)$ and the product of $n^{\prime}-1$ transpositions given by
$r=(2,n+3)(3,n+2)\cdots(n^{\prime},n^{\prime}+1)$ when $n$ is even and
$r=(2,n+3)(3,n+2)\cdots(n^{\prime},n^{\prime}+2)$ otherwise (see Figure 3
below). In particular, if ${\bf sgn}:\mathfrak{S}_{n+3}\rightarrow\\{\pm
1\,\\}$ denotes the signature, then one has

(94) ${\bf sgn}(c)=(-1)^{n}\qquad\mbox{ and }\qquad{\bf
sgn}(c)=(-1)^{n^{\prime}-1}\,.$

Figure 3. Axial symmetry of the $(n+3)$-gon when $n$ is even (left) and odd
(right).

Warning: The transformation denoted by $C$ here corresponds to the inverse of
the one unfortunately denoted the same in §4.1 which moreover is expressed
there with respect to another coordinates system on $\mathcal{M}_{0,n+3}$ than
the one used here.404040How the $x_{i}$’s of §4.1 and the $u_{k}$’s of this
section are related can be summarized by the equality
$[0,1,\infty,x_{1},\ldots,x_{n}]=[\infty,0,-1,-u_{1},\ldots,-u_{n}]$ (in
$\mathcal{M}_{0,n+3}$) which has to be thought of as identically satisfied. We
apologize to the reader for this inconvenience.

The map (72) gives rise to an isomorphism
$U\rightarrow\mathcal{M}_{0,n+3}^{\,>0}$ again denoted by $\varphi$. It is
straightforward (and left to the reader) to express $C$ and $R$ in the affine
coordinates $u_{1},\ldots,u_{n}$ on $U\subset]1,+\infty[^{n}$ explicitly.
Denoting (a bit abusively) by the same notation the automorphisms of $U$
corresponding to $R$ and $C$ (given by conjugating them by $\varphi$) and
agreeing that $u_{0}=1$ and $u_{-1}=0$, one has

$C(u)=\left(\,\frac{u_{n}}{u_{n}-u_{i-1}}\,\right)_{i=1}^{n}\qquad\quad\mbox{
and }\qquad\quad
R(u)=\left(\,\frac{u_{n}-u_{n-i-1}}{u_{n}-u_{n-1}}\,\right)_{i=1}^{n}\,.$

Actually, for our purpose here, we do not have to only consider $C$ and $R$,
which are automorphisms of $\mathcal{M}_{0,n+3}^{\,>0}$, but some lifts of
these to the oriented grassmannian $G_{2}^{or}({\mathbf{R}}^{n+3})$ as well,
and there is a subtlety about this. This subtle point has not be considered in
[Da2] although it is quite relevant regarding the invariant properties of
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ with respect to the action of
$\mathfrak{S}_{n+3}$.

Actually the lifts to the oriented grassmannian of automorphisms of
$\mathcal{M}_{0,n+3}$ considered by Damiano (either in [Da2] or in [Da1]) are
the simplest/most naive ones: given a permutation
$\sigma\in\mathfrak{S}_{n+3}\simeq{\rm Aut}(\mathcal{M}_{0,n+3})$, the lift
over it that Damiano considers is the automorphism $\widetilde{\sigma}\in{\rm
Aut}\big{(}G_{2}^{or}({\mathbf{R}}^{n+3})\big{)}$ induced by the natural
linear action of $\sigma$ on $\mathbf{R}^{n+3}$, namely

$\sigma\cdot
x=\sigma\cdot\big{(}x_{i}\big{)}_{i=1}^{n+3}=\big{(}x_{\sigma(i)}\big{)}_{i=1}^{n+3}\,.$

The nice feature of the naive lifting map $\sigma\mapsto\tilde{\sigma}$
considered by Damiano is that it gives rise to a group monomorphism
$\mathfrak{S}_{n+3}={\rm Aut}\big{(}\mathcal{M}_{0,n+3}\big{)}\rightarrow{\rm
Aut}\big{(}G_{2}^{or}(\mathbf{R}^{n+3})\big{)}$. But the problem with it is
clearly shown when considering the case of $C$: using the notations (74) and
(75), the associated permutation $c=(1\ldots n+3)$ is such that Damiano's
naive lift $\tilde{c}$ satisfies

$\tilde{c}\cdot\gamma(u)=\xi\Big{(}\,\tilde{c}\cdot
M_{u}\,\Big{)}=\xi\left(\,\begin{bmatrix}-u_{n}&1&0&-1&-u_{1}&\cdots&-u_{n-1}\\\
1&0&1&\,1&1&\cdots&1\end{bmatrix}\,\right)$

for any $u\in U$. Clearly, some of the $2\times 2$ minors of $\tilde{c}\cdot
M_{u}$ are positive, whereas others (namely all those involving the first
column of $\tilde{c}\cdot M_{u}$) are negative. This has the consequence that
the matrix $\tilde{c}\cdot M_{u}$ is not merely positive, which translates
into the fact that with Damiano's choice for the lift $\tilde{c}$, the image
of $\tilde{c}\cdot\gamma$ into the oriented grassmannian of 2-planes in
$\mathbf{R}^{n+3}$ does not coincide with the positive part
$G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$. Actually, the image
$\tilde{c}\cdot\gamma(U)$, which is $\tilde{c}\cdot
G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$, is disjoint from the positive part of the
oriented grassmannian: one has

$\tilde{c}\left(G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\,\right)\cap
G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}=\emptyset\,.$

In order to study the invariant properties of
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ with respect to the action of $C$, one has
to look for a lift $\hat{c}$ making the following diagram commutative:

(95)
$\textstyle{G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{c}}$$\textstyle{G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{C}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$
---

In particular, the lift $\hat{c}$ must let invariant the positive
grassmannian, which precisely Damiano's lift $\tilde{c}$ does not do.

##### Invariance properties of Euler's abelian relation $\mathcal{E}_{n}^{>0}$
with respect to pull-back under $C$.

The relevant lift of $c$ to deal when considering the notion of positivity on
the oriented grassmannian is well-known: it is the cyclic automorphism
$\hat{c}$ of $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}$ induced by the linear
isomorphism of $\mathbf{R}^{n+3}$ given by

(96)
$\big{(}x_{1},\ldots,x_{n+3}\big{)}\mapsto\big{(}-x_{n+3},x_{1},\ldots,x_{n+2}\,\big{)}\,$

whose matrix (written by blocs) with respect to the standard basis of
$\mathbf{R}^{n+3}$ is

$\begin{bmatrix}0&{\rm I}_{n+2}\\\ -1&0\end{bmatrix}\,.$

This matrix belongs to ${\rm O}_{n+3}(\mathbf{R})$ and has determinant equal
to $(-1)^{n-1}$. Hence $\hat{c}$ is not always in the special orthogonal group
but it is easily verified (using (66) for instance) that one has
$\hat{c}^{*}(E_{n})=E_{n}$ in any case. It follows that $E_{n}^{n}$ is left
invariant by the lift $\hat{c}$ as well. Consequently, it follows that the
pull-back of Euler's abelian relation $\boldsymbol{\mathcal{E}}_{n}^{>0}$
under $C$ coincides with $\boldsymbol{\mathcal{E}}_{n}^{>0}$ or with its
opposite, according whether $\hat{c}$ preserves the orientation of a generic
$H_{0}$-orbit in $G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$.414141We recall that
$H_{0}\simeq(\mathbf{R}_{>0})^{n+2}$ stands for the positive part of the
diagonal Cartan torus $H\subset{\rm SL}(\mathbf{R}^{n+3})$.

In order to determine the action of $\hat{c}$ on the orientation of the
$H_{0}$-orbits of elements in $G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$, let us
consider the following parametrization of the positive part of this oriented
grassmannian

(97) $\displaystyle
U\times(\mathbf{R}_{>0}\big{)}^{2}\times(\mathbf{R}_{>0}\big{)}^{n}$
$\displaystyle\longrightarrow G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$
$\displaystyle(u,k,h)=\Big{(}\,(u_{i})_{i=1}^{n},\big{(}k_{1},k_{2}\big{)},\big{(}h_{i}\big{)}_{i=1}^{n}\,\Big{)}$
$\displaystyle\longmapsto\begin{bmatrix}k_{1}&0&-1&-u_{1}h_{1}&\ldots&-u_{n}h_{n}\\\
0&k_{2}&\hskip 4.55254pt1&\hskip 4.55254pth_{1}&\ldots&\hskip
4.55254pth_{n}\end{bmatrix}\,$

(where $U$ is the set of $n$-tuples $(u_{i})_{i=1}^{n}\in\mathbf{R}^{n}$ such
that $1<u_{1}<\ldots u_{n-1}<u_{n}$). This map is an isomorphism hence the
$u_{i}$, $k_{s}$ and $h_{i}$ for $i=1,\ldots,n$ and $s=1,2$ form a system of
global coordinates on $G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$, which is
particularly useful for our purpose since relatively to these coordinates,
quotienting by the $H_{0}$-action corresponds exactly to the standard
projection $(u,k,h)\mapsto u$ (cf. Proposition 5.9 above). Let $\hat{C}$ be
the automorphism of
$U\times(\mathbf{R}_{>0}\big{)}^{2}\times(\mathbf{R}_{>0}\big{)}^{n}$ obtained
from the automorphism $\hat{c}$ of $G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$ by
means of the identification induced by (97). It is not difficult to give an
explicit formula for $\hat{C}$. Indeed setting $u_{0}=1$ and denoting by
$c_{i}(u)=u_{n}/(u_{n}-u_{i-1})$ the $i$-th component of $C(u)$ for
$i=1,\ldots,n$, straightforward computations (the details of which are left to
the reader) give us that

$\hat{C}(u,k,h)=\Big{(}\hat{u},\hat{k},\hat{h}\Big{)}$

with

$\displaystyle\hat{u}=$
$\displaystyle\,C(u)=\big{(}c_{i}(u)\big{)}_{i=1}^{n}=\left(\frac{u_{n}}{u_{n}-u_{i-1}}\right)_{i=1}^{n}\,,$
$\displaystyle\hat{k}=$
$\displaystyle\,\Big{(}\,\hat{k}_{1}\,,\,\hat{k}_{2}\,\Big{)}=\left(\frac{h_{n}}{k_{2}},\frac{k_{1}}{k_{2}u_{n}}\right)$
$\displaystyle\mbox{and }\quad\hat{h}=$
$\displaystyle\,\Big{(}\,\hat{h}_{i}\,\Big{)}_{i=1}^{n}=\left(\,\frac{1}{k_{2}\,c_{1}(u)}\,,\,\frac{h_{1}}{k_{2}\,c_{2}(u)}\,,\,\ldots\,,\,\frac{h_{n-1}}{k_{2}\,c_{n}(u)}\,\right)\,.$

The implicitly chosen (because natural) orientation on the $H_{0}$-orbits of
elements of $G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$ corresponds to the
$(n+2)$-form $dk\wedge dh=dk_{1}\wedge dk_{2}\wedge dh_{1}\wedge\cdots\wedge
dh_{n}$ in the coordinates $u_{i},k_{s}$ and $h_{i}$ ($i=1,\ldots,n$,
$s=1,2$). Using the expressions above for the components of $\hat{k}$ and
$\hat{h}$, one easily computes what is the $dk\wedge dh$-component of the
pull-back of this form under $\hat{C}$: setting $\partial k\wedge\partial
h=\partial_{k_{1}}\wedge\partial_{k_{2}}\wedge\partial_{h_{1}}\wedge\cdots\wedge\partial_{h_{n}}$,
one obtains that

$\big{(}\partial k\wedge\partial
h\big{)}\,{\lrcorner}\,\hat{C}^{*}\Big{(}\,dk\wedge
dh\,\Big{)}=(-1)^{n}\frac{\prod_{i=0}^{n-1}(u_{n}-u_{i})}{k_{2}^{n+3}u_{n}^{n+1}}\,.$

Since $u\in U$, one has $u_{n}-u_{i}>0$ for any $i=0,\ldots,n-1$ hence the
sign of the above expression is $(-1)^{n}$. On the other hand, $\hat{c}$ lets
$E_{n}^{n}$ invariant as we have seen above, hence we deduce the

######  Proposition 5.11.

The positive Euler's abelian relation is stable up to multiplication by
$(-1)^{n}$ under pull-back by $C$, i.e. identically on $U$, one has

$C^{*}\Big{(}\boldsymbol{\mathcal{E}}_{n}^{>0}\Big{)}=(-1)^{n}\,\boldsymbol{\mathcal{E}}_{n}^{>0}\,.$

When $n$ is even, we will state a more general result whose proof is much more
conceptual and which admits the above as a particular case (see Proposition
5.18 further below).

In any case, the preceding proposition has two interesting outcomes. The first
is that one recovers the fact that $\boldsymbol{\mathcal{E}}_{n}^{>0}$ is
entirely known as soon as one, say the first, of its components is known.
Indeed, the map $U_{n+3}(u)=(u_{1},\ldots,u_{n-1})$ is a first integral of the
$(n+3)$-th foliation of $\boldsymbol{\mathcal{W}}_{0,n+3}^{\,>0}$ on $U$ (that
foliation corresponding to forgetting the $(n+3)$-th point on
${\mathcal{M}}_{0,n+3}^{\,>0}$.

For $i=0,\ldots,n+2$, the rational map

$U_{n+3-i}=\big{(}C^{\circ i}\big{)}^{*}\big{(}U_{n+3}\big{)}$

(which is just given by taking the first $n-1$ components of the map obtained
by composing $C$ $i$-th times) is a rational first integral of the
$(n+3-i)$-th foliation of $\boldsymbol{\mathcal{W}}_{0,n+3}^{\,>0}$
(corresponding to forgetting the $(n+3-i)$-th point on
$\mathcal{M}_{0,n+3}^{>0}$). For any $i$, denote by $U_{i}^{k}$ for
$k=1,\ldots,n-1$ the components of $U_{i}$ and set
$dU_{i}=dU_{i}^{1}\wedge\cdots\wedge dU_{i}^{n-1}$. Since the
$U_{n+3}$-component of $\boldsymbol{\mathcal{E}}_{n}^{>0}$ is

$e_{n-1}\big{(}U_{n+3}\big{)}\,dU_{n+3}=e_{n-1}(u_{1},\ldots,u_{n-1})du_{1}\wedge\cdots\wedge
du_{n-1}\,,$

it follows from the above proposition that $\boldsymbol{\mathcal{E}}_{n}^{>0}$
is equivalent to the fact that the differential relation

(98)
$0=\sum_{i=1}^{n+3}{(-1)^{i\,n}}\,e_{n-1}(U_{i})\,dU_{i}^{1}\wedge\cdots\wedge
dU^{n-1}_{i}$

holds true identically for any $n$-tuple of real numbers
$(u_{1},\ldots,u_{n})\in U$.

######  Example 4.

When $n=2$, the maps $U_{i}$ are given by

$U_{5}=u_{1}\,,\quad U_{4}=\frac{u_{2}}{u_{2}-1}\,,\quad
U_{3}=\frac{u_{2}}{u_{1}}\,,\quad U_{2}=\frac{u_{2}-1}{u_{2}-u_{1}}\quad\mbox{
and }\quad U_{1}=\frac{u_{1}(u_{2}-1)}{u_{2}(u_{1}-1)}\,.$

Up to integration, (98) is equivalent to the fact that the relation

(99) ${\rm R}(u_{1})+{\rm R}\left(\frac{u_{2}}{u_{2}-1}\right)+{\rm
R}\left(\frac{u_{2}}{u_{1}}\right)+{\rm
R}\left(\frac{u_{2}-1}{u_{2}-u_{1}}\right)+{\rm
R}\left(\frac{u_{1}(u_{2}-1)}{u_{2}(u_{1}-1)}\right)=0$

holds true for any $u_{1},u_{2}$ such that $1<u_{1}<u_{2}$, where ${\rm R}$
stands for the function defined by

$\qquad{\rm
R}(u)=\frac{1}{2}\int_{u}^{+\infty}\left(\frac{\log(u-1)}{u}-\frac{\log(u)}{u-1}\right)du+\frac{\hskip
4.26773pt\pi^{2}}{10}\qquad\quad\big{(}\,u>1\,\big{)}\,.$

It can be verified that given $u>1$, one has ${\rm
R}(u)=R_{1}\big{(}(u-1)/u\big{)}$ where $R_{1}$ stands for the following
version of Rogers dilogarithm: $R_{1}(x)={\bf
L}i_{2}(x)+\log(x)\log(1-x)/2-\pi^{2}/15$ for $x\in]0,1[$.

One checks easily that (99) is indeed left invariant by the 5-cyclic
birational map

$C(u_{1},u_{2})=\left(\frac{u_{2}}{u_{2}-1},\frac{u_{2}}{u_{2}-u_{1}}\right)\,.$

The second interesting consequence of Proposition 5.11 is a nice
transformation formula for the function $e_{n-1}$ appearing in the components
of $\boldsymbol{\mathcal{E}}_{n}^{>0}$. Recall the rational functions
$\psi_{i}$ introduced just before Proposition 5.7, which are first integrals
for $\boldsymbol{\mathcal{W}}_{0,n+3}^{>0}$ as well. With respect to these,
the last two components of $\boldsymbol{\mathcal{E}}_{n}^{>0}$ in (82) are
written respectively

(100) $\displaystyle f_{n+2}(u_{1},\ldots,u_{n-2},u_{n}\big{)}\cdot
du_{1}\wedge\cdots\wedge du_{n-2}\wedge du_{n}$ and $\displaystyle
f_{n+3}(u_{1},\ldots,u_{n-1}\big{)}\cdot du_{1}\wedge\cdots\wedge
du_{n-2}\wedge du_{n-1}$

for two functions $f_{n+2}$ and $f_{n+3}$ globally defined (and analytic) on
$U$.

On the other hand, for any $i=1,\ldots,n+3$, the $i$-th component
$(-1)^{i-1}\psi_{i}^{*}\big{(}\mathcal{E}_{n-1}^{n}\big{)}$ of
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ can also be written
$\big{(}\delta_{i}\big{)}_{*}\big{(}\tau_{i}^{*}\big{(}\mathscr{E}_{n}^{n}\big{)}\big{)}$.
Then following carefully the computations of §5.1.4 for these expressions when
$i$ is $n+2$ or $n+3$ (with $m=n-1$), we obtain that the following holds true:

######  Fact 5.12.

As functions of $n-1$ variables, one has (up to multiplication by a non zero
constant)

(101) $f_{n+2}=f_{n+3}=e_{n-1}$

where $e_{n-1}$ stands of course for the function defined in (91) (in the case
when $m=n-1$).

On the other hand, one has:

(102) $\displaystyle C^{*}\Big{(}\,du_{1}\wedge\cdots\wedge
du_{n-1}\,\Big{)}=$
$\displaystyle\,\,d\left(\frac{u_{n}}{u_{n}-1}\right)\wedge
d\left(\frac{u_{n}}{u_{n}-u_{1}}\right)\wedge\cdots\wedge
d\left(\frac{u_{n}}{u_{n}-u_{n-2}}\right)$ $\displaystyle=$
$\displaystyle\,(-1)^{n-1}\frac{\big{(}u_{n}\big{)}^{n-2}}{\prod_{i=0}^{n-2}\big{(}u_{n}-u_{i}\big{)}^{2}}\cdot\Big{(}du_{1}\wedge\cdots\wedge
du_{n-2}\wedge du_{n}\Big{)}\,.$

Combining (100), (101) and (102), we deduce the

######  Proposition 5.13.

The function $e_{n-1}$ satisfies the following relation:

(103)
$e_{n-1}\left(\frac{u_{n}}{u_{n}-u_{0}},\frac{u_{n}}{u_{n}-u_{1}},\ldots,\frac{u_{n}}{u_{n}-u_{n-2}}\right)=\frac{\prod_{i=0}^{n-2}\big{(}u_{n}-u_{i}\big{)}^{2}}{(u_{n})^{n-2}}\cdot
e_{n-1}\Big{(}u_{1},\ldots,u_{n-2},u_{n}\Big{)}$

for any real numbers $u_{0},u_{1},\ldots,u_{n-2},u_{n}$ such that
$1=u_{0}<u_{1}<\ldots<u_{n-2}<u_{n}$.

Example 4 (continued). Up to multiplication by a non zero constant, one has
$e_{1}(u)=\log(u-1)/u-\log(u)/(u-1)$ for any $u>1$. Given such a $u$, one has

$\displaystyle e_{1}\left(\frac{u}{u-1}\right)=$
$\displaystyle\,-\frac{(u-1)\log(u-1)}{u}-(u-1)\big{(}\log\left(u\right)-\log\left({u-1}\right)\big{)}$
(104) $\displaystyle=$
$\displaystyle\,\left(-\frac{u-1}{u}+(u-1)\right)\log(u-1)-(u-1){\log\left(u\right)}$
$\displaystyle=$
$\displaystyle\,(u-1)^{2}\frac{\log(u-1)}{u}-(u-1)^{2}\frac{\log\left(u\right)}{u-1}=(u-1)^{2}e_{1}(u)\,.$

##### Invariance properties of Euler's abelian relation $\mathcal{E}_{n}^{>0}$
with respect to pull-back under $R$.

The case of the automorphism $R$ of $\mathcal{M}_{0,n+3}^{>0}$ is handled by
similar but somewhat more subtile arguments that we are going to discuss
below. The lift $\hat{r}$ of $R$ that we consider here is the automorphism of
the oriented grassmannian induced by the linear map
$(x_{i})_{i=1}^{n+3}\mapsto\big{(}-x_{1},x_{n+3},x_{n+2},\ldots,x_{2},x_{1}\big{)}$
whose matrix (written by blocs) with respect to the standard basis of
$\mathbf{R}^{n+3}$ is

$N_{n}=\begin{bmatrix}-1&0\\\ 0&{\rm J}_{n+2}\end{bmatrix}\,$

where ${\rm J}_{n+2}$ stands for the anti-diagonal $(n+2)\times(n+2)$ matrix.
The matrix $N_{n}$ is orthogonal with determinant $\delta_{n}=(-1)^{\lfloor
n/2\rfloor}$ thus does not always belong to ${\rm SO}_{n+3}(\mathbf{R})$.
Nevertheless, in any case (using for instance (66)) it can be proved that
$\hat{r}$ lets the Euler form $E_{n}$ invariant: one has
$\hat{r}^{*}(E_{n})=E_{n}$. But contrarily to $\hat{c}$, the map $\hat{r}$
does not give rise to an automorphism of the positive part of the oriented
grassmannian since, as it can be verified straightforwardly, one has

$\hat{r}\left(G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\right)=G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{<0}\,.$

In order to land into $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$, it is
necessary to post compose $\hat{r}$ with the change of orientation map $D$
defined in §5.1.1. One then obtains a commutative diagram

(105)
$\textstyle{G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D\circ\hat{r}}$$\textstyle{G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{R}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$
---

Because $D^{*}(E_{n})=-E_{n}$ (by Lemma 5.1.1), it follows that one has

(106) $(D\circ\hat{r})^{*}(E_{n}^{n})=(-1)^{n}E_{n}^{n}\,.$

Now one has to determine the action of $D\circ\hat{r}$ on the orientation of
the $H_{0}$-orbits in $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}^{>0}$. We will
proceed as in the case of $C$, by considering the map $\tilde{R}$ from
$U\times(\mathbf{R}_{>0})^{2}\times(\mathbf{R}_{>0})^{n}$ into itself
corresponding to $D\circ\hat{r}$. In the coordinates $u,k$ and $h$ on
$U\times(\mathbf{R}_{>0})^{2}\times(\mathbf{R}_{>0})^{n}$ considered above,
one has

$\tilde{R}(u,k,h)=\Big{(}\,\tilde{u}\,,\,\tilde{k}\,,\,\tilde{h}\,\Big{)}$

with

$\displaystyle\tilde{u}=R(u)=\left(\frac{u_{n}-u_{n-1-i}}{u_{n}-u_{n-1}}\right)_{i=1}^{n}\,,\quad\tilde{k}=\left(\frac{k_{1}}{h_{n-1}(u_{n}-u_{n-1})}\,,\,\frac{h_{n}}{h_{n-1}}\right)\quad\mbox{and
}\quad\tilde{h}=\left(\,\frac{h_{n-1-i}}{h_{n-1}}\,\,\right)_{i=1}^{n}$

where we use the following notation: $u_{-1}=0$, $u_{0}=1$, $h_{0}=1$ and
$h_{-1}=k_{2}$.

Using the expressions above for the components of $\tilde{k}$ and $\tilde{h}$,
one easily computes what is the $dk\wedge dh$-component of the pull-back of
this form under $\tilde{R}$: it is given by

$\big{(}\partial k\wedge\partial
h\big{)}\,{\lrcorner}\,\tilde{R}^{*}\Big{(}\,dk\wedge
dh\,\Big{)}=\frac{(-1)^{\lfloor
n/2\rfloor+1}}{(h_{n-1})^{n+3}(u_{n}-u_{n-1})}\,.$

The sign of the above expression on $U\times(\mathbf{R}_{>0})^{n+2}$ is that
of $(-1)^{\lfloor n/2\rfloor+1}$. Together with (106) and because $n+\lfloor
n/2\rfloor+1-\lfloor(n-1)/2\rfloor=2\lfloor n/2\rfloor+2$ is even for any
$n\geq 2$, this gives us the

######  Proposition 5.14.

Identically on $U$, one has

(107)
$R^{*}\Big{(}\boldsymbol{\mathcal{E}}_{n}^{>0}\Big{)}=(-1)^{\lfloor(n-1)/2\rfloor}\,\,\boldsymbol{\mathcal{E}}_{n}^{>0}\,.$

Example 4 (finished). When $n=2$, the birational involution $R$ is given by

$R(u_{1},u_{2})=\left(\frac{u_{2}-1}{u_{2}-u_{1}},\frac{u_{2}}{u_{2}-u_{1}}\right)\,.$

A straightforward verification gives that it lets the relation (99) entirely
invariant.

The ingredients needed in order to get a transformation formula for $e_{n-1}$
from (107) are similar but not completely the same according to the parity of
$n$. We now set $m=\lfloor n/2\rfloor$.

Let us first discuss the case when $n$ is odd. In this situation, $R$ lets
invariant the $((n+5)/2)$-th foliation of
$\boldsymbol{\mathcal{W}}_{0,n+3}^{>0}$. This foliation admits the map
$\psi_{(n+5)/2}(u)$ as first integral whose components are the $u_{i}$'s for
$i=1,\ldots,n$ distinct from $(n-1)/2$. The component of
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ with respect to this first integral can be
seen to be a multiple of

(108)
$e_{n-1}\Big{(}u_{1},\ldots,u_{(n-3)/2},\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{{u_{(n-1)/2}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-620.0pt]{4.30554pt}{620.0pt}}}{}}{0.5ex}}\stackon[1pt]{{u_{(n-1)/2}}}{\tmpbox},u_{(n+1)/2},\ldots,u_{n}\Big{)}\,\Omega_{{(n+5)/2}}$

with $\Omega_{{(n+5)/2}}=\wedge_{i\neq(n-1)/2}du_{i}=du_{1}\wedge\cdots\wedge
du_{(n-3)/2}\wedge du_{(n+1)/2}\wedge\cdots\wedge du_{n}$.

By a direct computation, one gets that

(109) $R^{*}\Big{(}\Omega_{{(n+5)/2}}\Big{)}=\frac{(-1)^{\lfloor
n/2\rfloor}}{\big{(}u_{n}-u_{n-1}\big{)}^{n}}\hskip
4.26773pt\Omega_{{(n+5)/2}}\,.$

Let us now deal with the case when $n$ is even. Setting $m=n/2$, one has
$n^{\prime}=\lceil(n+3)/2\rceil=m+2$ and the birational involution $R$
exchanges the $n^{\prime}$-th and the $(n^{\prime}+1)$-th foliations of
$\boldsymbol{\mathcal{W}}_{0,n+3}^{>0}$ (see Figure 3). The two corresponding
first integrals $\psi_{n^{\prime}}$ and $\psi_{n^{\prime}+1}$ have for
components the $u_{i}$'s for $i=1,\ldots,n$ distinct from $m-1$ and $m$
respectively. Accordingly, we set $\Omega_{n^{\prime}}=\wedge_{i\neq
m-1}du_{i}$ and $\Omega_{n^{\prime}+1}=\wedge_{i\neq m}du_{i}$. Then, up to a
common multiple, the components of $\boldsymbol{\mathcal{E}}_{n}^{>0}$ with
respect to these first integrals can be seen to be

On the other hand, a direct computation gives us that when $n$ is (even and)
strictly bigger than 2, one has

(110) $R^{*}\Big{(}\Omega_{n^{\prime}}\Big{)}=\frac{(-1)^{\lfloor
n/2\rfloor}}{\big{(}u_{n}-u_{n-1}\big{)}^{n}}\hskip
4.26773pt\Omega_{n^{\prime}+1}$

whereas when $n=2$, one has $n^{\prime}=3$ and the following relation holds
true:

(111)
$R^{*}\Big{(}\Omega_{3}\Big{)}=\frac{-du_{2}}{\big{(}u_{2}-1\big{)}^{2}}\,=\frac{-1}{\big{(}u_{2}-1\big{)}^{2}}\,\Omega_{4}\,.$

We recall that $m=\lfloor n/2\rfloor\geq 1$. We now have everything at hand to
get another transformation formula satisfied by $e_{n-1}$. Indeed, from (107)
together with (108) and (109) when $n$ is odd, and with (5.1.5) and (110) (or
(111) if $n=2$) when $n$ is even, we deduce the

######  Proposition 5.15.

1. The function $e_{1}$ satisfies the following relation for any $x_{2}>1$

(112)
$\frac{e_{1}\left(\frac{x_{2}}{x_{2}-1}\right)}{(x_{2}-1)^{2}}=e_{1}(x_{2})\,.$

2. For any $n$ strictly bigger than 2, the function $e_{n-1}$ satisfies the relation:

(113)
$\frac{e_{n-1}\left(\frac{u_{n}-u_{n-2}}{u_{n}-u_{n-1}},\ldots,\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\frac{u_{n}-u_{m}}{u_{n}-u_{n-1}}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-620.0pt]{4.30554pt}{620.0pt}}}{}}{0.5ex}}\stackon[1pt]{\frac{u_{n}-u_{m}}{u_{n}-u_{n-1}}}{\tmpbox},\ldots,\frac{u_{n}-1}{u_{n}-u_{n-1}},\frac{u_{n}}{u_{n}-u_{n-1}}\right)}{\big{(}u_{n}-u_{n-1}\big{)}^{n}}=e_{n-1}\Big{(}u_{1},\ldots,\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{u_{m}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-620.0pt]{4.30554pt}{620.0pt}}}{}}{0.5ex}}\stackon[1pt]{u_{m}}{\tmpbox},\ldots,u_{n}\Big{)}$

for any
$u_{1},\ldots,u_{m-1},\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{u_{m}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-620.0pt]{4.30554pt}{620.0pt}}}{}}{0.5ex}}\stackon[1pt]{u_{m}}{\tmpbox},u_{m+1},\ldots,u_{n}$
such that $1<u_{1}<\ldots<u_{m-1}<u_{m+1}<\ldots<u_{n}$.

Remark that in the case when $n=2$, the two functional relations given by
Proposition 5.13 and Proposition 5.15 that $e_{1}$ satisfies actually coincide
(equations (104) and (112) are the same!).

####  5.1.6. Euler's abelian relation ${\mathcal{E}}_{3}^{>0}$.

Let us write down (103) and (113) explicitly when $n=3$: the two corresponding
formulas are respectively

(114)
$\frac{e_{2}\left(\frac{u_{3}}{u_{3}-1},\frac{u_{3}}{u_{3}-u_{1}}\right)\,u_{3}}{(u_{3}-1)^{2}(u_{3}-u_{1})^{2}}=e_{2}(u_{1},u_{3})\qquad\mbox{
and
}\qquad\frac{e_{2}\left(\frac{u_{3}-1}{u_{3}-u_{2}},\frac{u_{3}}{u_{3}-u_{2}}\right)}{(u_{3}-u_{2})^{3}}=e_{2}(u_{2},u_{3})\,,$

and these two equalities are satisfied for any $u_{1},u_{2},u_{3}$ such that
$1<u_{1}<u_{2}<u_{3}$.

We have determined above the first terms of the Taylor expansion of $e_{2}$ at
a given base point $u^{*}$ and noticed that, up to multiplication by a non
zero constant, these terms were the same as those of the Taylor expansion at
$u^{*}$ of the rational function

$\tilde{e}_{2}:(u_{1},u_{2})\mapsto\frac{1}{u_{1}u_{2}(u_{2}-1)}\,.$

As elementary verifications show, the function $\tilde{e}_{2}$ satisfies both
identities (114) as well. That is not a coincidence.

For $i=1,\ldots,n+3$, denote by
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3}^{>0})[i]$ the subspace of
$\psi_{i}^{*}\Omega^{n-1}$ spanned by the $i$-th components of the ARs of
$\boldsymbol{\mathcal{W}}_{0,n+3}^{>0}$, with respect to the first integrals
$\psi_{i}$. From the results of §4.5, one gets that
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3}^{>0})[n+3]$ is exactly the
space of 2-forms $F(u_{1},u_{2})du_{1}\wedge du_{2}$ where $F$ ranges in the
vector space of rational functions spanned by the set

$\mathfrak{B}_{3}(u_{1},u_{2})=\left\\{\,F_{0}(u_{1},u_{2})=\frac{1}{u_{1}u_{2}}\,,\,F_{ij}(u_{1},u_{2})\,\Big{\lvert}\begin{tabular}[]{l}$i=2,3$,
\, $j=i+1,\ldots,5$\end{tabular}\right\\}\subset\mathbf{Q}(u_{1},u_{2})$

defined in (63). It is straightforward to verify that the function
$\tilde{e}_{2}$ defined above first belongs to
$\langle\mathfrak{B}_{3}(u_{1},u_{2})\rangle$; and secondly and most
importantly, is the unique element of this space (up to multiplication by a
non zero constant) satisfying the same identities as (114). Then taking into
account the determination of the order three jet of $e_{2}$ at $u^{*}$
computed above, we deduce that equality (93) is indeed indentically satisfied
on $U\subset]1,+\infty[^{2}$.

It is just a computational matter to write down (98) in explicit form: Euler's
abelian relation $\boldsymbol{\mathcal{E}}_{3}^{>0}$ corresponds to the
following identity

(116) $\sum_{i=1}^{6}(-1)^{i}\mathcal{E}_{3,i}=0$

where the six 2-forms $\mathcal{E}_{3,i}=\big{(}C^{\circ
6-i}\big{)}^{*}\big{(}\mathcal{E}_{3,6}\big{)}$'s are given by

$\mathcal{E}_{3,6}=\frac{du_{1}\wedge
du_{2}}{u_{1}u_{2}(u_{2}-1)}\,,\qquad\mathcal{E}_{3,5}=\frac{du_{1}\wedge
du_{3}}{u_{1}u_{3}(u_{3}-1)}\,,\qquad\mathcal{E}_{3,4}=\frac{du_{2}\wedge
du_{3}}{u_{2}u_{3}(u_{3}-1)}$

and

$\displaystyle\mathcal{E}_{3,3}=$ $\displaystyle\,\frac{-du_{1}\wedge
du_{2}}{u_{2}(u_{3}+u_{1})}+\frac{du_{1}\wedge
du_{3}}{u_{3}(u_{3}-u_{1})}-\frac{du_{2}\wedge
du_{3}}{u_{2}u_{3}(u_{3}-u_{1})}$ $\displaystyle\mathcal{E}_{3,2}=$
$\displaystyle\,\frac{du_{1}\wedge
du_{2}}{(u_{2}-1)(u_{3}u_{1})}-\frac{du_{1}\wedge
du_{3}}{(u_{3}-1)(u_{3}-u_{1})}+\frac{(u_{1}-1)\,du_{2}\wedge
du_{3}}{(u_{2}-1)(u_{3}-1)(u_{3}-u_{1})}$ $\displaystyle\mathcal{E}_{3,1}=$
$\displaystyle\,\frac{-u_{3}\,du_{1}\wedge
du_{2}}{u_{1}u_{2}(u_{2}-1)(u_{3}-u_{1})}+\frac{du_{2}\wedge
du_{3}}{u_{1}(u_{3}-1)(u_{3}-u_{1})}-\frac{(u_{1}-1)\,du_{2}\wedge
du_{3}}{u_{2}(u_{2}-1)(u_{3}-1)(u_{3}-u_{1})}\,.$

Obviously, $\boldsymbol{\mathcal{E}}_{3}^{>0}$ is rational. Actually, since
all the ARs of $\boldsymbol{\mathcal{W}}_{0,6}$ are combinatorial, one can
state the following striking result:

######  Proposition 5.16.

Euler's abelian relation $\boldsymbol{\mathcal{E}}_{3}^{>0}$ is combinatorial:
one has

$\boldsymbol{\mathcal{E}}_{3}^{>0}\in\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}\,.$

####  5.1.7. Euler's abelian relations and their behavior with respect to the
action of $\mathfrak{S}_{n+3}$

We now want discuss the questions (85) raised above.

Let us start by considering the first one: `Does the construction given above
of $\boldsymbol{\mathcal{E}}_{n}^{>0}$ depend on some choices? If yes, on
which ones?' The main object we used to construct
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ on ${\mathcal{M}}_{0,n+3}^{>0}$ in section
§5.1.3 is the map $\gamma=\xi\circ M:U\rightarrow
G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$ defined in diagram (75). Up to the
identification (72) between $U$ and $\mathcal{M}_{0,n+3}^{>0}$, $\gamma$
corresponds to a section over $\mathcal{M}_{0,n+3}^{>0}$ of the natural map
$\widehat{G}_{2}^{or}(\mathbf{R}^{n+3})\rightarrow\mathcal{M}_{0,n+3}$
obtained by composing the arrows from the top left corner to the bottom right
corner in diagram (65). Considering this diagram is helpful to better
understand $\boldsymbol{\mathcal{E}}_{n}^{>0}$: it shows that from an
intrinsic point of view, $\boldsymbol{\mathcal{E}}_{n}^{>0}$ is not defined on
$\mathcal{M}_{0,n+3}^{>0}$ but rather on one of the two components over
$\mathcal{M}_{0,n+3}^{>0}$ in $\mathcal{M}_{0,n+3}^{\,or}(\mathbf{R})$.

Let us write

$\nu^{-1}\Big{(}\mathcal{M}_{0,n+3}^{>0}\Big{)}=\mathcal{M}_{0,n+3}^{\,or,\,>0}\sqcup\mathcal{M}_{0,n+3}^{\,or\,,<0}\,$

this notation being justified by the fact that one has
$\mathcal{M}_{0,n+3}^{\,or,\,>0}(\mathbf{R})=G_{2}^{or}(\mathbf{R}^{n+3})^{>0}/H\subset\mathcal{M}_{0,n+3}^{\,or}(\mathbf{R})$
and similarly for the negative setting (corresponding to formally replacing
$>$ by $<$). Actually (72) is an identification between $U$ and
$\mathcal{M}_{0,n+3}^{\,or,\,>0}$ and $\gamma$ corresponds to a section of

$\pi_{H}:\widehat{G}_{2}^{or}(\mathbf{R}^{n+3})\longrightarrow\mathcal{M}_{0,n+3}^{\,or}(\mathbf{R})$

over the latter component. Since
$H\simeq(\mathbf{R}_{>0})^{n+2}\times(\mathbf{Z}_{2})^{n+2}$, there is no
canonical choice for such a section: the section $\gamma$ we have considered
is the one landing in $G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$. Any other section
is of the form $\gamma_{\underline{\varepsilon}}=\xi\circ
M_{\underline{\varepsilon}}$ for
$\underline{\varepsilon}=(\varepsilon_{i})_{i=1}^{n+2}\in\\{\pm 1\,\\}^{n+3}$
where $M_{\underline{\varepsilon}}:U\rightarrow\Omega_{2}$ is the map
$u\mapsto M_{u}\cdot{\rm Diag}(\varepsilon_{1},\ldots,\varepsilon_{n+3})$,
i.e. for $u\in U$, $M_{\underline{\varepsilon}}(u)$ is the matrix whose $i$-th
column is $\varepsilon_{i}$ times the corresponding column of $M_{u}$ (cf.
(74)). For each section $\gamma_{\underline{\varepsilon}}$, similar arguments
to those in §5.1.3 apply. But in view of Lemma 5.1.2, the Eulerian AR one
constructs using $\gamma_{\underline{\varepsilon}}$ actually coincides with
the one obtained by means of $\gamma$, which is
$\boldsymbol{\mathcal{E}}_{n}^{>0}$. This shows that this AR is well-defined
on $\mathcal{M}_{0,n+3}^{\,or,\,>0}$.

But a totally similar construction can be made on the other component
$\mathcal{M}_{0,n+3}^{\,or,\,<0}$ of the inverse image of
$\mathcal{M}_{0,n+3}^{\,>0}$ by $\nu$. One obtains a well-defined Eulerian AR
on this component, that we will denote accordingly by
$\boldsymbol{\mathcal{E}}_{n}^{<0}$. In order to construct it in coordinates,
that is on $U$, one just has to consider the map
$\check{\gamma}=D\circ\gamma:U\rightarrow G_{2}^{or}(\mathbf{R}^{n+3})^{<0}$.
Since the pull-back under $D$ of Euler's 2-form $E_{n}$ is $-E_{n}$, we obtain
that $D^{*}(E_{n}^{n})=(-1)^{n}E_{n}^{n}$, which gives us the following

######  Lemma 5.17.

Let $\nu^{>0}$ and $\nu^{<0}$ be the restrictions of $\nu$ to
$\mathcal{M}_{0,n+3}^{\,or,\,>0}$ and $\mathcal{M}_{0,n+3}^{\,or,\,<0}$
respectively.

1\. One has
$\big{(}\nu^{>0}\big{)}_{*}\big{(}\boldsymbol{\mathcal{E}}_{n}^{>0}\big{)}=(-1)^{n}\big{(}\nu^{<0}\big{)}_{*}\big{(}\boldsymbol{\mathcal{E}}_{n}^{<0}\big{)}$.

2\. Consequently, Euler's abelian relation $\boldsymbol{\mathcal{E}}_{n}^{>0}$
constructed on $\mathcal{M}_{0,n+3}^{>0}$ in §5.1.3 is

  * $a.$

well-defined when $n$ is even;

  * $b.$

only defined up to sign if $n$ is odd.

Albeit rather elementary, this result is of crucial importance regarding the
birational action of $\mathfrak{S}_{n+3}$ on the abelian relations of
$\boldsymbol{\mathcal{W}}_{0,n+3}$.

A first remark, is that when $n$ is odd, the notation
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ is a bit misleading since it may let the
reader think that there is a more natural/canonical Eulerian AR among
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ and $-\boldsymbol{\mathcal{E}}_{n}^{>0}$.
As it can be verified easily (using one of the two formulas (5.11) or (5.14)
for instance) this is not the case: there is no non arbitrary way to
distinguish one from the other. As we will see below, this has non trivial
consequences regarding the invariance properties satisfied (or not) by Euler's
abelian relation on $\mathcal{M}_{0,n+3}^{or,>}(\mathbf{R})$. Note however
that in spite of this, it does not make null and void some invariance results
stated in §5.1.5 (namely Proposition 5.11 and Proposition 5.15) when $n$ is
odd: it would just be necessary to make the statements of these propositions
more precise in this case, which is easy to do and is left to the reader.

Secondly, the construction above is in no way specific to the positive
component of $\mathcal{M}_{0,n+3}(\mathbf{R})$. It can be generalized to any
component of it: for any $\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}$, we
construct an Eulerian AR on $\mathcal{M}(\boldsymbol{\sigma})$ (of the web
$\boldsymbol{\mathcal{W}}_{0,n+3}$) which is well-defined, but only up to sign
when $n$ is odd. In any case, we denote by
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ this (when $n$ is even)
or one of these two (when $n$ is odd) abelian relations, the choice of
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ instead of its opposite
being arbitrary when $n$ is odd. The span
$\langle\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}\rangle$ is a well-
defined 1-dimensional vector subspace of the space
$\boldsymbol{AR}(\boldsymbol{\sigma})$ of abelian relations of the web
$\boldsymbol{\mathcal{W}}_{0,n+3}^{\,\boldsymbol{\sigma}}$. Since the
permutations in $\mathfrak{S}_{n+3}$ induce isomorphisms between the
$\boldsymbol{AR}(\boldsymbol{\sigma})$'s and because these spaces are pairwise
non-identical, it is necessary to have a fixed way to identify them all to one
of them, say
$\boldsymbol{AR}(\boldsymbol{1})=\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}^{\,>0}\big{)}$,
by means of fixed isomorphisms
$\Xi^{\boldsymbol{\sigma}}:\boldsymbol{AR}(\boldsymbol{\sigma})\rightarrow\boldsymbol{AR}(\boldsymbol{1})$,
one for each $\boldsymbol{\sigma}$.

Of course, it is natural to assume that these isomorphisms satisfy the
following properties: first, one should have $\Xi^{\boldsymbol{1}}={\rm Id}$.
Secondly, since the space
$\boldsymbol{AR}_{R}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ of
rational ARs of $\boldsymbol{\mathcal{W}}_{0,n+3}$ (on the whole moduli space
${\mathcal{M}}_{0,n+3}(\mathbf{R})$ or even on its complexification
${\mathcal{M}}_{0,n+3}$) naturally embeds into
$\boldsymbol{AR}(\boldsymbol{\sigma})$ for any $\boldsymbol{\sigma}$, a
natural property that the isomorphisms $\Xi^{\boldsymbol{\sigma}}$ must all
satisfy is that they coincide with the identity when taking their restriction
along $\boldsymbol{AR}_{R}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$.
Finally, if one expects the Euler ARs to give rise to a 1-dimensional
representation, then each
$\Xi^{\boldsymbol{\sigma}}(\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}})$
must be a non trivial multiple of $\boldsymbol{\mathcal{E}}_{n}^{>0}$ for any
$\boldsymbol{\sigma}$ (which makes sense even in the case when $n$ is odd).

To summarize, in order that the action of $\mathfrak{S}_{n+3}$ on the space(s)
of abelian relations of the web(s) under scrutiny give rise to a
representation with the expected properties, it is necessary that the
$\Xi^{\boldsymbol{\sigma}}$'s satisfy the following properties:

(120) $(i).$ one has $\Xi^{\boldsymbol{1}}={\rm
Id}_{\boldsymbol{AR}(\boldsymbol{1})}$;
---
$(ii).$ for any $\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}$, one has
$\Xi^{\boldsymbol{\sigma}}\lvert_{{$\boldsymbol{AR}_{R}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$}}={\rm
Id}_{{$\boldsymbol{AR}_{R}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$}}$;
$(iii).$ for any $\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}$, there exists
$\lambda^{\boldsymbol{\sigma}}\neq 0$ such that
$\Xi^{\boldsymbol{\sigma}}\big{(}\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}\big{)}=\lambda^{\boldsymbol{\sigma}}\,\boldsymbol{\mathcal{E}}_{n}^{>0}$.

We are going to going to consider the two cases according to the parity of $n$
separately, this because there exist isomorphisms satisfying (120) when $n$ is
even, whereas this does not seem to be the case when n is odd (which we prove
explicitly when n is equal to 3)

$\star$

##### Case when $n$ is even.

When $n$ is even, essentially all the results claimed in [Da2] are indeed
satisfied. Actually in this case, it is not necessary to work with oriented
grassmannians but everything can be obtained within the (let say) `classical'
Gelfand-MacPherson theory developed in [GM]. For this reason, the treatment
below of the case when $n$ is even is rather concise.

We then assume that $n\geq 2$ is even: hence $m=n/2$ is a positive integer. In
this case, one has $E_{n}^{n}=\big{(}E_{n}^{2}\big{)}^{m}$ and Euler's AR can
be constructed from $P_{1}=E_{n}^{2}$. This invariant 4-form represents the
first Pontryagin class of the oriented tautological bundle $\mathcal{T}^{or}$
hence is the pull-back under the 2-to-1 projection
$G_{2}^{or}(\mathbf{R}^{n+3})\rightarrow G_{2}(\mathbf{R}^{n+3})$ of the
invariant 4-form on $G_{2}(\mathbf{R}^{n+3})$, denoted by the same notation,
and which represents the first Pontryagin class of $\mathcal{T}$. Then the
$2n$-form $P_{1}^{m}$ on $G_{2}(\mathbf{R}^{n+3})$ is `leading' according to
the terminology of Gelfand-MacPherson's paper (cf. §1.3.4 therein). Then from
[GM, §1.3.2], one constructs the abelian relation
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ for any
$\boldsymbol{\sigma}$, which is well-defined on any component
$\mathcal{M}(\boldsymbol{\sigma})$ of $\mathcal{M}_{0,n+3}(\mathbf{R}^{n+3})$.
This explains more conceptually the point 2.a. of Lemma 5.17.

But the results of [GM] can also be used to investigate the invariant
properties of the Euler ARs. Contrarily to when dealing with oriented
grassmammians, in which case the choices of lifts of permutations can be a bit
subtle, here we will only consider the naive lifts $\tilde{\sigma}\in{\rm
Aut}\big{(}G_{2}(\mathbf{R}^{n+3})\big{)}$ of permutations
$\sigma\in\mathfrak{S}_{n+3}\simeq{\rm Aut}\big{(}\mathcal{M}_{0,n+3}\big{)}$,
whose definition we recall: $\tilde{\sigma}$ is the automorphism of the
grassmannian induced by the linear map
$\mathbf{R}^{n+3}\rightarrow\mathbf{R}^{n+3}$,
$(x_{i})_{i=1}^{n+3}\mapsto(x_{\sigma(i)})_{i=1}^{n+3}$, also denoted by
$\tilde{\sigma}$. It is well-known that the latter is an orthogonal
transformation. On the other hand, it is proved in [GM, Corollary 3.25] that
the 4-form $P_{1}$ on $G_{2}(\mathbf{R}^{n+3})$ not only is left invariant
under the action of ${\rm SO}_{n+3}(\mathbf{R})$, but also under that of the
full orthogonal group ${\rm O}_{n+3}(\mathbf{R})$. Thus one has
$\tilde{\sigma}_{*}(P_{1})=P_{1}$ hence
$\tilde{\sigma}_{*}(P_{1}^{m})=P_{1}^{m}$ for any permutation $\sigma$. On the
other hand, in the case when $\sigma$ is a transposition, one verifies easily
that $\tilde{\sigma}$ reverses the natural orientation of the $H_{0}$-orbits.

From all the preceding considerations, one deduces the

######  Proposition 5.18.

For any ${\sigma}\in\mathfrak{S}_{n+3}$:

  1. 1.

one has
$\tilde{\sigma}^{*}\big{(}\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}\big{)}={\bf
sgn}(\sigma)\,\boldsymbol{\mathcal{E}}_{n}^{>0}$;

  2. 2.

Euler's abelian relation $\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$
does not belong to $\boldsymbol{AR}_{C}({\boldsymbol{\mathcal{W}}}_{0,n+3})$
hence

(121)
$\boldsymbol{AR}({\boldsymbol{\sigma}})=\boldsymbol{AR}_{C}\big{(}{\boldsymbol{\mathcal{W}}}_{0,n+3}\big{)}\oplus\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}\big{\rangle}\,.$

###### Proof.

The first point follows from the arguments given just before the proposition
so let us deal with the second. If
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ belongs to
$\boldsymbol{AR}_{C}\big{(}{\mathcal{W}}_{0,n+3}\big{)}$ for one permutation
${\sigma}$, this holds true for all according to 1. We thus have a line
$\langle\boldsymbol{\mathcal{E}}_{n}^{>0}\rangle=\langle\boldsymbol{\mathcal{E}}_{n}^{>0}\rangle$
in the space of combinatorial ARs, which is invariant by $\mathfrak{S}_{n+3}$.
But this is impossible since, as proved by Damiano (see Corollary 4.9.1 here),
$\boldsymbol{AR}_{C}\big{(}{\boldsymbol{\mathcal{W}}}_{0,n+3}\big{)}$ is an
irreducible $\mathfrak{S}_{n+3}$-representation. Thus for any
$\boldsymbol{\sigma}$, one has
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}\not\in\boldsymbol{AR}_{C}\big{(}{\boldsymbol{\mathcal{W}}}_{0,n+3}\big{)}$
and the decomposition in direct sum (121) follows from dimensional
considerations. ∎

The first point of this proposition answers question (85).$ii.$ when $n$ is
even.

######  Remark 5.19.

1. When $n$ is odd, one cannot do as above and get an $(n-1)$-abelian relation for $\boldsymbol{\mathcal{W}}_{0,n+3}$ using the standard Gelfand-MacPherson theory applied to an invariant $2n$-form on the usual (i.e. non oriented) grassmannian $G_{2}(\mathbf{R}^{n+3})$. As explained above, when $n$ is even one can construct Euler's AR by considering the $(n/2)$-wedge power of an invariant representative of the first Pontryagin class. This is not possible when $n$ is odd: in this case, $2n$ is not a multiple of 4 hence one really has to work with an invariant 2-form at some point. And if $n=2m+1$, the thing is that there is no such invariant 2-form on $G_{2}(\mathbf{R}^{n+3})$ as it follows from the well known description

$\boldsymbol{H}^{*}\Big{(}G_{2}\big{(}\mathbf{R}^{n+3}\big{)},\mathbf{R}\Big{)}\simeq\frac{\mathbf{R}\Big{[}\,p_{1},\overline{p}_{1},\ldots,\overline{p}_{m+1}\,\Big{]}}{\big{(}\,p\,\overline{p}-1\,\big{)}}$

of its cohomology ring over $\mathbf{R}$, where $p_{1}$ stands for the first
Pontryagin class of the tautological bundle $\mathcal{T}$ on
$G_{2}(\mathbf{R}^{n+3})$, the $\overline{p}_{k}$'s are the Pontryagin classes
of the rank $n+1$ cotautological bundle $\overline{\mathcal{T}}$, and where
$p$ and $\overline{p}$ denote the corresponding Pontryagin characters.434343We
recall that $\mathcal{T}$ and $\overline{\mathcal{T}}$ fit into the following
short exact sequence
$0\rightarrow\mathcal{T}\longrightarrow\mathbf{R}^{n+3}\longrightarrow\overline{\mathcal{T}}\rightarrow
0$ of fiber bundles over $G_{2}(\mathbf{R}^{n+3})$, where $\mathbf{R}^{n+3}$
stands for the trivial bundle of rank $n+3$. Also, the associated Pontryagin
characters are the elements $p=1+p_{1}$ and
$\overline{p}=1+\overline{p}_{1}+\cdots+\overline{p}_{\lfloor(n+3)/2\rfloor-1}$
of the cohomology ring with real coefficients of
$G_{2}\big{(}\mathbf{R}^{n+3}\big{)}$.

At this point, we believe it is interesting to recall how the cohomology ring
of the oriented grassmannian $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}$ is
related to that of the standard one: via the injection of the former
cohomology ring into the latter induced by the 2-1 covering
$G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}\rightarrow G_{2}(\mathbf{R}^{n+3})$,
one has

(122)
$\boldsymbol{H}^{*}\Big{(}G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)},\mathbf{R}\Big{)}\simeq\frac{\boldsymbol{H}^{*}\big{(}G_{2}\big{(}\mathbf{R}^{n+3}\big{)},\mathbf{R}\big{)}\big{[}\,e\,\big{]}}{\big{(}\,p_{1}-e^{2}\,\big{)}}=\frac{\mathbf{R}\Big{[}e,p_{1},\overline{p}_{1},\ldots,\overline{p}_{n/2}\Big{]}}{\big{(}\,p\,\overline{p}-1\,,\,p_{1}-e^{2}\,\big{)}}$

when $n$ is even, and

(123)
$\quad\boldsymbol{H}^{*}\Big{(}G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)},\mathbf{R}\Big{)}\simeq\frac{\boldsymbol{H}^{*}\big{(}G_{2}\big{(}\mathbf{R}^{n+3}\big{)},\mathbf{R}\big{)}\big{[}\,e,\overline{e}\,\big{]}}{\big{(}\,e\,\overline{e}\,,\,p_{1}-e^{2}\,,\,\overline{p}_{m+1}-\overline{e}^{2}\,\big{)}}=\frac{\mathbf{R}\Big{[}e,p_{1},\overline{e}\,,\,\overline{p}_{1},\ldots,\overline{p}_{m+1}\Big{]}}{\big{(}\,p\,\overline{p}-1\,,\,e\,\overline{e}\,,\,p_{1}-e^{2}\,,\,\overline{p}_{m+1}-\overline{e}^{2}\,\big{)}}$

when $n$ is odd and is written $n=2m+1$ for a positive integer $m$, where in
both cases $e$ (resp. $\overline{e}$) stands here for the Euler class of the
oriented tautological bundle $\mathcal{T}^{or}$ (resp. the cotautological
bundle $\overline{\mathcal{T}}^{or}$) on
$G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}$, with $p_{1}$ and the
$\overline{p}_{k}$'s being the associated Pontryagin classes. 444444The
descriptions of the cohomology rings above are very classical, see [Sad] or
[He] and the references therein.

When working with oriented grassmannians, one can consider (an invariant
representative of) the Euler class independently of the parity of $n$, which
has the convenience to allow to build Euler's abelian relation
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ in an uniform manner.

2. From (122) and (123), it follows that as an $\mathbf{R}$-algebra, the cohomology ring of $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}$ is generated by $e$ when $n$ is odd, and by $e$ and $\overline{e}$ but not by $e$ alone when $n$ is odd (cf. the remark in [GMZ, p. 267]). This shows that the claim that `the real cohomology ring of $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}$ is generated by the Euler class' p. 1341 of [Da2] is incorrect when $n$ is odd. However, this is irrelevant in what concerns the validity of the proof of [Da2, Theorem 5.1] given in the sixth section of [Da2]: since its degree is $n+1$, the class $\overline{e}$ does not belong to the set of characteristic classes considered in the statement of this theorem and because its square is non zero and has degree $2n+2=\dim\big{(}G_{2}^{or}(\mathbf{R}^{n+3})\big{)}$, it is a non trivial multiple of the $(n+1)$-th wedge power of $e$. This remark, which is missing in [Da2, §6] from our point of view, shows that no case has been forgotten and that the proof given there is complete.

For each class
$\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}=\mathfrak{S}_{n+3}/D_{0,n+3}$,
thanks to the decomposition in direct sum (121), one constructs an isomorphism
$\Xi^{\boldsymbol{\sigma}}:\boldsymbol{AR}({\boldsymbol{\sigma}})\rightarrow\boldsymbol{AR}(\mathcal{W}_{0,n+3}^{\,>0})$
uniquely characterized by the properties (120) with
$\lambda^{\boldsymbol{\sigma}}=1$ in $(iii)$. We then consider the following
map:

(124) $\displaystyle\mathfrak{S}_{n+3}$ $\displaystyle\longrightarrow{\rm
Aut}\Big{(}\boldsymbol{AR}\big{(}{\boldsymbol{\mathcal{W}}}_{0,n+3}^{\,>0}\big{)}\Big{)}$
$\displaystyle\sigma$
$\displaystyle\longmapsto\quad\Xi^{\boldsymbol{\sigma}}\circ\sigma^{*}:\hskip
7.11317pt\Psi\longmapsto\Xi^{\boldsymbol{\sigma}}\Big{(}\sigma^{*}\big{(}\Psi\big{)}\Big{)}\,.$

######  Corollary 5.20.

The above map makes of

(125)
$\boldsymbol{AR}\left({\boldsymbol{\mathcal{W}}}_{0,n+3}^{\,>0}\right)=\boldsymbol{AR}_{C}\left(\boldsymbol{\mathcal{W}}_{0,n+3}\right)\oplus\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{>0}\big{\rangle}\,.$

a $\mathfrak{S}_{n+3}$-module with
$\boldsymbol{AR}_{C}\big{(}{\boldsymbol{\mathcal{W}}}_{0,n+3}\big{)}$ and
$\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{>0}\big{\rangle}$ as irreducible
components, with Young symbols $[221^{n-1}]$ and $[1^{n+3}]$ respectively. In
particular, $\mathfrak{S}_{n+3}$ acts as the signature representation on the
1-dimensional component
$\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{>0}\big{\rangle}$.

To conclude this discussion of the case when $n$ is even, we recall that when
$n=2$, considering the local system (with non trivial monodromy) of complex
abelian relations of $\boldsymbol{\mathcal{W}}_{0,n+3}$ on the complex moduli
space $\mathcal{M}_{0,n+3}(\mathbf{C})$, we have constructed in Example 3 (via
a method different from that discussed just above) a complex
$\mathfrak{S}_{5}$-representation on the complexification of
$\boldsymbol{AR}\big{(}{\boldsymbol{\mathcal{W}}}_{0,5}^{>0}\big{)}$. The
latter has two irreducible components: one is the complexification of the
natural $\mathfrak{S}_{5}$-representation on
$\boldsymbol{AR}_{C}\big{(}{\boldsymbol{\mathcal{W}}}_{0,5}\big{)}$ and the
other, denoted by $\langle\boldsymbol{{\mathcal{A}}b}\rangle$ in Example 3, is
the trivial representation.

We believe that this generalizes to any $n$ even: considering the general form
for the components of $\boldsymbol{\mathcal{E}}_{n}^{>0}$ we conjecture and
have proved this conjecture to be correct for $n$ even and less than 12 (cf.
(137) and Proposition 5.28 further), that the complexification
$\boldsymbol{\mathcal{E}}_{n}^{\mathbf{C}}$ of
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ extends to a global multivalued AR on the
whole complexified moduli space ${\mathcal{M}}_{0,n+3}(\mathbf{C})$, with
logarithmic hence unipotent monodromy. Using the same elementary arguments as
in Example 3, this would give a 1-step filtration on
$\boldsymbol{AR}\big{(}{\boldsymbol{\mathcal{W}}}_{0,n+3}^{\,\mathbf{C}}\big{)}$
inducing a decomposition into irreducible $\mathfrak{S}_{n+3}$-modules

(126) ${\bf
Gr}\boldsymbol{{AR}}^{\bullet}\left(\boldsymbol{\mathcal{W}}_{0,n+3}^{\,\mathbf{C}}\right)=\boldsymbol{{AR}}_{C}\left(\boldsymbol{\mathcal{W}}_{0,n+3}^{\,\mathbf{C}}\right)\oplus\Big{\langle}\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}\Big{\rangle}$

with the 1-dimensional component
$\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}\big{\rangle}$ being
the trivial complex $\mathfrak{S}_{n+3}$-representation.

This might seem a bit surprising at first sight, a natural (but naive) guess
would be that (126) coincides with the complexification of the real
$\mathfrak{S}_{n+3}$-representation (125). But this cannot be the case since
the 1-dimensional component of the former would be the trivial representation,
whereas that of the latter is the signature representation. Here is a (not
fully rigorous) explanation of this apparent inconsistency: focusing only on
the components of dimension 1, one can say that (125) is obtained by comparing
the Euler's abelian relations $\boldsymbol{\mathcal{E}}_{n}^{>0}$ and
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ at two fixed points
$u^{>0}$ and $u^{\sigma}$ of two connected components
$\mathcal{M}_{0,n+3}^{>0}$ and $\mathcal{M}(\boldsymbol{\sigma})$ for any
$\sigma\in\mathfrak{S}_{0,n+3}$.

It is sufficient to only consider the case when $\sigma$ is a transposition in
order to understand the complex action on
$\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}\big{\rangle}$. One
identifies $\mathcal{M}_{0,n+3}$ with $\mathbf{R}^{n}\setminus A_{n}$ and
accordingly $\mathcal{M}_{0,n+3}(\mathbf{C})$ with $\mathbf{C}^{n}\setminus
A_{n}$ and for $i,j$ such that $1\leq i<j\leq n+3$, one denotes by $H_{i,j}$
the hyperplane of the braid arrangement corresponding to the limits obtained
by making coinciding the $i$-th and $j$-th components of configurations of
$n+3$ points on the Riemann sphere. Then the action of $\sigma=(i,j)$ on
$\boldsymbol{\mathcal{E}}_{n}^{\mathbf{C}}$ can be understood as follows: one
can find a smooth loop $\gamma:[0,1]\rightarrow\mathbf{C}^{n}\setminus
A_{n}\simeq\mathcal{M}_{0,n+3}(\mathbf{C})$, based at
$\gamma(0)=\gamma(1)=u^{>0}$ and such that $\gamma(1/2)=u^{\sigma}$, which is
of trivial index with respect to all the hyperplanes of the braid arrangement
$A_{n}$ at the exception of the hyperplane $H_{i,j}$, with respect to which it
is of index 1. For $\epsilon=0,1$, one sets
${\gamma}_{\epsilon}:[0,1]\rightarrow\mathbf{C}^{n}\setminus A_{n}$,
$t\mapsto\gamma(t/2+\epsilon/2)$. For any path
$\overline{\gamma}:[0,1]\rightarrow\mathbf{C}^{n}\setminus A_{n}$ and any germ
of holomorphic object $F$ at $\overline{\gamma}(0)$ one denotes by
$\overline{\gamma}\cdot F$ the germ at $\overline{\gamma}(1)$ obtained after
analytic continuation of $F$ along $\overline{\gamma}$ (in case its exists of
course).

Since the loop $\gamma$ is the concatenation $\gamma_{1}\,\gamma_{0}$, one has

$\sigma\,\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}=(i,j)\,\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}=\gamma\cdot\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}=\gamma_{1}\cdot\Big{(}\gamma_{0}\cdot\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}\Big{)}$

as germs of holomorphic ARs at $u^{>0}$. Up to sign, the complex analytic germ
of abelian relation
$\gamma_{0}\cdot\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}$ coincides with
(the germ at $u^{\sigma}$ of) the complexification of
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$, denoted by
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma},\mathbf{C}}$. Since the
orientation of $H_{0}$-orbits does not change along continuous deformations,
one has
$\gamma_{0}\cdot\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}=-\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma},\mathbf{C}}$.
Applying the same reasoning to $\gamma_{1}$, one gets that the following fact
holds true:

> As $\mathfrak{S}_{n+3}$-representations, the irreducible components of
> dimension 1 in (125) and in (126) are related in the following manner: the
> latter
> $\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}\big{\rangle}$
> identifies with the square of the complexification
> $\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{>0}\big{\rangle}^{\mathbf{C}}$
> of the former. Mathematically, one has
>
>
> $\Big{\langle}\boldsymbol{\mathcal{E}}_{n}^{\,\mathbf{C}}\Big{\rangle}\simeq\left(\big{\langle}\boldsymbol{\mathcal{E}}_{n}^{>0}\big{\rangle}^{\mathbf{C}}\right)^{\otimes
> 2}\,.$

Since it has not yet been proved that Euler' AR
$\boldsymbol{\mathcal{E}}_{n}^{\mathbf{C}}$ has unipotent monodromy for all
$n\geq 2$ even, the existence of a decomposition (126) with a trivial
irreducible component of dimension 1 is not known in full generality, so the
previous statement is only conjectural for the moment.

$\star$

##### Case when $n$ is odd.

When $n$ is odd, the situation is quite different from that in the former
(even) case. Indeed, since the Eulerian abelian relations
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$'s for
$\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}$ are only well-defined up to sign in
this case, looking for isomorphisms $\Xi^{\boldsymbol{\sigma}}$ satisfying
(120) actually requires to have beforehand suitably chosen one of the two
representatives $\pm\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ of
Euler's AR on $\mathcal{M}(\boldsymbol{\sigma})$ for any $\sigma$.

Since $\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ is
of maximal dimension $(n+1)(n+2)/2$,
$\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ is combinatorial for any
$\sigma\in\mathfrak{S}_{n+3}$. Since
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ is
irreducible as a $\mathfrak{S}_{n+3}$-representation when $n$ is odd
(according to Theorem 4.2), one immediately gets the following negative
result:

######  Proposition 5.21.

There is no set
$\big{\\{}\,{}^{\star}\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}\,\lvert\,\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}\,\\}$
of representatives of Euler's abelian relations such that for all permutations
$\sigma\in\mathfrak{S}_{n+3}$,
$\sigma_{*}\,\big{(}{}^{\star}\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{1}}\big{)}$
is colinear with
${}^{\star}\boldsymbol{\mathcal{E}}_{n}^{\boldsymbol{\sigma}}$ (as elements of
$\boldsymbol{AR}(\boldsymbol{\sigma})$).

This result implies in particular that there is no way to build a
1-dimensional $\mathfrak{S}_{n+3}$-representation from the Euler's abelian
relations following the lines considered by Damiano in [Da2].

It is worth considering the simplest case when this occurs, namely when $n=3$.
Let us discuss a concrete example of how the
$\boldsymbol{\mathcal{E}}_{3}^{\boldsymbol{\sigma}}$'s are related with the
abelian relation $\boldsymbol{\mathcal{E}}_{3}^{>0}$ considered in §5.1.6. Let
$S$ be the involutive automorphism of $\boldsymbol{\mathcal{W}}_{0,6}$ given
by $S(u_{1},u_{2},u_{3})=(u_{2},u_{1},u_{3})$ in the coordinates
$u_{1},u_{2},u_{3}$. This map can be seen to be the realization as a
birational automorphism of the transposition $s=(45)\in\mathfrak{S}_{6}$. It
induces an isomorphism between $\mathcal{M}_{0,6}^{\,>0}$ and
$\mathcal{M}(\boldsymbol{s})\simeq\\{(u_{i})_{i=1}^{3}\,\big{\lvert}\,1<u_{2}<u_{1}<u_{3}\,\\}$.
Using the explicit expression (116) for $\boldsymbol{\mathcal{E}}_{3}^{>0}$,
one gets that its pull-back under $S$ is an AR on
$\mathcal{M}(\boldsymbol{s})$ whose sixth component is

$S^{*}\big{(}\mathcal{E}_{3,6}\big{)}=S^{*}\left(\frac{du_{1}\wedge
du_{2}}{u_{1}u_{2}(u_{2}-1)}\right)=-\frac{du_{1}\wedge
du_{2}}{u_{1}u_{2}(u_{1}-1)}\,.$

On the other hand, by direct computations, one can compute the first terms of
the Taylor expansion of the Eulerian AR
$\boldsymbol{\mathcal{E}}_{3}^{\boldsymbol{s}}$ and verify that one has
$S^{*}(\boldsymbol{\mathcal{E}}_{3}^{>0})=-\,\boldsymbol{\mathcal{E}}_{3}^{\boldsymbol{s}}$.
However, as rational abelian relations of $\boldsymbol{\mathcal{W}}_{0,6}$,
$\boldsymbol{\mathcal{E}}_{3}^{>0}$ and
$\boldsymbol{\mathcal{E}}_{3}^{\boldsymbol{s}}$ are easily seen to be linearly
independent.

By direct computation, one verifies that
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,6})=\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$
is spanned by the $\mathfrak{S}_{6}$-orbit of
$\boldsymbol{\mathcal{E}}_{3}^{>0}$ which is consistent with the fact that the
space of ARs of $\boldsymbol{\mathcal{W}}_{0,6}$ is an irreducible
$\mathfrak{S}_{6}$-representation.

####  5.1.8. An explicit formula for Euler's abelian relation when $n$ is odd.

By means of direct computations, we have been able, for $n$ odd and less than
or equal to 9, to give a closed formula for Euler's abelian relation. Indeed,
using Proposition 4.12 (together with Remark 4.13), it is just a matter of
linear algebra to determine the functions $F$ depending on
$u^{\prime}=U_{1}(u)=(u_{1},\ldots,u_{n-1})$ satisfying the following
properties on the whole domain $U$:

(129) $\bullet$ the function $F$ satisfies the same functional identities as
(103) and (113);
---
$\bullet$ the identity
$0=\sum_{i=1}^{n+3}(-1)^{i}\left(C^{\circ(i-1)}\right)^{*}\Big{(}F(u^{\prime})\,du_{1}\wedge\cdots\wedge
du_{n-1}\Big{)}$ holds true identically.

We obtained that the set of such functions is a vector space of dimension one,
spanned by an explicit rational function. In particular, it gives us an
explicit closed rational formula for the function $e_{n-1}(u^{\prime})$ of
Proposition 5.9.

######  Proposition 5.22.

Let $n$ be an odd integer.

  1. 1.

Euler's abelian relation $\boldsymbol{\mathcal{E}}_{n}^{>0}$ is combinatorial
(hence rational): one has

$\boldsymbol{\mathcal{E}}_{n}^{>0}\in\boldsymbol{AR}_{C}\Big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\Big{)}\,.$

  2. 2.

If $n$ is moreover assumed to be less than or equal to 11 then as a function
of $u^{\prime}$, the function corresponding to the $U_{1}$-component of
Euler's AR $\boldsymbol{\mathcal{E}}_{n}^{>0}$ is rational and given by

(130) $\tilde{e}_{n-1}=F_{0}+\sum_{2\leq i<j\leq
n}(-1)^{i+j}F_{ij}+\sum_{i=2}^{n}(-1)^{i}F_{i,n+2}\,.$

Consequently, up to a non zero multiplicative constant, the function $e_{n-1}$
defined via the integral formula (91) coincides with the above rational
function $\tilde{e}_{n-1}$.

We conjecture that the previous statement actually is satisfied for any odd
integer $n\geq 3$.

######  Remark 5.23.

1\. Since it is irrelevant regarding its content, the preceding proposition
has been stated without mentioning the subtlety that Euler's abelian relation
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ on $\mathcal{M}_{0,n+3}^{\,>0}$ actually
is only defined up to sign (cf. Lemma 5.17).

2\. It turns out that both conditions in (129) have to be assumed in order to
get a space of solutions of dimension 1. Indeed, for $n$ odd less than $11$,
we have verified that the elements of the space of functions spanned by
$\boldsymbol{\mathfrak{B}}_{n}(u^{\prime})$ (see (63)) and satisfying only one
of these two conditions has dimension $(n+1)/2$. We have also verified (by
means of direct computations as well) that for an element
$F\in\langle\boldsymbol{\mathfrak{B}}_{n}(u^{\prime})\rangle$, satisfying the
functional identity (113) is automatic if one assumes that $F$ satisfies
(103). We conjecture that these facts hold true in full generality.

####  5.1.9. An explicit formula for Euler's abelian relation when $n$ is
even.

If the results of §5.1.7 (especially Corollary 5.20) give a clear picture of
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}^{\,>0}\big{)}$ as a
vector space (and even as a $\mathfrak{S}_{n+3}$-module) when $n$ is even,
they have the defect of not being explicit. However, as in the case when $n$
is odd (cf. the proposition just above), it is possible to make (the
components of) Euler's abelian relation explicit when $n$ is odd as well, at
least for small values of $n$ (we give a conjectural closed formula in the
general case).

We assume that $n$ is even in what follows. Since
$\boldsymbol{\mathcal{E}}_{n}^{>0}$ does not belong to
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$, one
cannot use a similar approach to that used just above to get Proposition 5.22.
The path we follow below is the following. We first deal with the $n=4$ case:
applying a variation of Abel's method for solving functional equations, we
determine in explicit form (one of) the components of
$\boldsymbol{\mathcal{E}}_{4}^{>0}$. Then we deduce from it a conjectural
closed formula for the function $e_{n-1}$ in (91) for $n\geq 2$ arbitrary.
Then, by means of explicit computations, we verify that this formula is indeed
correct for all even integers less than or equal to $12$.

##### The case $n=4$ via Abel's method.

It is now well-known that Abel's method for solving functional equations, by
means of successive differentiations and eliminations in order to get a
(partial) differential equation eventually, gives rise to a powerful and
efficient method to determine the abelian relations of 1-codimensional
webs.454545For instance, see [P4, §1.5] and the references there. Facing the
problem of making the components of $\boldsymbol{\mathcal{E}}_{4}^{>0}$
explicit, we have realised that it could be handled by adapting Abel's method
again. Since it could be useful in dealing with other (curvilinear) webs, we
say below a few words on how Abel's method works in the case under scrutiny.

We work on $\mathbf{C}^{4}$ and $x_{1},\ldots,x_{4}$ stand for the standard
variables on it. We denote by $\partial_{x_{1}},\ldots,\partial_{x_{4}}$ the
associated constant vector fields. The first integrals we are going to work
with are the $U_{1},\ldots,U_{7}$ given in (48), and
$\Omega_{1},\ldots,\Omega_{7}$ stand for the corresponding normals (see (49)).
Our goal is to apply an appropriate adaptation of Abel's method to determine,
for all $i$, the general form of the functions ${\rm e}_{i}$ of three
variables appearing in the following identity:

(131) $\sum_{i=1}^{7}{\rm e}_{i}(U_{i})\,\Omega_{i}=0\,.$

Thanks to the action of the cyclic map $C$ defined in (50), just dealing with
${\rm e}_{1}$ is enough.

For $s=1,2,3$ and any $i$, we denote by $\delta_{s}({\rm e}_{i})$ the partial
derivative of ${\rm e}_{i}$ with respect to the $s$-th variable and we also
consider the following rational vector fields on $\mathbf{C}^{4}$:

$X_{i}=\partial_{x_{i}}\hskip 8.5359pt(i=1,\ldots,4)\,,\quad
X_{5}=\sum_{i=1}^{4}(x_{i}-1)\,{\partial_{x_{i}}}\,,\quad
X_{6}=\sum_{i=1}^{4}x_{i}\,{\partial_{x_{i}}}\quad\mbox{ and }\quad
X_{7}=\sum_{i=1}^{4}x_{i}(x_{i}-1)\,{\partial_{x_{i}}}\,.$

Each $X_{i}$ defines the foliation which admits $U_{i}$ as a primitive first
integral. Consequently one has identically $X_{i}(U_{i})=0$ whereas for $i\neq
j$, $X_{i}(U_{j})\neq 0$ holds true (generically) thanks to the transversality
of the leaves of the foliations of $\boldsymbol{\mathcal{W}}_{0,7}$.

One has $\Omega_{i}=\wedge_{k\neq i}dx_{k}$ for $i=1,\ldots,4$ hence
$(\Omega_{i})_{i=1}^{4}$ is a (essentially the canonical) basis of
$\Omega^{4}(\mathbf{C}^{4})$. Setting
$\omega_{i}=\Omega_{i}(\partial_{x_{2}},\partial_{x_{3}},\partial_{x_{4}}\big{)}$
for $i=1,\ldots,7$, and considering only the $\Omega_{1}$ components of each
term of the sum in (131), we deduce that the following functional identity
with scalar coefficients holds true

(132) ${\rm e}_{1}(U_{1})+{\rm e}_{5}(U_{5})\,\omega_{5}+{\rm
e}_{6}(U_{6})\,\omega_{6}+{\rm e}_{7}(U_{7})\,\omega_{7}=0\,.$

Dividing (132) by $\omega_{7}^{1}$ and applying $X_{7}$ makes ${\rm
e}_{7}(U_{7})$ disappear and gives us the following `partial functional-
differential equation':

$\displaystyle{\rm
e}_{1}(U_{1})\,X_{7}\left(\frac{1}{\omega_{7}}\right)+\sum_{s=1}^{3}\delta_{s}({\rm
e}_{1})(U_{1})\,X_{7}(U_{1,s})+\sum_{k=5,6}\left({\rm
e}_{k}(U_{k})\,X_{7}\left(\frac{\omega_{k}}{\omega_{7}}\right)+\sum_{s=1}^{3}\delta_{s}({\rm
e}_{k})(U_{k})\,\frac{\omega_{k}X_{7}(U_{k,s})}{\omega_{7}}\right)=0\,.$

Dividing the LHS by $X_{7}(U_{k,3})\omega_{6}/\omega_{7}$ and applying $X_{6}$
makes the term involving $\delta_{3}({\rm e}_{6})$ disappear. One can then
continue this process of division-derivation-elimination following an
algorithmic process quite similar to that described in [P2, §2.2.2], to
eventually get that any function ${\rm e}_{1}={\rm e}_{1}(x_{2},x_{3},x_{4})$
appearing in an identity of the form (131) necessarily also satisfies

$\displaystyle\bigg{(}x_{34}x_{24}x_{23}\,{D_{234}}-$
$\displaystyle\,x_{23}(x_{2}-2\,x_{4}+x_{3})\,{D_{23}}+x_{24}(x_{4}+2x_{3}+x_{2})\,{D_{24}}$
$\displaystyle-$
$\displaystyle\,x_{34}(x_{3}-2x_{2}+x_{4})\,{D_{34}}+2x_{34}\,{D_{2}}-2x_{24}\,{D_{3}}+2x_{23}\,{D_{4}}\bigg{)}\cdot{\rm
e}_{1}=0$

with $x_{ij}=x_{i}-x_{j}$ for any $i,j=1,\ldots,4$, where we use the following
notations: $D_{234}$ stands for the order three partial derivative $\partial
x_{2}\partial x_{3}\partial x_{4}$, $D_{23}$ for $\partial x_{2}\partial
x_{3}$, etc.

Actually, changing the order in which the terms involving the ${\rm e}_{i}$'s
for $i\neq 1$ (and their partial derivatives) are eliminated gives rise to
other partial differential equations. What we get at the end is an explicit
system of partial differential equations, which we denote $(S{{\rm e}_{1}})$
(it is formed by several PDE's similar to the one above and there is no point
in making it explicit here). Contrarily to so many similar computations we did
in the past to determine the ARs of given planar webs, we have not been able
to solve $(S{{\rm e}_{1}})$ using a computer algebra system without making
additional assumptions. Looking for solutions of a certain kind464646We were
looking for solutions of $(S{{\rm e}_{1}})$ which are Laurent polynomials in
the expressions $x_{i}$, $x_{i}-1$, $x_{ij}$ ($i,j=2,3,4$) and their
logarithms. It is not relevant to elaborate more on this here., we have been
able to find one of these in explicit form.

We consider the following function, defined on the set
$X=\big{\\{}(x_{i})_{i=2}^{4}\in\mathbf{R}^{3}\,\big{\lvert}\,1<x_{2}<x_{3}<x_{4}\,\big{\\}}$:

$\displaystyle\varepsilon_{3}(x_{2},x_{3},x_{4})=$
$\displaystyle\,\frac{\mathit{x_{2}}^{2}\ln|\mathit{x_{2}}|}{\mathit{x_{3}}\mathit{x_{4}}(\mathit{x_{2}}-1)(\mathit{x_{2}}-\mathit{x_{3}})(\mathit{x_{2}}-\mathit{x_{4}})}-\frac{\mathit{x_{3}}^{2}\ln|\mathit{x_{3}}|}{\mathit{x_{2}}\mathit{x_{4}}(\mathit{x_{2}}-\mathit{x_{3}})(\mathit{x_{3}}-1)(\mathit{x_{3}}-\mathit{x_{4}})}$
$\displaystyle\,+\frac{\mathit{x_{4}}^{2}\ln|\mathit{x_{4}}|}{\mathit{x_{2}}\mathit{x_{3}}(\mathit{x_{4}}-1)(\mathit{x_{2}}-\mathit{x_{4}})(x_{3}-\mathit{x_{4}})}-\frac{(\mathit{x_{2}}-1)^{2}\ln|\mathit{x_{2}}-1|}{\mathit{x_{2}}(\mathit{x_{2}}-\mathit{x_{3}})(\mathit{x_{2}}-\mathit{x_{4}})(\mathit{x_{3}}-1)(\mathit{x_{4}}-1)}$
(133)
$\displaystyle\,-\frac{(\mathit{x_{2}}-\mathit{x_{3}})^{2}\ln|\mathit{x_{2}}-\mathit{x_{3}}|}{\mathit{x_{2}}\mathit{x_{3}}(\mathit{x_{2}}-1)(\mathit{x_{2}}-\mathit{x_{4}})(\mathit{x_{3}}-1)(x_{3}-\mathit{x_{4}})}+\frac{(\mathit{x_{2}}-\mathit{x_{4}})^{2}\ln|\mathit{x_{2}}-\mathit{x_{4}}|}{\mathit{x_{2}}\mathit{x_{4}}(\mathit{x_{2}}-1)(\mathit{x_{4}}-1)(\mathit{x_{2}}-\mathit{x_{3}})(x_{3}-\mathit{x_{4}})}$
$\displaystyle\,+\frac{(\mathit{x_{3}}-1)^{2}\ln|\mathit{x_{3}}-1|}{\mathit{x_{3}}(\mathit{x_{2}}-1)(\mathit{x_{4}}-1)(\mathit{x_{2}}-\mathit{x_{3}})(x_{3}-\mathit{x_{4}})}-\frac{(x_{3}-\mathit{x_{4}})^{2}\ln|x_{3}-\mathit{x_{4}}|}{\mathit{x_{3}}\mathit{x_{4}}(\mathit{x_{4}}-1)(\mathit{x_{3}}-1)(\mathit{x_{2}}-\mathit{x_{3}})(\mathit{x_{2}}-\mathit{x_{4}})}$
$\displaystyle\,-\frac{(\mathit{x_{4}}-1)^{2}\ln|\mathit{x_{4}}-1|}{\mathit{x_{4}}(\mathit{x_{2}}-1)(\mathit{x_{3}}-1)(\mathit{x_{2}}-\mathit{x_{4}})(x_{3}-\mathit{x_{4}})}\,.\vskip
6.0pt plus 2.0pt minus 2.0pt$

######  Remark 5.24.

If one restricts to $X\simeq\mathcal{M}_{0,6}^{>0}(\mathbf{R})$, one can drop
all the absolute values in (5.1.9). However the presence of absolute values in
the above definition is useful since it allows to extend $\varepsilon_{3}$
straightforwardly to a function defined on the whole complement
$\mathbf{R}^{3}\setminus A_{3}\simeq\mathcal{M}_{0,6}(\mathbf{R})$ where the
affine space has $x_{2},x_{3},x_{4}$ as coordinates with $A_{3}$ being the
braid arrangement cut out by

$x_{2}x_{3}x_{4}(x_{2}-1)(x_{3}-1)(x_{4}-1)(x_{2}-x_{3})(x_{2}-x_{4})(x_{3}-x_{4})=0.$

By direct computations, we then get the following result:

######  Proposition 5.25.

The function $\varepsilon_{3}$ defined just above:

  1. 1. is a solution of the system of PDEs $(S{{\rm e}_{1}})$;

  2. 2. satisfies the functional relations corresponding to (103) and (113): identically on $X$, one has

$\displaystyle\varepsilon_{3}\left(\frac{x_{4}}{x_{4}-1},\frac{x_{4}}{x_{4}-x_{2}},\frac{x_{4}}{x_{4}-x_{3}}\right)=$
$\displaystyle\,\varepsilon_{3}\big{(}x_{2},x_{3},x_{4}\big{)}\,(x_{4}-1)^{2}(x_{4}-x_{2})^{2}(x_{4}-x_{3})^{2}{x_{4}}^{-2}\vspace{0.2cm}$
$\displaystyle\mbox{ and
}\quad\varepsilon_{3}\left(\frac{x_{4}-x_{2}}{x_{4}-x_{3}},\frac{x_{4}-1}{x_{4}-x_{3}},\frac{x_{4}}{x_{4}-x_{3}}\right)=$
$\displaystyle\,\varepsilon_{3}\big{(}x_{2},x_{3},x_{4}\big{)}\,(x_{4}-x_{3})^{4}\,;\vskip
6.0pt plus 2.0pt minus 2.0pt$

  3. 3. is such that the following identity holds true on $X$:

(134)
$0=\sum_{i=1}^{7}\Big{(}C^{\circ(i-1)}\Big{)}^{*}\Big{(}\,\varepsilon_{3}(x_{2},x_{3},x_{4})\,dx_{2}\wedge
dx_{3}\wedge dx_{4}\,\Big{)}$

3. where $C$ stands for the cyclic birational map defined in (50).474747In the case under consideration, one has $C(x)=\Big{(}(x_{2}-1)/(x_{2}-x_{1}),(x_{3}-1)/(x_{3}-x_{1}),(x_{4}-1)/(x_{4}-x_{1}),{1}/{x_{1}}\Big{)}$.

The relation (134) means that the 7-tuple

(135)
$\bigg{(}\,\Big{(}C^{\circ(i-1)}\Big{)}^{*}\Big{(}\,\varepsilon_{3}(x_{2},x_{3},x_{4})\,dx_{2}\wedge
dx_{3}\wedge dx_{4}\ \Big{)}\,\bigg{)}_{i=1}^{7}$

is an AR for $\boldsymbol{\mathcal{W}}_{0,7}^{\,>0}$. Since its components
involve logarithms, it is clear that this AR does not belong to
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,7})$ and is colinear to
$(\boldsymbol{\mathcal{E}}_{4}^{>0})$ modulo the combinatorial ARs. Because we
know an explicit basis for
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,7})$ (cf. Theorem 4.2), one
can verify that the properties 2. and 3. of the preceding proposition
characterize a subspace of dimension 1 of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,7})$. We thus get the

######  Corollary 5.26.

Up to multiplication by a non zero constant:

1. the identity (134) corresponds to Euler's abelian relation $\boldsymbol{\mathcal{E}}_{4}^{>0}$;

2. the function $\varepsilon_{3}$ defined above coincides with the function $e_{3}$ of Proposition 5.9.

An interesting feature of $\varepsilon_{3}$, which follows from the fact that
the arguments of all the logarithms involved in its definition are absolute
values, is that $\varepsilon_{3}$ extends to $\mathbf{R}^{3}\setminus A_{4}$
(see Remark 5.24 above) and one can verify that (134) actually holds true
identically on the whole complement $\mathbf{R}^{4}\setminus A_{4}$ where
$A_{4}$ stands for the braid arrangement of type $A_{4}$.484848$A_{4}$ is cut
out by the polynomial equation $\prod_{i=1}^{4}x_{i}(x_{i}-1)\cdot\prod_{1\leq
i<j\leq 4}(x_{i}-x_{j})=0$. Thus up to the standard identification
$\mathbf{R}^{4}\setminus A_{4}\simeq\mathcal{M}_{0,7}(\mathbf{R})$, for any
$\boldsymbol{\sigma}\in\mathfrak{K}_{7}$ the restriction of (135) on any
connected component $\mathcal{M}(\boldsymbol{\sigma})$ of
$\mathcal{M}_{0,7}(\mathbf{R})$ is an AR which can be proved to coincide with
Euler's one $\boldsymbol{\mathcal{E}}_{4}^{\boldsymbol{\sigma}}$.

But $\varepsilon_{3}$ has another characteristic that is most important for
our purpose, which relies on the fact that it is a function of several
variables. Indeed, in clear contrast with the single-variable formula (3) for
Rogers' dilogarithm, it is rather easy to guess from the quite specific type
of formula (5.1.9), what might be its generalization for $n\geq 2$ even
arbitrary. Indeed, if one sets

$x_{i0}=x_{i}\,,\qquad x_{i1}=x_{i}-1\qquad\mbox{and}\qquad
x_{ij}=x_{i}-x_{j}$

for all $i,j$ such that $2\leq i<j\leq 4$ and if $M_{0,6}(x_{2},x_{3},x_{4})$
denotes the polynomial which cuts out $A_{3}$ in Remark 5.24, then
$\varepsilon_{3}$ can be written as follows

$\varepsilon_{3}(x_{2},x_{3},x_{4})=\frac{\sum_{i=2}^{4}p_{i0}(x_{i})^{3}\,{\rm
Log}\,\lvert x_{i0}\lvert+\sum_{i=2}^{4}p_{i1}(x_{i1})^{3}\,{\rm Log}\,\lvert
x_{i1}\lvert+\sum_{2\leq i<j\leq 4}^{n}p_{ij}(x_{ij})^{3}\,{\rm Log}\,\lvert
x_{ij}\lvert}{M_{0,6}(x_{2},x_{3},x_{4})}$

for some polynomials $p_{i0}$, $p_{i1}$ and $p_{ij}$ which it is not difficult
to make explicit (see below). And the nice feature of the preceding expression
is that it can be generalized to any even integer $n\geq 2$ quite
straightforwardly.

######  Remark 5.27.

  1. (1)

Of course, being closed, the 3-form $E_{4}$ admits a (local) not unique
primitive $\int E_{4}$ which can be explicitly computed and seen to carry many
dilogarithmic terms. Since the expression of $\int E_{4}$ is quite involved
and because we will not use it here, we will not elaborate on this.

  2. (2)

A direct computation gives
${C}^{*}\big{(}\varepsilon_{3}(U_{1})\,\Omega_{1}\big{)}=-\varepsilon_{3}(U_{2})\,\Omega_{2}$
hence the first two components of $(\boldsymbol{\mathcal{E}}_{4})$ are
$\varepsilon_{3}(x_{2},x_{3},x_{4})dx_{2}\wedge dx_{3}\wedge dx_{4}$ and
$-\varepsilon_{3}(x_{1},x_{3},x_{4})dx_{1}\wedge dx_{3}\wedge dx_{4}$. It
follows that the involution consisting in exchanging the variables $x_{1}$ and
$x_{2}$ transforms $(\boldsymbol{\mathcal{E}}_{4})$ into its opposite. This is
coherent with the first point of Proposition 5.18.

##### An explicit formula for $e_{n-1}$ for any $n\geq 2$ even.

Here $n\geq 2$ stands for a fixed even integer. First we set some notation
that we will use to define a function $\varepsilon_{n-1}$ through an explicit
formula. Then we will discuss how this function is meaningful regarding the
Euler's abelian relation of $\boldsymbol{\mathcal{W}}_{0,n+3}^{\,>0}$.

We denote here by $X$ the open domain in $\mathbf{R}^{n}$ whose elements are
$n$-tuples $x=(x_{i})_{i=1}^{n}$ such that $1<x_{1}<\ldots<x_{n}$ (recall that
there is a natural identification
$X\simeq\mathcal{M}_{0,n+3}^{\,>0}(\mathbf{R})$).

Here $C$ stands for the cyclic birational map defined in (50) again. It is
useful to work with the following first integrals for
$\boldsymbol{\mathcal{W}}_{0,n+3}^{\,>0}$, which behave well regarding the
cyclic symmetry induced by $C$: one sets
$V_{1}=U_{1}:\mathbf{R}^{n}\rightarrow\mathbf{R}^{n-1},\,x\mapsto(x_{2},\ldots,x_{n})$
and

$V_{i}=(V_{i}^{2},\ldots,V_{i}^{n})=V_{1}\circ\Big{(}C^{\circ(i-1)}\Big{)}$

for $i=1,\ldots,n+3$. The associated normals are denoted by $\Gamma_{i}$: one
has

$\Gamma_{1}=\Omega_{1}=dx_{2}\wedge\ldots\wedge dx_{n}\quad\mbox{ and
}\quad\Gamma_{i}=\Big{(}C^{\circ(i-1)}\Big{)}^{*}\big{(}\Omega_{1}\big{)}=dV_{i}^{2}\wedge\ldots\wedge
dV_{i}^{n}\quad\big{(}i=2,\ldots,n+3\big{)}\,.$

Looking for an explicit function
$\varepsilon_{n-1}(V_{1})=\varepsilon_{n-1}(x_{2},\ldots,x_{n})$ such that the
following relation

(136)
$\sum_{i=1}^{n+3}\Big{(}C^{\circ(i-1)}\Big{)}^{*}\Big{(}\varepsilon_{n-1}(V_{1})\,dx_{2}\wedge\ldots\wedge
dx_{n}\Big{)}=\sum_{i=1}^{n+3}\varepsilon_{n-1}(V_{i})\,\Gamma_{i}=0$

holds true identically on $X$, one sets for any
$x=(x_{1},\ldots,x_{n})\in\mathbf{R}^{n}$:

  * •

$x^{\prime}=V_{1}(x)=(x_{2},\ldots,x_{n})\in\mathbf{R}^{n-1}$;

  * •

$x_{n+1}=0$ and $x_{n+2}=1$;

  * •

$\tilde{x}=(x_{1},\ldots,x_{n},x_{n+1},x_{n+2})\in\mathbf{R}^{n+2}$;

  * •

$\tilde{x}^{\prime}=(x_{2},\ldots,x_{n},x_{n+1},x_{n+2})\in\mathbf{R}^{n+1}$;

  * •

${\rm M}_{n}(x)=\prod_{1\leq i<j\leq n}(x_{i}-x_{j})$;

  * •

$M_{0,n+3}(x)={\rm
M}_{n+2}(\tilde{x})=\prod_{i=1}^{n}x_{i}\,\big{(}x_{i}-1\big{)}\prod_{1\leq
k<\ell\leq n}\big{(}x_{k}-x_{\ell}\big{)}$;

  * •

for $i,j=1,\ldots,n$, one sets $x_{ij}=x_{i}-x_{j}$ (thus
$\tilde{x}_{i,n+1}=x_{i}$ and $\tilde{x}_{i,n+2}=x_{i}-1$ for any $i\leq n$);

  * •

for $i=2,\ldots,n$ and $j=i+1,\ldots,n+2$, one denotes by
$x_{\widehat{\imath\jmath}}$ the $(n-1)$-tuple obtained from $\tilde{x}$ by
removing from it, in the same step: its first, $i$-th and $j$-th coefficients.

The notation $M_{0,n+3}$ is not absolutely needed but we use it because it is
enlightening: indeed, for $\mathbf{K}=\mathbf{R}$ or $\mathbf{C}$, the
equation $M_{0,n+3}(z)=0$ for $z\in\mathbf{K}^{n}$ cuts out the braid
arrangement $A_{n}$ in $\mathbf{K}^{n}$ which is such that there is a natural
identification between $\mathcal{M}_{0,n+3}(\mathbf{K})$ and
$\mathbf{K}^{n}\setminus A_{n}$.

Then, with these notations at hand, for any
$x^{\prime}=(x_{2},\ldots,x_{n})\in\mathbf{R}^{n-1}\setminus A_{n-1}$, one
sets:

(137)
$\varepsilon_{n-1}(x^{\prime})=\frac{1}{M_{0,n+2}\big{(}x^{\prime}\big{)}}\sum_{\begin{subarray}{c}i=2,\ldots,n\\\
j=i+1,\ldots,n+2\end{subarray}}(-1)^{j-i}\,{\rm
M}_{n-1}\big{(}x_{\widehat{\imath\jmath}}\big{)}\,\big{(}\tilde{x}_{ij}\big{)}^{n-1}\,{\rm
Log}\,\lvert\,\tilde{x}_{ij}\,\lvert\,.$

For $n=4$, one recovers exactly the function defined in (5.1.9).

By direct computations, we get the following generalization of Proposition
5.25:

######  Proposition 5.28.

Let $n\geq 2$ be an even integer less than or equal to 12.

  1. 1.

The function $\varepsilon_{n-1}$ satisfies the functional relations
corresponding to (103) and (113): identically on $X$, one has

$\displaystyle\varepsilon_{n-1}\left(\frac{x_{n}}{x_{n}-1},\frac{x_{n}}{x_{n}-x_{2}},\ldots,\frac{x_{n}}{x_{n}-x_{n-1}}\right)=$
$\displaystyle\,\varepsilon_{n-1}\big{(}x^{\prime}\big{)}\,{{{\rm
M}_{n+1}\big{(}\tilde{x}^{\prime}\big{)}}^{2}}{(x_{n})^{-n}}$
$\displaystyle\mbox{ and
}\quad\varepsilon_{n-1}\left(\frac{x_{n}-x_{n-2}}{x_{n}-x_{n-1}},\ldots,\frac{x_{n}-x_{2}}{x_{n}-x_{n-1}},\frac{x_{n}-1}{x_{n}-x_{n-1}},\frac{x_{n}}{x_{n}-x_{n-1}}\right)=$
$\displaystyle\,\varepsilon_{n-1}\big{(}x^{\prime}\big{)}\,\big{(}x_{n}-x_{n-1}\big{)}^{n}\,;\vskip
6.0pt plus 2.0pt minus 2.0pt$

  2. 2.

Identity (136) holds true identically and consequently it corresponds to
Euler's abelian relation $\boldsymbol{\mathcal{E}}_{n}^{>0}$ and
$\varepsilon_{n-1}$ coincides with the function $e_{n-1}$ of Proposition 5.9
(up to multiplication by a non zero constant).

Of course, we conjecture that this proposition is actually satisfied for any
even integer $n\geq 2$.

##### An Eulerian basis of the space of combinatorial abelian relations for
any $n\geq 2$ even?

The identity (136) has an interesting feature which comes from the fact that
there are logarithmic terms in the definition of $\varepsilon_{n-1}$: these
being transcendent over any field of rational functions, one can construct
many rational ARs from Euler's one. In this paragraph, we formalise this in
precise terms and we show that, at least for the first values (namely
$2,4,6,8$) of $n$, this approach is relevant regarding the question of
constructing a basis of
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$.