&=\mathcal{S}^{k}_{\sigma}\left(\Psi^{\rm
dec}_{\sigma}(\mathbf{y})\right)\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus\\{\sigma(s)\\}}}]{{\scriptscriptstyle}}{+}\left(\mathbf{0}^{{\underline{n}}}_{p_{\sigma(s)}\bigl{(}\mathcal{S}^{k}_{\sigma}\left(\Psi^{\rm
dec}_{\sigma}(\mathbf{y})\right)\bigr{)}}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus\\{\sigma(t)\\}}}]{{\scriptscriptstyle}}{+}\mathcal{S}^{\sigma(J_{k+1})}\left(\left(\Psi_{\sigma}^{\rm
dec}\right)^{{\underline{n}}}_{J_{k+1}}(\mathbf{z})\right)\right)\\\
&=\mathcal{S}^{k+1}_{\sigma}\left(\Psi^{\rm
dec}_{\sigma}(\mathbf{x})\right)\end{split}$

follows from the induction hypothesis and (42). ∎

The following is a more general version of the second statement of the
previous theorem, which is the first step of the proof a symmetric
decomposition of a symmetric $n$-fold vector bundle.

######  Proposition B.2.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be a symmetric
$n$-fold vector bundle, and consider an $l$-partition $\rho$ of
${\underline{n}}$ for $1\leq l\leq n$. Consider for each partition
$\rho^{\prime}\in\\{\sigma(\rho)\mid\sigma\in S_{n}\\}$ a linear splitting
$\Sigma^{\rho^{\prime}}\colon\overline{\mathbb{E}^{\rho^{\prime}}}\to\mathbb{E}^{\rho^{\prime}}$.
For each coarsement $\underline{\rho}$ in $(l-1)$ subsets of a partition
$\rho^{\prime}$ in the $S_{n}$-orbit of $\rho$ take a linear decomposition
$\mathcal{S}^{\underline{\rho}}\colon(\mathbb{E}^{\rm
dec})^{\underline{\rho}}\to\mathbb{E}^{\underline{\rho}}$ of the $(l-1)$-core
$\mathbb{E}^{\underline{\rho}}$ of $\mathbb{E}$. Assume that:

  1. (1)

For each $\sigma\in S_{n}$, the diagram

${\overline{\mathbb{E}^{\rho}}}$${\mathbb{E}^{\rho}}$${\overline{\mathbb{E}^{\sigma(\rho)}}^{\sigma}}$${(\mathbb{E}^{\sigma(\rho)})^{\sigma}}$$\scriptstyle{\Sigma^{\rho}}$$\scriptstyle{\Psi^{\rm
dec}_{\sigma}}$$\scriptstyle{\Phi_{\sigma}}$$\scriptstyle{(\Sigma^{\sigma(\rho)})^{\sigma}}$

commutes.

  2. (2)

For each $\sigma\in S_{n}$ and each coarsement $\underline{\rho}$ of $\rho$

$\Phi_{\sigma}\circ\mathcal{S}^{\underline{\rho}}=\mathcal{S}^{\sigma(\underline{\rho})}\circ\Psi^{\rm
dec}_{\sigma}\colon(\mathbb{E}^{\rm
dec})^{\underline{\rho}}\to\left(\mathbb{E}^{\sigma(\underline{\rho})}\right)^{\sigma}.$

  3. (3)

For two coarsements $\rho_{1}$ and $\rho_{2}$ of $\rho$ in $l-1$ subsets the
two decompositions $\mathcal{S}^{\rho_{1}}$ and $\mathcal{S}^{\rho_{2}}$ are
compatible as in (21).

  4. (4)

$\Sigma^{\rho}$ is compatible with $\mathcal{S}^{\underline{\rho}}$ as in (22)
for each coarsement $\underline{\rho}$ of $\rho$ in $l-1$ subsets.

Then Proposition A.1 yields for each $\sigma\in S_{n}$ a decomposition

$\mathcal{S}^{\sigma(\rho)}\colon(\mathbb{E}^{\rm
dec})^{\sigma(\rho)}\to\mathbb{E}^{\sigma(\rho)}$

of the $l$-core $\mathbb{E}^{\sigma(\rho)}$ of $\mathbb{E}$, that restricts to
$\Sigma^{\sigma(\rho)}$ and the decompositions of the appropriate
$(l-1)$-cores of $\mathbb{E}$. These decompositions satisfy

$\Phi_{\sigma}(I)\circ\mathcal{S}^{\rho}(I)=\mathcal{S}^{\sigma(\rho)}(\sigma(I))\circ\Psi^{\rm
dec}_{\sigma}(I)\colon(\mathbb{E}^{\rm
dec})^{\rho}(I)\to\mathbb{E}^{\sigma(\rho)}(\sigma(I))$

for all $\sigma\in S_{n}$ and all $I\in\operatorname{Obj}(\lozenge^{\rho})$.

###### Proof.

The proof is very similar to the proof of the second statement in Proposition
B.1. Write $\rho=\\{I_{1},\ldots,I_{l}\\}$ and build all sets

$J_{1},\ldots,J_{\binom{l}{2}}\in\operatorname{Obj}\left(\lozenge^{\rho}\right),$

each consisting in the union of exactly two elements of $\rho$. Then for each
$\sigma\in S_{n}$

$J_{1}^{\sigma}:=\sigma(J_{1}),\ldots,J^{\sigma}_{\binom{l}{2}}:=\sigma\left(J_{\binom{l}{2}}\right)$

is such a list of subsets for the partition
$\sigma(\rho)=\\{\sigma(I_{1}),\ldots,\sigma(I_{l})\\}$ of ${\underline{n}}$.
Each such subset $J_{i}^{\sigma}$ defines a coarsement $\underline{\rho}$ of
$\sigma(\rho)$, namely the one with $J_{i}^{\sigma}$ replacing the two
elements of $\sigma(\rho)$ it is a union of. The corresponding splitting

$\mathcal{S}^{\underline{\rho}}\colon\mathbb{E}^{{\rm
dec},\underline{\rho}}\to\mathbb{E}^{\underline{\rho}}$

which is chosen in the hypotheses of the theorem is named
$\mathcal{S}^{J_{i}^{\sigma}}$ in the following.

As in Section A these lists yield for each $\sigma\in S_{n}$ and each
$k=1,\ldots,\binom{l}{2}$ a family
$\mathcal{A}^{k}_{\sigma}=(B^{k,\sigma}_{I})_{I\in\operatorname{Obj}(\lozenge^{\sigma(\rho)})}$,

$\begin{split}B^{k,\sigma}_{I}&=\left\\{\begin{array}[]{cl}A_{\\#I}&\quad\text{
if }I\in\sigma(\rho)\text{ or if there is $i\leq k$ such that
}\sigma(J_{i})\subseteq I;\\\ M&\quad\text{
otherwise.}\end{array}\right.\end{split}$

A chain of decomposed $l$-fold vector bundles
$\lozenge^{\sigma(\rho)}\to{\mathbf{Man}^{\infty}}$

$\overline{\mathbb{E}^{\sigma(\rho)}}=:\mathbb{E}^{0}_{\sigma}\hookrightarrow\mathbb{E}^{1}_{\sigma}\hookrightarrow\ldots\hookrightarrow\mathbb{E}^{\binom{l}{2}}_{\sigma}:=\mathbb{E}^{{\rm
dec},\sigma(\rho)},$

is defined by the families $\mathcal{A}^{k}_{\sigma}$,
$k=0,\ldots,\binom{l}{2}$, as well as a family of monomorphisms

$\mathcal{S}^{0}_{\sigma}=\Sigma^{\sigma(\rho)}\colon\overline{\mathbb{E}^{\sigma(\rho)}}\to\mathbb{E}^{\sigma(\rho)},\quad\mathcal{S}^{1}_{\sigma}\colon\mathbb{E}^{1}_{\sigma}\to\mathbb{E}^{\sigma(\rho)},\quad\ldots,\quad\mathcal{S}^{\binom{l}{2}}_{\sigma}=\mathcal{S}^{\sigma(\rho)}\colon\mathbb{E}^{{\rm
dec},\sigma(\rho)}\to\mathbb{E}^{\sigma(\rho)}$

of $l$-fold vector bundles
$\lozenge^{\sigma(\rho)}\to{\mathbf{Man}^{\infty}}$, where the first and last
morphisms are $\Sigma^{\sigma(\rho)}$ and $\mathcal{S}^{\sigma(\rho)}$,
respectively.

Consider

$\mathbf{x}=(x_{J})_{\begin{subarray}{c}J\subseteq I\\\
J\in\operatorname{Obj}(\lozenge^{\rho})\end{subarray}}\in\mathbb{E}^{k}_{\operatorname{id}}(I)$

for some $k\in\\{1,\ldots,\binom{l}{2}\\}$ and for
$\sigma=\operatorname{id}\in S_{n}$. Then as before there is an $m\in M$ such
that for all $J\subseteq I$ with $J\in\operatorname{Obj}(\lozenge^{\rho})$:

$x_{J}\left\\{\begin{array}[]{ll}=0^{A_{\\#J}}_{m}&\text{ if
}J\not\in\rho\text{ and there exists no }i\in\\{1,\ldots,k\\}\text{ with
}J_{i}\subseteq J\\\ \in A_{\\#J}(m)&\text{ otherwise}.\end{array}\right.$

For $\sigma\in S_{n}$ the image $\mathbf{x}^{\prime}$ of this point
$\mathbf{x}$ under $\Psi^{\rm dec}_{\sigma}$ lives in $\mathbb{E}^{\rm
dec,\sigma(\rho)}(\sigma(I))$ and its coordinates satisfy

$\begin{split}x^{\prime}_{J}&=\epsilon(\sigma^{-1},J)x_{\sigma^{-1}(J)}\\\
&\left\\{\begin{array}[]{ll}=0^{A_{\\#J}}_{m}&\text{ if
}\sigma^{-1}(J)\not\in\rho\text{ and there exists no
}i\in\\{1,\ldots,k\\}\text{ with }J_{i}\subseteq\sigma^{-1}(J)\\\ \in
A_{\\#J}(m)&\text{ otherwise}\end{array}\right.\end{split}$

for all $J\subseteq\sigma(I)$ with
$J\in\operatorname{Obj}(\lozenge^{\sigma(\rho)})$. Hence

$\begin{split}x^{\prime}_{J}&=\epsilon(\sigma^{-1},J)x_{\sigma^{-1}(J)}\\\
&\left\\{\begin{array}[]{ll}=0^{A_{\\#J}}_{m}&\text{ if
}J\not\in\sigma(\rho)\text{ and there exists no }i\in\\{1,\ldots,k\\}\text{
with }\sigma(J_{i})\subseteq J\\\ \in A_{\\#J}(m)&\text{
otherwise}\end{array}\right.\end{split}$

for all $J\subseteq\sigma(I)$ with
$J\in\operatorname{Obj}(\lozenge^{\sigma(\rho)})$, which shows that
$\mathbf{x}^{\prime}=\Psi^{\rm
dec}_{\sigma}(\mathbf{x})\in\mathbb{E}^{k}_{\sigma}(\sigma(I))$, and so that
$\Psi^{\rm dec}_{\sigma}$ restricts to a natural isomorphism

$\Psi^{\rm
dec}_{\sigma}\colon\mathbb{E}^{k}\to(\mathbb{E}^{k}_{\sigma})^{\sigma}$

of the functors
$\mathbb{E}^{k},(\mathbb{E}^{k}_{\sigma})^{\sigma}\colon\lozenge^{\rho}\to{\mathbf{Man}^{\infty}}$.

Now since $\mathbb{E}^{\binom{l}{2}}=(\mathbb{E}^{\rm dec})^{\rho}$,
$\mathbb{E}^{\binom{l}{2}}_{\sigma}=(\mathbb{E}^{\rm dec})^{\sigma(\rho)}$ and
$\mathcal{S}^{\binom{l}{2}}_{\operatorname{id}}=\mathcal{S}^{\rho}$ as well as
$\mathcal{S}^{\binom{l}{2}}_{\sigma}=\mathcal{S}^{\sigma(\rho)}$, it suffices
to show inductively that for $k=0,\ldots,\binom{l}{2}$ and for $\sigma\in
S_{n}$ the diagram

${\mathbb{E}^{k}_{\operatorname{id}}}$${\mathbb{E}^{\rho}}$${\left(\mathbb{E}_{\sigma}^{k}\right)^{\sigma}}$${(\mathbb{E}^{\sigma(\rho)})^{\sigma}}$$\scriptstyle{\mathcal{S}^{k}_{\operatorname{id}}}$$\scriptstyle{\Psi^{\rm
dec}_{\sigma}}$$\scriptstyle{\Phi^{\rho}_{\sigma}}$$\scriptstyle{(\mathcal{S}^{k}_{\sigma})^{\sigma}}$

of morphisms of $l$-fold vector bundles commutes. It is enough to check this
at the top level, i.e. to check that

${\mathbb{E}^{k}_{\operatorname{id}}({\underline{n}})}$${\mathbb{E}^{\rho}({\underline{n}})}$${\mathbb{E}^{k}_{\sigma}({\underline{n}})}$${\mathbb{E}^{\sigma(\rho)}({\underline{n}})}$$\scriptstyle{\mathcal{S}^{k}_{\operatorname{id}}({\underline{n}})}$$\scriptstyle{\Psi^{\rm
dec}_{\sigma}({\underline{n}})}$$\scriptstyle{\Phi^{\rho}_{\sigma}({\underline{n}})}$$\scriptstyle{\mathcal{S}^{k}_{\sigma}({\underline{n}})}$

commutes. For $k=0$ this is given by the first assumption in the theorem.
Assume that the diagram commutes for a
$k\in\left\\{0,\ldots,\binom{l}{2}-1\right\\}$, and choose

$\mathbf{x}=(x_{I})_{I\in\operatorname{Obj}(\lozenge^{\rho})}\in\mathbb{E}^{k+1}_{\operatorname{id}}({\underline{n}})$

over a base point $m\in M$. Define
$\mathbf{y}:=(y_{I})_{I\in\operatorname{Obj}(\lozenge^{\rho})}\in\mathbb{E}^{k}_{\operatorname{id}}({\underline{n}})\subseteq\mathbb{E}^{\rm
dec,\rho}({\underline{n}})$ by

$y_{I}=\left\\{\begin{array}[]{cl}x_{I}&\text{ if $I\in\rho$ or there is
$i\leq k$ such that $J_{i}\subseteq I$},\\\ 0^{A_{\\#I}}_{m}&\text{
otherwise.}\end{array}\right.$

and
$\mathbf{z}:=(z_{I})_{I\in\operatorname{Obj}(\lozenge^{\rho})}\in\mathbb{E}^{{\rm
dec},\rho_{J_{k+1}}}({\underline{n}})$ by

$z_{I}=\left\\{\begin{array}[]{cl}y_{I}&\text{ whenever
}I\subseteq{\underline{n}}\setminus J_{k+1},\\\ x_{I}&\text{ whenever
$J_{k+1}\subseteq I$ and there is no $i\leq k$ with $J_{i}\subseteq I$,}\\\
0^{A_{\\#I}}_{m}&\text{ otherwise. }\end{array}\right.$

Here, $\rho_{J_{k+1}}$ is the $(l-1)$-coarsement of $\rho$ obtained by
replacing by $J_{k+1}$ the two sets $J_{k+1}$ is the union of. Then writing
$J_{k+1}=I_{s}\cup I_{t}$ with $1\leq s<t\leq l$, it is easy to check that

$\mathbf{x}=\mathbf{y}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{s}}}]{{\scriptscriptstyle}}{+}\left(\mathbf{0}^{{\underline{n}}}_{p^{{\underline{n}}}_{{\underline{n}}\setminus
I_{s}}(\mathbf{y})}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{t}}}]{{\scriptscriptstyle}}{+}\mathbf{z}\right)=\mathbf{y}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{t}}}]{{\scriptscriptstyle}}{+}\left(\mathbf{0}^{{\underline{n}}}_{p^{{\underline{n}}}_{{\underline{n}}\setminus
I_{t}}(\mathbf{y})}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{s}}}]{{\scriptscriptstyle}}{+}\mathbf{z}\right)\,,$

and $\mathcal{S}^{k+1}_{\operatorname{id}}(\mathbf{x})$ is then defined by

$\mathcal{S}^{k+1}_{\operatorname{id}}(\mathbf{x})=\mathcal{S}^{k}_{\operatorname{id}}(\mathbf{y})\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{s}}}]{{\scriptscriptstyle}}{+}\left(\mathbf{0}^{{\underline{n}}}_{p^{{\underline{n}}}_{{\underline{n}}\setminus
I_{s}}\bigl{(}\mathcal{S}^{k}_{\operatorname{id}}(\mathbf{y})\bigr{)}}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{t}}}]{{\scriptscriptstyle}}{+}\mathcal{S}_{\operatorname{id}}^{J^{\operatorname{id}}_{k+1}}(\mathbf{z})\right)$

As above $\mathbf{x}^{\prime}:=\Psi^{\rm dec}_{\sigma}(\mathbf{x})$ is then an
element of $\mathbb{E}^{k+1}_{\sigma}({\underline{n}})$ and the corresponding
objects $\mathbf{y}^{\prime}\in\mathbb{E}^{k}_{\sigma}({\underline{n}})$ and
$\mathbf{z}^{\prime}\in{\mathbb{E}^{{\rm
dec},\rho_{J^{\sigma}_{k+1}}}}({\underline{n}})$ are then $\Psi^{\rm
dec}_{\sigma}(\mathbf{y})$ and $\Psi^{{\rm dec}}_{\sigma}(\mathbf{z})$,
respectively. Then

$\begin{split}\Phi_{\sigma}\left(\mathcal{S}^{k+1}_{\operatorname{id}}(\mathbf{x})\right)&=\Phi_{\sigma}\left(\mathcal{S}^{k}_{\operatorname{id}}(\mathbf{y})\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{s}}}]{{\scriptscriptstyle}}{+}\left(\mathbf{0}^{{\underline{n}}}_{p^{{\underline{n}}}_{{\underline{n}}\setminus
I_{s}}\bigl{(}\mathcal{S}^{k}_{\operatorname{id}}(\mathbf{y})\bigr{)}}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus
I_{t}}}]{{\scriptscriptstyle}}{+}\mathcal{S}_{\operatorname{id}}^{J^{\operatorname{id}}_{k+1}}(\mathbf{z})\right)\right)\\\
&=\Phi_{\sigma}\left(\mathcal{S}_{\operatorname{id}}^{k}(\mathbf{y})\right)\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus\sigma(I_{s})}}]{{\scriptscriptstyle}}{+}\left(\mathbf{0}^{{\underline{n}}}_{p^{{\underline{n}}}_{{\underline{n}}\setminus\sigma(I_{s})}\bigl{(}\Phi_{\sigma}\left(\mathcal{S}_{\operatorname{id}}^{k}(\mathbf{y})\right)\bigr{)}}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus\sigma(I_{t})}}]{{\scriptscriptstyle}}{+}\Phi_{\sigma}\left(\mathcal{S}^{J^{\operatorname{id}}_{k+1}}(\mathbf{z})\right)\right)\\\
&=\mathcal{S}^{k}_{\sigma}\left(\Psi^{\rm
dec}_{\sigma}(\mathbf{y})\right)\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus\sigma(I_{s})}}]{{\scriptscriptstyle}}{+}\left(\mathbf{0}^{{\underline{n}}}_{p^{{\underline{n}}}_{{\underline{n}}\setminus\sigma(I_{s})}\bigl{(}\mathcal{S}^{k}_{\sigma}\left(\Psi^{\rm
dec}_{\sigma}(\mathbf{y})\right)\bigr{)}}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle{\underline{n}}\setminus\sigma(I_{t})}}]{{\scriptscriptstyle}}{+}\mathcal{S}^{J^{\sigma}_{k+1}}\left(\Psi_{\sigma}^{\rm
dec}(\mathbf{z})\right)\right)\\\ &=\mathcal{S}^{k+1}_{\sigma}\left(\Psi^{\rm
dec}_{\sigma}(\mathbf{x})\right)\end{split}$

follows from the induction hypothesis and (42). ∎

######  Proposition B.3.

Let $\mathbb{E}\colon\square^{n}\to{\mathbf{Man}^{\infty}}$ be a symmetric
$n$-fold vector bundle and consider an $l$-partition $\rho$ of
${\underline{n}}$. Assume that

  1. (1)

For each $\rho^{\prime}$ in the $S_{n}$-orbit of $\rho$ and for each
$(l-1)$-coarsement $\underline{\rho}$ of $\rho^{\prime}$ there is a
decomposition $\mathcal{S}^{\underline{\rho}}\colon\mathbb{E}^{{\rm
dec},{\underline{\rho}}}\to\mathbb{E}^{\underline{\rho}}$.

  2. (2)

For each pair $\rho_{1},\rho_{2}$ of $(l-1)$-coarsements of a partition
$\rho^{\prime}$ in the $S_{n}$-orbit of $\rho$, the decompositions
$\mathcal{S}^{\rho_{1}}$ and $\mathcal{S}^{\rho_{2}}$ are compatible as in
(21).

  3. (3)

For each $\sigma\in S_{n}$ and each coarsement $\underline{\rho}$ of $\rho$

$\Phi_{\sigma}\circ\mathcal{S}^{\underline{\rho}}=(\mathcal{S}^{\sigma(\underline{\rho})})^{\sigma}\circ\Psi^{\rm
dec}_{\sigma}\colon(\mathbb{E}^{\rm
dec})^{\underline{\rho}}\to\mathbb{E}^{\sigma(\underline{\rho})}.$

Then for each $\rho^{\prime}$ in the $S_{n}$-orbit of $\rho$ there exists a
linear splitting

$\Sigma^{\rho^{\prime}}\colon\overline{\mathbb{E}^{\rho^{\prime}}}\to\mathbb{E}^{\rho^{\prime}}$

of $\mathbb{E}^{\rho^{\prime}}$ that is compatible as in (22) with all
decompositions $\mathcal{S}^{\underline{\rho}}$ for all coarsements
$\underline{\rho}$ of $\rho^{\prime}$, and such that

$\Phi_{\sigma}\circ\Sigma^{\rho}=(\Sigma^{\sigma(\rho)})^{\sigma}\circ\Psi_{\sigma}.$

for all $\sigma\in S_{n}$.

###### Proof.

Recall that the following claim is proved in [HL20, Theorem 3.5], see also
Section 4.4: Given an $n$-fold vector bundle $\mathbb{E}$, with linear
splittings $\Sigma_{I}$ of $\mathbb{E}^{I,\emptyset}$ for all
$I\subsetneq{\underline{n}}$ with $\\#I=n-1$, such that
$\Sigma_{I_{1}}(J)=\Sigma_{I_{2}}(J)$ whenever $J\subseteq I_{1}\cap I_{2}$,
then there exists a linear splitting $\Sigma$ of $\mathbb{E}$ with
$\Sigma(J)=\Sigma_{I}(J)$ whenever $J\subseteq I\subseteq{\underline{n}}$.

For a partition $\rho$ of ${\underline{n}}$ each
$I\in\operatorname{Obj}(\lozenge^{\rho})$ defines a face $\mathbb{E}^{\rho,I}$
of $\mathbb{E}^{\rho}$. It is the restriction of $\mathbb{E}$ to the full
subcategory of $\lozenge^{\rho}$ with objects contained in $I$.

Choose an $l$-partition $\rho$ of ${\underline{n}}$. Take an $l$-partition
$\rho^{\prime}=\\{I_{1},\ldots,I_{l}\\}$ of ${\underline{n}}$ in the
$S_{n}$-orbit of $\rho$ and choose an $(l-1)$-coarsement $\underline{\rho}$ of
$\rho^{\prime}$. Then there exist $1\leq s<t\leq l$ such that

$\underline{\rho}=\\{I_{s}\cup
I_{t},I_{1},\ldots,I_{s-1},I_{s+1},\ldots,I_{t-1},I_{t+1},\ldots,I_{l}\\}=:\rho_{st}.$

The decomposition $\mathcal{S}^{\underline{\rho}}$ of
$\mathbb{E}^{\underline{\rho}}$ defines then a decomposition of the face
$\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{t})}$ of
$\mathbb{E}^{\rho^{\prime}}$ since
$\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup
I_{t})}=\mathbb{E}_{\underline{\rho},{\underline{n}}\setminus(I_{s}\cup
I_{t})}$ is also a face of $\mathbb{E}^{\underline{\rho}}$. Denote by
$\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{t})}$ the induced
linear splitting of
$\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{t})}$.

Fix $s\in\\{1,\ldots,l\\}$. Then as above the $(l-2)$-fold vector bundles
$\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{t})}$, for
$t\in{\underline{l}}\setminus\\{s\\}$, are all sides of the $(l-1)$-fold
vector bundle $\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus I_{s}}$.
Choose $t,r\in{\underline{l}}\setminus\\{s\\}$. Then

$\rho_{st}\sqcap\rho_{sr}=\\{I_{s}\cup I_{r}\cup I_{t}\\}\cup\\{I_{x}\mid
x\in{\underline{l}}\setminus\\{r,s,t\\}\\}.$

Since $\mathcal{S}^{\rho_{st}}$ and $\mathcal{S}^{\rho_{sr}}$ coincide on the
common core $\mathbb{E}^{\rho_{st}\sqcap\rho_{sr}}$ of
$\mathbb{E}^{\rho_{st}}$ and $\mathbb{E}^{\rho_{sr}}$ by (21), they coincide
in particular on its face
$\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{r}\cup
I_{t})}$. Hence
$\mathcal{S}^{\rho_{st}}\arrowvert_{{\underline{n}}\setminus(I_{s}\cup
I_{r}\cup
I_{t})}=\mathcal{S}^{\rho_{sr}}\arrowvert_{{\underline{n}}\setminus(I_{s}\cup
I_{r}\cup I_{t})}$ and the splittings
$\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{t})}$ and
$\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{r})}$ satisfy
consequently

$\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup
I_{t})}(I)=\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup
I_{r})}(I)$

for all $I\in\operatorname{Obj}(\lozenge^{\rho^{\prime}})$ with
$I\subseteq{\underline{n}}\setminus(I_{s}\cup I_{t}\cup I_{r})$.

By the claim proved in [HL20, Theorem 3.5] there is consequently a linear
splitting $\Sigma_{\rho^{\prime},{\underline{n}}\setminus I_{s}}$ of
$\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus I_{s}}$, such that

(43) $\Sigma_{\rho^{\prime},{\underline{n}}\setminus
I_{s}}\arrowvert_{{\underline{n}}\setminus(I_{s}\cup
I_{t})}=\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup I_{t})}$

for all $t\in{\underline{l}}\setminus\\{s\\}$.

Apply this to each $l$-partition $\rho^{\prime}$ in the $S_{n}$-orbit of
$\rho$ and obtain linear splittings of the $(l-1)$-sides of
$\mathbb{E}^{\rho^{\prime}}$, that are compatible with the given linear
splittings of its $(l-2)$-sides. By the condition (2) in the hypotheses of the
theorem,

(44) $\Phi_{\sigma}(J)\circ\Sigma_{\rho^{\prime},{\underline{n}}\setminus
I}(J)=\Sigma_{\sigma(\rho^{\prime}),{\underline{n}}\setminus\sigma(I)}(\sigma(J))\circ\Psi_{\sigma}(J)$

follows immediately for all $\sigma\in S_{n}$ and for all
$J\in\operatorname{Obj}(\lozenge^{\rho^{\prime}})$ with
$J\subsetneq{\underline{n}}\setminus I$.

Now choose $\rho^{\prime}$ in the $S_{n}$-orbit of $\rho$ and for
$I\in\rho^{\prime}$ define

$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}\colon\overline{\mathbb{E}^{\rho^{\prime}}}^{{\underline{n}}\setminus
I}\to\mathbb{E}^{\rho^{\prime},{\underline{n}}\setminus I},$

by

$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}(J)=\Sigma_{\rho^{\prime},{\underline{n}}\setminus I}(J)$

for $J\in\operatorname{Obj}(\lozenge^{\rho^{\prime}})$ with
$J\subsetneq{\underline{n}}\setminus I$ and

$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus I)=\frac{1}{n!}\cdot_{K}\sum^{K}_{\sigma\in
S_{n}}\left(\Phi_{\sigma^{-1}}\circ\Sigma_{\sigma(\rho^{\prime}),{\underline{n}}\setminus\sigma(I)}\circ\Psi_{\sigma}\right)({\underline{n}}\setminus
I)$

for any $K\in\rho^{\prime}\setminus\\{I\\}$, where the scalar multiplication
$\cdot_{K}$ and the addition $\sum^{K}$ are the scalar multiplication and the
addition in the vector bundle

$p^{{\underline{n}}\setminus I}_{{\underline{n}}\setminus(I\cup
K)}\colon\mathbb{E}^{\rho^{\prime}}({\underline{n}}\setminus
I)\to\mathbb{E}^{\rho^{\prime}}({\underline{n}}\setminus(I\cup K)).$

This sum is well-defined since for all such $K$ and for all $\sigma\in S_{n}$

$\begin{split}&p^{{\underline{n}}\setminus I}_{{\underline{n}}\setminus(I\cup
K)}\circ\Phi_{\sigma^{-1}}({\underline{n}}\setminus\sigma(I))\circ\Sigma_{\sigma(\rho^{\prime}),{\underline{n}}\setminus\sigma(I)}({\underline{n}}\setminus\sigma(I))\circ\Psi_{\sigma}({\underline{n}}\setminus
I)\\\ &\,\,=\Phi_{\sigma^{-1}}({\underline{n}}\setminus\sigma(I\cup K))\circ
p^{{\underline{n}}\setminus\sigma(I)}_{{\underline{n}}\setminus\sigma(I\cup
K)}\circ\Sigma_{\sigma(\rho^{\prime}),{\underline{n}}\setminus\sigma(I)}({\underline{n}}\setminus\sigma(I))\circ\Psi_{\sigma}({\underline{n}}\setminus
I)\\\
&\overset{\eqref{comp_obtained_splittings}}{=}\Phi_{\sigma^{-1}}({\underline{n}}\setminus\sigma(I\cup
K))\circ\Sigma_{\sigma(\rho^{\prime}),{\underline{n}}\setminus\sigma(I)}({\underline{n}}\setminus\sigma(I\cup
K))\circ
p^{{\underline{n}}\setminus\sigma(I)}_{{\underline{n}}\setminus\sigma(I\cup
K)}\circ\Psi_{\sigma}({\underline{n}}\setminus I)\\\
&\overset{\eqref{condition_comp_sym_level-2}}{=}\Sigma_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus(I\cup
K))\circ\Psi_{\sigma^{-1}}({\underline{n}}\setminus\sigma(I\cup
K))\circ\Psi_{\sigma}({\underline{n}}\setminus(I\cup K))\circ
p^{{\underline{n}}\setminus I}_{{\underline{n}}\setminus(I\cup K)}\\\
&\,\,=\Sigma_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus(I\cup K))\circ p^{{\underline{n}}\setminus
I}_{{\underline{n}}\setminus(I\cup K)},\end{split}$

which does not depend on $\sigma$.

For
$\mathbf{x}\in\overline{\mathbb{E}^{\rho^{\prime}}}^{{\underline{n}}\setminus
I}({\underline{n}}\setminus I)$ and $K\neq J\in\rho^{\prime}\setminus\\{I\\}$

$\begin{split}&n!\cdot_{J}\sum^{K}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\\\
&=\underset{n!-\text{times}}{\underbrace{\left(\sum^{K}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\right)\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
J}}]{{\scriptscriptstyle}}{+}\ldots\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
J}}]{{\scriptscriptstyle}}{+}\left(\sum^{K}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\right)}}.\\\
\end{split}$

Since all summands can be added with each other over ${\underline{n}}\setminus
K$ and over ${\underline{n}}\setminus J$, a reordering of the summands over
${\underline{n}}\setminus K$ in each sum and the interchange law131313For
$n=2$ this is
$\left(\mathbf{x}_{\operatorname{id}}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
K}}]{{\scriptscriptstyle}}{+}\mathbf{x}_{(12)}\right)\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
J}}]{{\scriptscriptstyle}}{+}\left(\mathbf{x}_{(12)}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
K}}]{{\scriptscriptstyle}}{+}\mathbf{x}_{\operatorname{id}}\right)=\left(\mathbf{x}_{\operatorname{id}}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
J}}]{{\scriptscriptstyle}}{+}\mathbf{x}_{(12)}\right)\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
K}}]{{\scriptscriptstyle}}{+}\left(\mathbf{x}_{(12)}\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
J}}]{{\scriptscriptstyle}}{+}\mathbf{x}_{\operatorname{id}}\right)$.  yield

$\begin{split}&\underset{n!-\text{times}}{\underbrace{\left(\sum^{J}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\right)\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
K}}]{{\scriptscriptstyle}}{+}\ldots\stackrel{{\scriptstyle[}}{{\scriptscriptstyle
K}}]{{\scriptscriptstyle}}{+}\left(\sum^{J}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\right)}}\\\
&=n!\cdot_{K}\left(\sum^{J}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\right).\end{split}$

Thus by multiplying both sides with $\frac{1}{n!}$ over
${\underline{n}}\setminus K$ and with $\frac{1}{n!}$ over
${\underline{n}}\setminus J$:

$\begin{split}\frac{1}{n!}\cdot_{K}\left(\sum^{K}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\right)=\frac{1}{n!}\cdot_{J}\left(\sum^{J}_{\sigma\in
S_{n}}(\Phi_{\sigma^{-1}}\circ\tilde{\Sigma}_{\sigma(\rho),\sigma(I)}\circ\Psi_{\sigma})(\mathbf{x})\right).\end{split}$

Hence the definition of
$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus I}$ does not depend on
the choice of $K$, and so for each $K\in\rho^{\prime}\setminus\\{I\\}$ the map

$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus
I)\colon\overline{\mathbb{E}^{\rho^{\prime}}}({\underline{n}}\setminus
I)\to\mathbb{E}^{\rho^{\prime}}({\underline{n}}\setminus I)$

is a vector bundle morphism over

$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus(I\cup
K))\colon\overline{\mathbb{E}^{\rho^{\prime}}}({\underline{n}}\setminus(I\cup
K))\to\mathbb{E}^{\rho^{\prime}}({\underline{n}}\setminus(I\cup K)).$

Assume that

$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus
I)\left((a_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}\right)=\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus
I)\left((b_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}\right)$

for $\left((a_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}\right)$ and
$\left((b_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}\right)$ in
$\overline{\mathbb{E}^{\rho^{\prime}}}({\underline{n}}\setminus I)$. Then for
each $K\in\rho^{\prime}\setminus\\{I\\}$

$\begin{split}\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I\cup
K)}({\underline{n}}\setminus(I\cup
K))\left((a_{J})_{J\in\rho^{\prime}\setminus\\{I,K\\}}\right)&=\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus(I\cup
K))\left((a_{J})_{J\in\rho^{\prime}\setminus\\{I,K\\}}\right)\\\
&=p^{{\underline{n}}\setminus I}_{{\underline{n}}\setminus(I\cup
K)}\left(\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus
I)\left((a_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}\right)\right)\\\
&=p^{{\underline{n}}\setminus I}_{{\underline{n}}\setminus(I\cup
K)}\left(\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus
I)\left((b_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}\right)\right)\\\
&=\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}({\underline{n}}\setminus(I\cup
K))\left((b_{J})_{J\in\rho^{\prime}\setminus\\{I,K\\}}\right)\\\
&=\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I\cup
K)}({\underline{n}}\setminus(I\cup
K))\left((b_{J})_{J\in\rho^{\prime}\setminus\\{I,K\\}}\right).\end{split}$

Since $\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I\cup K)}$ is a linear
splitting of $\mathbb{E}_{\rho^{\prime},{\underline{n}}\setminus(I\cup K)}$,
hence a monomorphism of vector bundles,

$(a_{J})_{J\in\rho^{\prime}\setminus\\{I,K\\}}=(b_{J})_{J\in\rho^{\prime}\setminus\\{I,K\\}}.$

But since $K\in\rho^{\prime}\setminus\\{I\\}$ was arbitrary, this shows

$(a_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}=(b_{J})_{J\in\rho^{\prime}\setminus\\{I\\}}.$

This shows that $\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus I}$ is
a monomorphism of $(l-1)$-fold vector bundles, hence a linear splitting of the
$(l-1)$-fold vector bundle $\mathbb{E}^{\rho^{\prime},I}$.

By construction the obtained collection of linear splittings of the sides of
the $l$-fold vector bundles $\mathbb{E}^{\rho^{\prime}}$ for all
$\rho^{\prime}$ in the $S_{n}$-orbit of $\rho$

$\Phi_{\sigma}(J)\circ\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I}(J)=\tilde{\Sigma}_{\sigma(\rho^{\prime}),{\underline{n}}\setminus\sigma(I)}(\sigma(J))\circ\Psi_{\sigma}(J)$

for all $I\in\rho^{\prime}$,
$J\in\operatorname{Obj}(\lozenge^{\rho^{\prime}})$ with
$J\subseteq{\underline{n}}\setminus I$ and all $\sigma\in S_{n}$. In addition
for $I_{1}\neq I_{2}\in\rho^{\prime}$ and
$J\in\operatorname{Obj}(\lozenge^{\rho^{\prime}}$ such that

$J\subseteq({\underline{n}}\setminus I_{1})\cap({\underline{n}}\setminus
I_{2})={\underline{n}}\setminus(I_{1}\cup I_{2}),$
$\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus
I_{1}}(J)=\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{1}\cup
I_{2})}(J)=\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus I_{2}}(J).$

As a consequence, there exists a linear splitting

$\Sigma^{\rho^{\prime}}\colon\overline{\mathbb{E}^{\rho^{\prime}}}\to\mathbb{E}^{\rho^{\prime}}$

of $\mathbb{E}^{\rho^{\prime}}$ with

$\Sigma^{\rho^{\prime}}\arrowvert_{{\underline{n}}\setminus
I}=\tilde{\Sigma}_{\rho^{\prime},{\underline{n}}\setminus I}$

for all $I\in\rho^{\prime}$. As above, an averaging of the obtained family of
splittings (for all $\rho^{\prime}$ in the $S_{n}$-orbit of $\rho$) shows that
these linear splittings can be chosen such that

(45)
$\Phi_{\sigma}(J)\circ\Sigma^{\rho^{\prime}}(J)=\Sigma^{\sigma(\rho^{\prime})}(\sigma(J))\circ\Psi_{\sigma}(J)$

for all $J\in\operatorname{Obj}(\lozenge^{\rho^{\prime}})$ and all $\sigma\in
S_{n}$.

The compatibility in (22) with the decompositions of the highest order cores
is immediate by construction: For $I_{s},I_{t}\in\rho^{\prime}$ as above and
$J\in\operatorname{Obj}(\lozenge^{\rho_{st}})\subseteq\operatorname{Obj}(\lozenge^{\rho^{\prime}})$

$\Sigma^{\rho^{\prime}}(J)=\tilde{\Sigma}_{\rho,{\underline{n}}\setminus
I_{s}}(J)=\Sigma_{\rho^{\prime},{\underline{n}}\setminus(I_{s}\cup
I_{t})}(J)=\mathcal{S}^{\rho_{st}}\circ\iota(J).\qed$

###  B.1. Proof of Theorem 5.10

Finally Theorem 5.10 can be proved. This is the subject of the remainder of
this section. For simplicity, the action $\Psi^{\rm dec}$ of $S_{n}$ on the
decomposed $n$-fold vector bundle $\mathbb{E}^{\rm dec}$ is simply written
$\Psi$ in this proof. First consider all $2$-cores of $\mathbb{E}$. As
explained in Proposition 4.4, these are indexed by all possible partitions of
${\underline{n}}$ in two subsets. Consider such a partition
$\rho=\\{I,{\underline{n}}\setminus I\\}$ for $\emptyset\neq
I\subsetneq{\underline{n}}$ and choose a linear decomposition

${{E_{I}^{I}\times_{M}E^{{\underline{n}}\setminus I}_{{\underline{n}}\setminus
I}\times_{M}E_{{\underline{n}}}^{{\underline{n}}}}}$${{\mathbb{E}^{\rho}({\underline{n}})}}$${{E_{{\underline{n}}\setminus
I}^{{\underline{n}}\setminus I}}}$${{E_{{\underline{n}}\setminus
I}^{{\underline{n}}\setminus
I}}}$${{E_{I}^{I}}}$${{E_{I}^{I}}}$${M}$${M}$$\scriptstyle{\widetilde{\mathcal{S}^{\rho}}}$$\scriptstyle{\operatorname{id}\qquad}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\operatorname{id}}$

of the $2$-core $\mathbb{E}^{\rho}$, which is a double vector bundle with
sides $E_{I}^{I}$ and $E^{{\underline{n}}\setminus
I}_{{\underline{n}}\setminus I}$ and with core
$E_{{\underline{n}}}^{{\underline{n}}}$. After such a decomposition
$\mathcal{S}^{\rho^{\prime}}$ of $\mathbb{E}^{\rho^{\prime}}$ has been chosen
for each partition $\rho^{\prime}$ of ${\underline{n}}$ in two sets, choose
again a fixed such partition $\rho=\\{I,{\underline{n}}\setminus I\\}$ for
$\emptyset\neq I\subsetneq{\underline{n}}$ and define

$\mathcal{S}^{\rho}\colon\mathbb{E}^{{\rm dec},\rho}\to\mathbb{E}^{\rho}$

by

(46)
$\mathcal{S}^{\rho}({\underline{n}}):=\frac{1}{n!}\cdot_{E_{I}^{I}}\sum_{\sigma\in
S_{n}}^{E_{I}^{I}}\left(\Phi_{\sigma^{-1}}\circ\widetilde{\mathcal{S}^{\sigma(\rho)}}\circ\Psi_{\sigma}\right)({\underline{n}}),$

as well as

$\mathcal{S}^{\rho}(I)=\operatorname{id}_{E^{I}_{I}}\qquad\mathcal{S}^{\rho}({\underline{n}}\setminus
I)=\operatorname{id}_{E^{{\underline{n}}\setminus I}_{{\underline{n}}\setminus
I}}$

and $\mathcal{S}^{\rho}(\emptyset)=\operatorname{id}_{M}$, respectively. (46)
is well-defined since for all $\sigma\in S_{n}$,

$\begin{split}p^{{\underline{n}}}_{I}\circ\Phi_{\sigma^{-1}}({\underline{n}})\circ\widetilde{\mathcal{S}^{\sigma(\rho)}}({\underline{n}})\circ\Psi_{\sigma}({\underline{n}})&=\Phi_{\sigma^{-1}}(\sigma(I))\circ
p^{{\underline{n}}}_{\sigma(I)}\circ\widetilde{\mathcal{S}^{\sigma(\rho)}}({\underline{n}})\circ\Psi_{\sigma}({\underline{n}})\\\
&=\Phi_{\sigma^{-1}}(\sigma(I))\circ\operatorname{id}_{E_{\sigma(I)}^{\sigma(I)}}\circ
p^{{\underline{n}}}_{\sigma(I)}\circ\Psi_{\sigma}({\underline{n}})\\\
&=\Phi_{\sigma^{-1}}(\sigma(I))\circ\Psi_{\sigma}(I)\circ
p^{{\underline{n}}}_{I}\\\
&=\left(\epsilon(\sigma,I)\cdot\operatorname{id}_{E_{I}^{I}}\right)\circ\left(\epsilon(\sigma,I)\cdot\operatorname{id}_{E_{I}^{I}}\right)\circ
p^{{\underline{n}}}_{I}=p_{I}^{{\underline{n}}},\end{split}$

since $\Phi_{\sigma^{-1}}(\sigma(I))$ and $\Psi_{\sigma}(I)$, being the
restrictions to
$\mathbb{E}^{\sigma(\rho)}(\sigma(I))=E_{\sigma(I)}^{\sigma(I)}$ and
$\mathbb{E}^{{\rm dec},\rho}(I)=E_{I}^{I}$ of $\Phi_{\sigma^{-1}}(\sigma(I))$
and $\Psi_{\sigma}(I)$, must equal
$\epsilon(\sigma^{-1},\sigma(I))\cdot\operatorname{id}_{E_{\sigma(I)}^{\sigma(I)}}=\epsilon(\sigma,I)\cdot\operatorname{id}_{E_{I}^{I}}$
and $\epsilon(\sigma,I)\cdot\operatorname{id}_{E_{I}^{I}}$, respectively.
Further, by the interchange law, (46) does not change if $I$ is replaced by
${\underline{n}}\setminus I$.

By construction,

(47)
$\Phi_{\sigma}\circ\mathcal{S}^{\rho}=\mathcal{S}^{\sigma(\rho)}\circ\Psi_{\sigma}\colon\mathbb{E}^{{\rm
dec},\rho}\rightarrow(\mathbb{E}^{\sigma(\rho)})^{\sigma}$

for all $\sigma\in S_{n}$ and all partitions $\rho$ of ${\underline{n}}$ in
two subsets. All the $2$-partitions of ${\underline{n}}$ have the unique
common $1$-coarsement $\underline{\rho}=\\{{\underline{n}}\\}$. Since the
restriction to the ultracore of each decomposition of a $2$-core is the
identity $\operatorname{id}_{\mathbb{E}^{{\underline{n}}}_{{\underline{n}}}}$,
all the obtained $2$-core decompositions are compatible as in (21).

By Proposition B.3 there exist linear splittings of the $3$-cores of
$\mathbb{E}$, which are symmetrically compatible and also compatible with all
previously chosen decompositions of the $2$-cores. By Proposition B.2 there
exist then symmetrically compatible decompositions of all the $3$-cores of
$\mathbb{E}$, that are compatible as in (21). A recursive use of Propositions
B.3 and B.2 yields then for each $l=3,\ldots,n$ symmetrically compatible
decompositions of all $l$-cores, which are compatible as in (21). The claim is
then proved at the last step $l=n$.

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