\section{Notation and preliminaries}\label{sec:02-notation}
\begin{setup}\label{setup:standing}
Throughout the paper, $g,n\geq0$ satisfy $2g-2+n>0$, and we call such $(g,n)$ a stable pair. Let $\Mbar_{g,n}$ denote the
smooth proper moduli stack of stable curves with ordered markings. The
symmetric group $\Sn_n$ acts by relabelling.
Rational cohomology of a finite quotient stack is the invariant part
of the rational cohomology of its cover, and averaging makes
finite-group invariants exact; in particular the multiplicity functors
$\operatorname{Hom}_{\mathbb Q[\Sn_n]}(V_\lambda,-)$ are exact.

All tensor products of multiplicity spaces are over $\mathbb Q$, followed by scalar
extension to $\mathbb Q_\ell$ in an $\ell$-adic realization.
\end{setup}
\begin{notation}\label{def:motivic-pieces}
The objects $\Lmot,\Soneone,\Pmot,\Sonesix$, their Hodge types, and the
modules $\Khook_n$ and $V_\lambda$ are as in Section~\ref{sec:01-introduction}.
\end{notation}
\begin{notation}\label{not:symmetric-functions}
Let $\Lambda$ be the ring of symmetric functions over $\mathbb Q$, with
the power-sum, complete homogeneous and elementary symmetric functions
$p_d,h_d,e_d$ ($d\ge1$).
Set $h_0=e_0=1$ and $h_d=0$ for $d<0$.
One has $\Lambda=\mathbb Q[p_1,p_2,\ldots]$, graded by $\deg p_d=d$.
If $\Lambda_n$ denotes its degree-$n$ component, write
$\widehat\Lambda=\prod_{n\ge0}\Lambda_n$ for the completion by degree,
and $[F]_n$ for the degree-$n$ component of $F\in\widehat\Lambda$.
All infinite symmetric-function sums below are interpreted in this
completion. In particular, set
\[
 E=\sum_{d\ge0}h_d,\qquad E_{\ge j}=\sum_{d\ge j}h_d\quad(j\ge0).
\]
\end{notation}
\begin{notation}\label{not:characters}
For a partition $\lambda=(\lambda_1\ge\cdots\ge\lambda_\ell>0)$ of $n$,
let $V_\lambda$ be the corresponding rational Specht module and
let $s_\lambda\in\Lambda_n$ be the Schur function.
It is homogeneous of degree $n$; in particular,
$s_{(d)}=h_d$ and $s_{(1^d)}=e_d$ for $d\ge1$.
Write $p_\lambda=\prod_{j=1}^{\ell}p_{\lambda_j}$, with
$p_{\varnothing}=1$. For $\sigma\in\Sn_n$, the partition
$\operatorname{type}(\sigma)$ records its cycle lengths.
For a symmetric function $f$, write $f^\perp$ for the adjoint of
multiplication by $f$ under the Hall inner product, in which the
Schur functions are orthonormal.
\end{notation}
The Frobenius characteristic of a rational $\Sn_n$-module $V$ is
\[
\ch_n(V)=\frac1{n!}\sum_{\sigma\in\Sn_n}
\tr(\sigma\mid V)p_{\operatorname{type}(\sigma)}.
\]
It satisfies $\ch_n(V_\lambda)=s_\lambda$. Thus the coefficient
$[s_\lambda]\ch_n(V)$ is the multiplicity of $V_\lambda$ in $V$.

For convenience, we set $s_{(a,1^r)}=0$ when $a\le0$.
Plethysm uses $\Psi_r(p_i)=p_{ri}$ and $\Psi_r(\Lmot)=\Lmot^r$;
$\partial$ denotes $\partial/\partial p_1$.

\begin{notation}\label{def:common-series}
Set
\[
\begin{aligned}
H&=\sum_{a\geq10}\bigl(s_{(a-10,1^{10})}+s_{(a-9,1^9)}\bigr),\\
G&=\sum_{a\geq9}\bigl(s_{(a-9,1^9)}+s_{(a-8,1^8)}\bigr),\\
U&=Es_{(2,1^{10})}+E_{\geq3}H.
\end{aligned}
\]
We use $h_2[E]=(E^2+\Psi_2(E))/2$ and
$e_2[E]=(E^2-\Psi_2(E))/2$.
\end{notation}
\begin{setup}\label{setup:intersection-conventions}
For a finite marking set $I$, we write $\Sn_I$ for its permutation
group and $\Mbar_{h,I}$ for the moduli stack with those markings.
For a stable pair $(h,|I|)$, let
\(\pi:\mathcal C_{h,I}\to\Mbar_{h,I}\)
be the universal curve, with sections $s_i$ and section divisors $D_i$,
and with $\omega_\pi$ the relative dualizing sheaf.
We use
\[
 \psi_i=c_1(s_i^*\omega_\pi),\qquad
 \kappa_1=\pi_*c_1\left(\omega_\pi\left(\sum_{i\in I}D_i\right)\right)^2.
\]
We assign polynomial degree one to each of $\psi_i$ and $\kappa_1$.
A homogeneous polynomial of degree $d$ therefore represents a class in
$H^{2d}(\Mbar_{h,I};\mathbb Q)$.
Integration means evaluation on the fundamental class of the relevant
proper moduli stack, with the normalization
$\int_{[\mathrm{pt}/A]}1=1/|A|$ for a finite group $A$.

Write $S=\Soneone$ and put
\[
 J(u,v)=\int_{\Mbar_{1,[11]}}u\smile v.
\]
The form $J$ is alternating and nondegenerate. In formulas for underlying
rational vector spaces we suppress the common Tate factor of a Gysin map.
\end{setup}

\begin{definition}[Stable graphs and clutching classes]\label{def:stable-graphs}
Recall that the strata correspond to stable graphs $\Gamma$: a vertex
set $V$ with genus function $g:V\to\mathbb Z_{\ge0}$, a half-edge set
$H$ with vertex map $v:H\to V$ and involution $\iota$; the edges $E$
are the $2$-cycles of $\iota$ (self-edges allowed), the legs $L$ its
fixed points, in bijection with the markings; $(V,E)$ is connected,
and $2g(v)-2+|F(v)|>0$ for every vertex, $F(v)$ being the flags at $v$.
The total genus of $\Gamma$ is
$
 \sum_vg(v)+|E(\Gamma)|-|V(\Gamma)|+1.
$
Graph automorphisms fix each ordinary leg label. Set
$$
 X_\Gamma=\prod_{v\in V(\Gamma)}\Mbar_{g(v),F(v)},\qquad
 \xi_\Gamma:X_\Gamma\longrightarrow\Mbar_{g,n},\qquad
 [\Gamma,\alpha]=\frac{\xi_{\Gamma*}\alpha}{|\Aut(\Gamma)|}.
$$
Here $\xi_\Gamma$ is the labelled clutching map. This normalization
agrees with integration from the quotient stack for invariant classes:
on an invariant class $\alpha$, quotient-stack pushforward is
$\xi_{\Gamma*}\alpha/|\Aut(\Gamma)|$, so that half-edge exchanges and
vertex exchanges are already incorporated and there is no further
factor of two, and the ordinary pushforward uses the complex normal
orientation, not an edge-order determinant.
Let $\mathcal G^{(1)}_{g,n}$ denote the isomorphism classes of stable
graphs of type $(g,n)$ with one unordered edge.
\end{definition}

\begin{definition}[Contractions and common refinements]\label{def:graph-contractions}\label{def:common-refinements}
For a stable graph $\Gamma$ and a subset $S\subseteq E(\Gamma)$, let
$\Gamma/S$ be the graph obtained by contracting the edges in $S$.
Each connected component $C$ of the spanning subgraph $(V(\Gamma),S)$
is replaced by a vertex of genus
\[
 \sum_{v\in V(C)}g(v)+b_1(C),\qquad
 b_1(C)=|E(C)|-|V(C)|+1.
\]
The remaining edges and labelled legs are attached to the corresponding
vertices of $\Gamma/S$.

A contraction $f:\Gamma\to\Gamma_0$ consists of a subset
$S_f\subseteq E(\Gamma)$ together with a specified isomorphism
\[
 \phi_f:\Gamma/S_f\xrightarrow{\sim}\Gamma_0
\]
of stable graphs, preserving incidence, vertex genus labels, and leg
labels. An edge is \emph{retained by $f$} if it does not belong to $S_f$.

Fix stable graphs $A$ and $B$ of the same type. A \emph{common refinement}
is a stable graph $\Gamma$ equipped with contractions
\[
 A\xleftarrow{f}\Gamma\xrightarrow{g}B.
\]
We call $(\Gamma,f,g)$ \emph{generic} if
$S_f\cap S_g=\varnothing$. The set of edges retained by both maps is
\[
 H(f,g)=E(\Gamma)\setminus(S_f\cup S_g).
\]

Two common refinements $(\Gamma,f,g)$ and $(\Gamma',f',g')$ are
isomorphic if there is an isomorphism of stable graphs
$\alpha:\Gamma\xrightarrow{\sim}\Gamma'$ such that
\[
 f=f'\circ\alpha,\qquad g=g'\circ\alpha.
\]
We write $\mathcal R(A,B)$ for the set of isomorphism classes of
generic common refinements.
\end{definition}