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%% define your title in the usual way
\title{Shard polytopes}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Arnau Padrol, Vincent Pilaud and Julian Ritter]{Arnau Padrol\thanks{\href{mailto:arnau.padrol@imj-prg.fr}{arnau.padrol@imj-prg.fr}. Supp. by ANR grant CAPPS~17\,CE40\,0018.}\addressmark{1},
Vincent Pilaud\thanks{\href{mailto:vincent.pilaud@lix.polytechnique.fr}{vincent.pilaud@lix.polytechnique.fr}. Supp. by ANR CAPPS~17\,CE40\,0018 and CHARMS~19\,CE40\,0017.}\addressmark{2}, \and Julian Ritter\thanks{\href{mailto:julian.ritter@lix.polytechnique.fr}{julian.ritter@lix.polytechnique.fr}. Supp. by ANR CAPPS~17\,CE40\,0018.}\addressmark{3}
}

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\address{
	\addressmark{1}Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Universit\'e, Paris, France \\
	\addressmark{2}CNRS \& LIX, \'Ecole Polytechnique, Palaiseau, France \\
	\addressmark{3}LIX, \'Ecole Polytechnique, Palaiseau, France
}

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\abstract{
For any lattice congruence of the weak order on permutations, N.~Reading proved that glueing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called quotient fan, and V.~Pilaud and F.~Santos showed that it is the normal fan of a polytope, called quotientope. We provide an alternative simpler approach based on Minkowski sums of elementary polytopes, called shard polytopes, which have remarkable combinatorial and geometric properties. In contrast to the original construction of quotientopes, our approach extends to type $B$.
}

%% put your French abstract here, or comment this out if you don't have one
\resumetitle{R\'esum\'e}
\resume{
Pour toute congruence de treillis de l'ordre faible sur les permutations, N.~Reading a montr\'e que recoller ensemble les c\^ones de l'\'eventail de tresses qui appartiennent \`a une m\^eme classe de congruence d\'efinit un \'eventail complet, appel\'e \'eventail quotient, et V.~Pilaud et F.~Santos ont montr\'e que cet \'eventail quotient est l'\'eventail normal d'un polytope, appel\'e quotientope. Nous pr\'esentons une approche alternative bas\'ee sur des sommes de Minkowski de polytopes \'el\'ementaires, appel\'es polytopes de tessons, avec des propri\'et\'es combinatoires et g\'eom\'etriques remarquables. Contrairement \`a la construction originale des quotientopes, notre approche s'\'etend au type~$B$.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{weak order, lattice quotients, shards, type cone, deformed permutahedra}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

The \defn{weak order} is a fundamental lattice structure on the set~$\fS_n$ of permutations of~$[n]$, defined by inclusion of inversion sets: $\sigma \le \tau$ if and only if $\inv(\sigma) \subseteq \inv(\tau)$ where $\inv(\sigma) \eqdef \set{(\sigma(i), \sigma(j))}{i < j \text{ and } \sigma(i) > \sigma(j)}$.
See \cref{fig:weakOrder4} for illustrations when~$n = 4$.
Its Hasse diagram can be seen geometrically as:
\begin{compactitem}
\item the dual graph of the \defn{braid fan}~$\Fan_n$, defined by the hyperplanes~$\set{\b{x} \in \R^n}{x_i = x_j}$ for all~$1 \le i < j \le n$, directed from the region $x_1 < \dots < x_n$ to the opposite one,
\item or the graph of the \defn{permutahedron} $\Perm \eqdef \conv \set{\big( \sigma^{-1}(1), \dots, \sigma^{-1}(n) \big)}{\sigma \in \fS_n}$, oriented in the linear direction~$\gamma \eqdef (-n+1, -n+3, \dots, n-3, n-1)$.
\end{compactitem}

\begin{figure}
	\capstart
	\centerline{\includegraphics[scale=.5]{weakOrderLeft4} \; \includegraphics[scale=.49]{braidFanLeft4} \; \includegraphics[scale=.49]{permutahedronLeft4}}
	\caption{The Hasse diagram of the weak order on~$\fS_4$ (left) can be seen as the dual graph of the braid fan~$\Fan_4$ (middle) or as the graph of the permutahedron~$\Perm[4]$ (right).}
	\label{fig:weakOrder4}
\end{figure}

\begin{figure}[b]
	\capstart
	\centerline{\includegraphics[scale=.5]{weakOrderCongruence4} \; \includegraphics[scale=.39]{TamariLattice4} \includegraphics[scale=.49]{associahedron4}}
	\caption{The quotient of the weak order by the sylvester congruence (left) is the Tamari lattice (middle), and its quotient fan is the normal fan of the associahedron (right).}
	\label{fig:Tamari4}
\end{figure}

Here, we discuss similar geometric realizations for lattice quotients of the weak order.
A \defn{lattice congruence} is an equivalence relation~$\equiv$ that respects meets and joins ($x \equiv x'$ and~$y \equiv y'$ implies $x \meet y \, \equiv \, x' \meet y'$ and~$x \join y \, \equiv \, x' \join y'$).
It defines a \defn{lattice quotient}~$\fS_n/{\equiv}$ on its equivalence classes.
The prototype is the classical \defn{Tamari lattice} on binary trees with $n$ nodes~\cite{Tamari}, seen as the quotient of the weak order on~$\fS_n$ by the \defn{sylvester congruence}.
Its Hasse diagram is the graph of the classical \defn{associahedron}~$\Asso[n]$ which admits elegant descriptions by vertices~\cite{Loday}, by facets~\cite{ShniderSternberg}, or as a Minkowski sum of the faces of the standard simplex~\cite{Postnikov}. See \cref{fig:Tamari4}.
In general, for any lattice congruence~$\equiv$ of the weak order, the Hasse diagram of the lattice quotient~$\fS_n/{\equiv}$ can be seen geometrically~as:
\begin{compactitem}
\item the dual graph of the \defn{quotient fan}~$\Fan_\equiv$ of~\cite{Reading-HopfAlgebras}, obtained by glueing together the chambers of the braid fan~$\Fan_n$ that belong to the same congruence class of~$\equiv$,
\item or the graph of a \defn{quotientope} of~\cite{PilaudSantos-quotientopes}, oriented in the direction~$\gamma$ defined above.
\end{compactitem}
The quotientopes of~\cite{PilaudSantos-quotientopes} were obtained by a careful but slightly obscur choice of right-hand sides defining an inequality normal to each ray of the braid fan.
We present here an alternative approach based on Minkowski sums of elementary polytopes. %, with remarkable combinatorial and geometric properties.

Our construction is based on arcs and shards.
An \defn{arc} is a quadruple~$\arc \eqdef (a, b, A, B)$ where~$1 \le a < b \le n$ and~$A \sqcup B = {]a,b[}$.
It is represented by a curve wiggling around points on the horizontal axis, joining~$a$ to~$b$ while passing above the points of~$A$ and below the points of~$B$.
The set of all arcs is denoted by~$\arcs_n$.
Each lattice congruence~$\equiv$ of~$\fS_n$ corresponds to an upper ideal~$\arcs_\equiv$ of the \defn{forcing order} on arcs, defined by ${(a, b, A, B) \prec (a', b', A', B')}$ if~$a \le a' < b' \le b$ and~$A' \subseteq A$ and~$B' \subseteq B$.
For instance, the sylvester congruence defining the Tamari lattice corresponds to the ideal of up arcs~$\set{(a, b, {]a,b[}, \varnothing)}{1 \le a < b \le n}$.
Geometrically, an arc~$\arc \eqdef (a, b, A, B)$ defines a \defn{shard}~$\shard(\arc)$, given by the piece of the hyperplane~$\b{x}_a = \b{x}_b$ satisfying the inequalities~${\b{x}_{a'} \le \b{x}_a = \b{x}_b \le \b{x}_{b'}}$ for all~$a' \in A$ and~$b' \in B$.
The union of the walls of the quotient fan~$\Fan_\equiv$ is the union of shards~$\shard(\arc)$ over all arcs of~$\arcs_\equiv$. See \cref{fig:shards4}.

\begin{figure}
	\capstart
	\centerline{\includegraphics[scale=.48]{braidArrangement4} \; \includegraphics[scale=.48]{shards4} \; \includegraphics[scale=.48]{LodayFan4}}
	\caption{The stereographic projection of the braid fan~$\Fan_4$ from the pole~$4321$ (left),~the corresponding shards (middle), the quotient fan by the sylvester congruence~(right).}
	\label{fig:shards4}
\end{figure}

The central idea of our work is to realize the quotient fan~$\Fan_\equiv$ as a Minkowski sum where each summand is responsible for some shards of~$\arcs_\equiv$ to appear in the normal fan.
To illustrate this idea, let us start with a simple construction.
For any arc~$\arc$, denote by~$\arcs_\arc$ the arc ideal generated by~$\arc$.
The corresponding congruence~$\equiv_\arc$ is a \defn{Cambrian congruence}~\cite{Reading-CambrianLattices}, and the quotient fan~$\Fan_\arc$ is the normal fan of the \defn{$\arc$-associahedron}~$\Asso[\arc]$~\cite{HohlwegLange}.

\begin{theorem}
\label{thm:main1}
Consider an arbitrary congruence~$\equiv$ of the weak order, and let~$\arc_1, \dots, \arc_p$ denote the arcs generating the ideal~$\arcs_\equiv$. Then the quotient fan~$\Fan_\equiv$ is
\begin{compactitem}
\item the common refinement of the Cambrian fans~$\Fan_{\arc_1}, \dots, \Fan_{\arc_p}$, and
\item the normal fan of the Minkowski sum of the associahedra~$\Asso[\arc_1], \dots, \Asso[\arc_p]$.
\end{compactitem}
\end{theorem}

This construction was already used for certain specific quotients but never exploited for arbitrary lattice congruences.
In contrast to the intricate construction of~\cite{PilaudSantos-quotientopes}, \cref{thm:main1} has the advantage to transfer the geometric difficulty into the construction of the $\arc$-associahedron~$\Asso[\arc]$, already done in~\cite{HohlwegLange}.
Here, each $\arc_i$-associahedron $\Asso[\arc_i]$ is responsible for the shards of the ideal~$\arcs_{\arc_i}$ to appear in the normal fan of the Minkowski~sum.

We already mentioned that the classical associahedron decomposes as the Minkowski sum of faces of the standard simplex~\cite{Postnikov}.
In general, the $\arc$-associahedron can be decomposed further into Minkowski sums of more elementary pieces.
The main idea of our work is to push this idea forward, looking for the most possible elementary pieces.

\begin{theorem}
\label{thm:main2}
For each arc~$\alpha$, there exists a Minkowski indecomposable polytope~$\shardPolytope$, called \defn{shard polytope}, such that the union of the walls of the normal fan of the shard polytope~$\shardPolytope$ contains the shard~$\shard(\arc)$ and is contained in the union of the shards~$\shard(\arc')$ for all arcs~$\arc'$ forcing~$\arc$.
\end{theorem}

We describe these polytopes by a simple combinatorial construction in \cref{subsec:definitionShardPolytopes}, and as matroid polytopes of series-parallel graphs in \cref{subsec:matroidPolytopes}.

As a consequence of their normal fan property, we can use shard polytopes to construct polytopal realizations of quotient fans, in a similar way as we used $\arc$-associahedra in \cref{thm:main1}.
Each shard polytope~$\shardPolytope$ is now responsible for the shard~$\shard(\arc)$ to appear in the normal fan, and the remaining of the normal fan of~$\shardPolytope$ does not mess up the picture since it is contained in the union of the shards~$\shard(\arc')$ for all arcs~$\arc'$ forcing~$\arc$.

\begin{theorem}
\label{thm:main3}
For any lattice congruence~$\equiv$ of the weak order and positive coefficients~$\coeffSP_\arc \in \R_{>0}^{\arcs_\equiv}$, the quotient fan~$\Fan_\equiv$ is the normal fan of the Minkowski sum~$\shardPolytope[\arcs_\equiv] \eqdef \sum_{\arc \in \arcs_\equiv} \coeffSP_\arc \, \shardPolytope$.
\end{theorem}

Already setting the coefficients~$\coeffSP_\arc = 1$, this construction recovers (up to translation) relevant realizations of some specific quotient fans mentioned above: the classical associahedron of~\cite{ShniderSternberg, Loday, Postnikov} for the sylvester congruence and the $\arc$-associahedron of~\cite{HohlwegLange} for the $\arc$-Cambrian congruence.
More generally any quotientope of~\cite{PilaudSantos-quotientopes} is a Minkowski sum of dilated shard polytopes (up to translation).
More details are given in \cref{subsec:quotientopes}.

In fact, if we allow for Minkowski sums and differences, the family of shard polytopes provides a relevant Minkowski basis of the space of \defn{deformed permutahedra} of~\cite{Postnikov,PostnikovReinerWilliams} (or ``generalized permutahedra'', those polytopes whose normal fan coarsens the braid fan). % up to translation.

\begin{theorem}
\label{thm:main4}
Up to translation, any deformed permutahedron has a unique decomposition as a Minkowski combination~$\deformedPermutahedronSP \eqdef \sum_{\arc \in \arcs_n} \coeffSP_\arc \, \shardPolytope[\arc]$ of shard polytopes, with~$\coeffSP_\arc \in \R$ for~$\arc \in \arcs_n$.
\end{theorem}

This statement and the exchange matrix with the classical basis of faces of the standard simplex~\cite{ArdilaBenedettiDoker} are presented in \cref{subsec:MinkowskiBasis}, and used to the compute the (mixed) volumes of shard polytopes as reported in \cref{subsec:mixedVolumes}.

Finally, our long term objective is to extend the construction of shard polytopes to lattices of regions of hyperplane arrangements beyond the braid arrangement (see~\cite{Reading-chapters} for an introduction to the topic).
We achieve the first step in this perspective by constructing shard polytopes for the type~$B$ Coxeter group in \cref{sec:typeB}.
They provide elementary pieces for the first construction of quotientopes for all lattice quotients of the type~$B$ weak order, and the first natural Minkowski basis for type~$B$ deformed permutahedra.

Many details and all proofs are omitted in this extended abstract for space reason, but a complete treatment can be found in the long version of this work~\cite{PadrolPilaudRitter}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Shard polytopes}
\label{sec:shard polytopes}

%%%%%%%

\subsection{Definition and basic properties}
\label{subsec:definitionShardPolytopes}

\begin{definition}
For an arc~$\arc \eqdef (a, b, A, B)$, we define
\begin{compactitem}
\item an \defn{$\arc$-alternating matching} as a (possibly empty) sequence~$M = \{a_1, b_1, \dots, a_k, b_k\}$ with ${a \le a_1 < b_1 < \dots < a_k < b_k \le b}$ and~$a_i \! \in \! \{a\} \cup A$ while~$b_i \! \in \! B \cup \{b\}$ for~${i \in [k]}$,
\item the \defn{characteristic vector} of the $\arc$-alternating matching~$M$ as~$\chi(M) = \sum_{i \in [k]} \b{e}_{a_i} - \b{e}_{b_i}$,
\item an \defn{$\arc$-fall} (resp.~\defn{$\arc$-rise}) as a position~$j \in [a,b[$ such that~$j \in \{a\} \cup A$ and~$j+1 \in B \cup \{b\}$ (resp.~such that~$j \in \{a\} \cup B$ and~$j+1 \in A \cup \{b\}$).
\end{compactitem}
\end{definition}

\begin{proposition}
\label{prop:shardPolytope}
The \defn{shard polytope}~$\shardPolytope$ of an arc~$\arc$ is the polytope defined equivalently as
\begin{compactitem}
\item the convex hull of the characteristic vectors of all $\arc$-alternating matchings,
\item the subset of the hyperplane~$\hyp \eqdef \bigset{\b{x} \in \R^n}{\sum_{i \in [n]} \b{x}_i = 0}$ defined by
	\begin{compactitem}[$\circ$]
	\item $\b{x}_i = 0$ for any~$i \in [n] \ssm [a,b]$, $\b{x}_{a'} \ge 0$ for any~$a' \in A$, and $\b{x}_{b'} \le 0$ for any~$b' \in B$, 
	\item $\sum_{i \le f} \b{x}_i \le 1$ for any $\arc$-fall~$f$ and~$\sum_{i \le r} \b{x}_i \ge 0$ for any $\arc$-rise~$r$.
	\end{compactitem}
\end{compactitem}
\end{proposition}

\cref{fig:shardPolytopes} shows some shard polytopes and illustrates the following elementary properties of their vertices, edges, faces and facets, and their behavior by central symmetry.

\begin{figure}[b]
	\capstart
	\centerline{\includegraphics[width=\textwidth]{shardPolytopes}}
	\vspace{-.3cm}
	\caption{Some shard polytopes with~$n = 4$ (by convention $\bullet = 1$, $\cdot = 0$ and $\circ = -1$).}
	\label{fig:shardPolytopes}
\end{figure}

\begin{proposition}
\label{prop:elemPropShardPolytope}
For any arc~$\arc \eqdef (a, b, A, B)$,
\begin{compactitem}
\item the shard polytope~$\shardPolytope$ has dimension~$b-a$,
\item the vertices of~$\shardPolytope$ are precisely all characteristic vectors of $\arc$-alternating matchings,
\item two $\arc$-alternating matchings~$M, M'$ form an edge of~$\shardPolytope$ if and only if~${|M \symdif M'| = 2}$,
\item any face of~$\shardPolytope$ is a Cartesian product of shard polytopes,
\item the facets of~$\shardPolytope$ are precisely defined by the inequalities of \cref{prop:shardPolytope}.
\end{compactitem}
\end{proposition}

\begin{proposition}
\label{prop:symmetriesShardPolytope}
Consider the central symmetries on arcs~$\theta(a,b,A,B) = (\bar b, \bar a, \bar B, \bar A)$ and on vectors~$\Theta(\b{e}_i) = -\b{e}_{\bar i}$ where~$\bar i \eqdef n+1-i$.
Then~$\shardPolytope[\theta(\arc)] = \Theta(\shardPolytope)$ for any arc~$\arc$.
\end{proposition}

Finally, the central property of shard polytopes is the following.

\begin{proposition}
\label{prop:shardPolytopeFan}
For any arc~$\arc$, the union of the walls of the normal fan of the shard polytope $\shardPolytope$ contains the shard~$\shard(\arc)$ and is contained in the union of the shards~$\shard(\arc')$ for~$\arc'$ forcing~$\arc$.
\end{proposition}

%%%%%%%

\subsection{Shard polytopes as matroid polytopes}
\label{subsec:matroidPolytopes}

Let $\mat$ be a matroid on the ground set $[n]$ (see~\cite{Oxley} for an introduction to matroid theory).
Its \defn{matroid polytope}~$\polytope{P}_{\mat}\subset\R^n$ is the convex hull of the characteristic vectors of its bases.
The following characterization gives a geometric axiomatization of matroids, which provides directly the proof that shard polytopes are actually matroid polytopes.

\begin{theorem}[{\cite[Thm.~4.1]{GelfandGoreskyMacPhersonSerganova1987}}]
A polytope is a matroid polytope if and only if all its vertices have $0/1$ coordinates and all its edges are translations of some vectors~$\b{e}_i-\b{e}_j$ with $i \ne j$.
\end{theorem}

\begin{corollary}
The translated shard polytope~$\translatedShardPolytope \eqdef \shardPolytope + \one_{B \cup \{b\}}$ is a matroid polytope for any arc~$\arc\eqdef (a, b, A, B)$.
\end{corollary}

We can give a precise description of these matroids, which are actually certain connected series-parallel graphic matroids.
Let us recall some terminology.
A graph is \defn{series-parallel} if it can be obtained from a single edge with distinct endpoints via the operations of series extension (replacing an edge by a path of length~$2$) and parallel extension (replacing an edge by two parallel edges with the same endpoints).
The \defn{(cycle) matroid} of a connected graph $G \eqdef (V,E)$ is the matroid on $E$ whose bases are the edge sets of spanning trees of $G$.
A matroid is \defn{graphic} if it is the cycle matroid of a graph, and \defn{series-parallel} if it is the cycle matroid of a series-parallel graph, see~\cite[Sect.~5.4]{Oxley}.

\begin{definition}
\label{def:shardgraph}
For an arc $\arc \eqdef (a, b, A, B) \in \arcs_n$, let ${\{a\} \cup A = \{a = a_1 < \dots < a_{|A|+1}\}}$ and $B \cup \{b\} = \{b_1 < \dots < b_{|B|+1} = b\}$, and set $b_{0} \eqdef a-1$ for convenience.
Define the \defn{shard graph} $\Gamma_\arc$ to be the (multi-)graph with vertex set $[0, |B|+1]$ and
\begin{compactitem}
\item for each $1 \le i \le |A|+1$, an edge labeled $a_i$ joining vertex~$k$ to vertex~$|B|+1$, where ${0 \le k \le |B|}$ is such that ${b_k < a_i < b_{k+1}}$,
\item for each $1 \le j \le |B|+1$, an edge labeled $b_j$ joining vertex~$j-1$ to vertex~$j$,
\item for each $k \in [n] \ssm [a,b]$, a loop labeled by~$k$ on vertex~$|B|+1$.
\end{compactitem}
The \defn{shard matroid} of the arc~$\arc$ is the cycle matroid $\mat_\arc$ of $\Gamma_\arc$, whose ground set is $[n]$.

\begin{figure}[b]
	\capstart
	\centerline{\input{seriesParallel}}
	\caption{The graph~$\Gamma_{\arc}$ for the arc~$\arc \eqdef (1, 14, \{2, 4, 6, 9, 10, 13\}, \{3, 5, 7, 8, 11, 12\})$.}
	\label{fig:seriesParallel}
\end{figure}
\end{definition}

\begin{proposition}
\label{prop:2connected}
The graph~$\Gamma_\arc$ stripped of loops is a $2$-connected series-parallel graph.
\end{proposition}

\begin{proposition}
\label{prop:SPisMP}
The matroid polytope of the shard matroid $\mat_\arc$ is the translated shard polytope~${\translatedShardPolytope \eqdef \shardPolytope + \one_{B \cup \{b\}}}$.
\end{proposition}

%%%%%%%

\subsection{Quotientopes from shard polytopes}
\label{subsec:quotientopes}

We now construct polytopal realizations of quotient fans in the same spirit as in \cref{thm:main1}, but using shard polytopes rather than associahedra as elementary summands.
The following statements immediately follow from \cref{prop:shardPolytopeFan,prop:symmetriesShardPolytope}.

\begin{figure}[b]
	\capstart
	\centerline{\includegraphics[width=.99\textwidth]{shardPolytopeSums3}}
	\caption{Minkowski sums~$\shardPolytope[\arcs]$ for all arc ideals~$\arcs \subseteq \arcs_3$ containing the basic~arcs.}
	\label{fig:shardPolytopeSums3}
\end{figure}

\begin{proposition}
\label{prop:MinkowskiSumShardPolytopes}
For any lattice congruence~$\equiv$ of the weak order, the quotient fan~$\Fan_\equiv$ is the normal fan of the Minkowski sum~$\shardPolytope[\arcs_\equiv] \eqdef \sum_{\arc \in \arcs_\equiv} \shardPolytope$.
\end{proposition}

\begin{proposition}
\label{coro:symmetriesMinkowskiSumShardPolytopes}
If an arc ideal~$\arcs$ is centrally symmetric, then~$\shardPolytope[\arcs] = \Theta(\shardPolytope[\arcs])$.
\end{proposition}

Observe that the quotient fan~$\Fan_\equiv$ is actually the normal fan of any Minkowski sum~$\sum_{\arc \in \arcs_\equiv} \coeffSP_\arc \, \shardPolytope$ with~$\coeffSP_\arc > 0$ for any~$\arc \in \arcs_\equiv$, see \cref{thm:main3}.
We stick with coefficients~$\coeffSP_\arc = 1$ as this convention recovers the original constructions of~\cite{Loday, HohlwegLange} as described in \cref{exm:LodayAssoMinkowskiSum,exm:HohlwegLangeAssoMinkowskiSum}.
The following four examples are illustrated in \cref{fig:shardPolytopeSums3}.

\begin{example}
\label{exm:ZonoMinkowskiSum}
For basic arcs, the $(i, i+1, \varnothing, \varnothing)$-alternating matchings are~$\varnothing$ and~${\{i, i+1\}}$, thus the shard polytope $\shardPolytope[i, i+1, \varnothing, \varnothing]$ is just the segment~$[0, \b{e}_i - \b{e}_{i+1}]$.
The Minkowski sum~$\shardPolytope[\set{(i, i+1, \varnothing, \varnothing)}{i \in [n-1]}]$ is thus the parallelotope~$\sum_{i \in [n-1]} [0, \b{e}_i - \b{e}_{i+1}]$.
\end{example}

\begin{example}
\label{exm:LodayAssoMinkowskiSum}
For up arcs, the~$(a, b, ]a,b[, \varnothing)$-alternating matchings are~$\varnothing$ and~$\{i, b\}$ for ${a \le i < b}$, thus the shard polytope $\shardPolytope[{a, b, ]a,b[, \varnothing}]$ is the translate of the standard simplex~$\simplex_{[a,b]}$ by the vector~$-\b{e}_b$.
The Minkowski sum~$\shardPolytope[\set{(a, b, {]a,b[}, \varnothing)}{1 \le a < b \le n}]$ is thus the translate by the vector~$-\sum_{i \in [n]} \b{e}_i$ of the classical associahedron of~\cite{ShniderSternberg,Loday,Postnikov}.
\end{example}

\begin{example}
\label{exm:HohlwegLangeAssoMinkowskiSum}
For the $\arc$-Cambrian congruence, the Minkowski sum~$\shardPolytope[\arcs_\arc]$ is actually the translate by the vector~$-\sum_{i \in [a,b]} \b{e}_i$ of the $\arc$-associahedron~$\Asso[\arc]$ of \cite{HohlwegLange}.
\end{example}

\begin{example}
\label{exm:weirdPermMinkowskiSum}
For the ideal of all arcs~$\arcs_n$, the Minkowski sum of all shard polytopes gives a realization of the braid fan~$\Fan_n$.
Although it is not the convex hull of all permutations of a given point as the classical permutahedron~$\Perm$, the resulting polytope has clearly centrally symmetry by~\cref{coro:symmetriesMinkowskiSumShardPolytopes}.
\end{example}

Besides these specific examples, the following statement shows that our construction strictly contains that of~\cite{PilaudSantos-quotientopes}.

\begin{proposition}
\label{prop:quotientopesMinkowskiSumsShardPolytopes}
Any quotientope of~\cite{PilaudSantos-quotientopes} is a Minkowski sum of dilated shard polytopes.
\end{proposition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Minkowski geometry of shard polytopes}

%%%%%%%

\subsection{Type cones and shard polytopes}
\label{subsec:typeCone}

A \defn{weak Minkowski summand} of a polytope~$\polytope{P}$ is a polytope $\polytope{Q}$ satisfying the following equivalent conditions:
\begin{compactitem}
\item there are a real $\lambda \ge 0$ and a polytope~$\polytope{R}$ such that ${\polytope{Q} + \polytope{R} = \lambda \polytope{P}}$,
\item the normal fan of~$\polytope{Q}$ coarsens the normal fan of~$\polytope{P}$,
\item $\polytope{Q}$ is obtained from~$\polytope{P}$ by parallelly translating its facets without passing vertices.
\end{compactitem}
The set of weak Minkowski summands of a polytope~$\polytope{P}$ has the structure of a polyhedral cone,
which is sometimes called (closed) \defn{type cone}~\cite{McMullen-typeCone} or \defn{deformation cone}~\cite{Postnikov, PostnikovReinerWilliams} of the polytope~$\polytope{P}$.
This cone has dimension equal to the number~$N$ of facets of~$\polytope{P}$, but it has a lineality space of dimension equal to the dimension~$n$ of~$\polytope{P}$ (corresponding to translations), so its intrinsic dimension is~$N-n$.
An important property is that Minkowski sums of weak Minkowski summands translate to positive combinations in the type cone.
Thus, the rays of the type cone represent \defn{Minkowski indecomposable} summands of~$\polytope{P}$.

The weak Minkowski summands of the classical permutahedron~$\Perm$ form a particularly interesting family, studied under the name \defn{generalized permutahedra} in~\cite{Postnikov,PostnikovReinerWilliams}. Here, we prefer to use the more explicit name \defn{deformed permutahedra}. They are usually described by submodular inequalities~\cite{Postnikov} or as Minkowski sums and differences of faces~$\triangle_J \eqdef \conv \set{\b{e}_j}{j \in J}$ of the standard simplex~$\triangle_{[n]}$~\cite{ArdilaBenedettiDoker} as follows.

\begin{proposition}[\cite{Postnikov}]
\label{prop:deformedPermutahedraZ}
Any deformed permutahedron can be represented as
\[
\deformedPermutahedronRHS \eqdef \bigset{\b{x} \in \R^n}{ \dotprod{\one}{\b{x}} = \b{z}_{[n]} \text{ and } \dotprod{\one_R}{\b{x}} \ge \b{z}_R \text{ for all } \varnothing \ne R \subsetneq [n]}
\]
for some~$\b{z} \in \R^{2^{[n]}}$ with $\b{z}_\varnothing = 0$ and $\b{z}_R + \b{z}_S \le \b{z}_{R \cup S} + \b{z}_{R \cap S}$ for all~$R, S \in 2^{[n]}$.
Moreover, this representation is unique if all inequalities~$\dotprod{\one_R}{\b{x}} \ge \coeffRHS_R$ are tight (as always implicitly~assumed).
\end{proposition}

\begin{proposition}[\cite{ArdilaBenedettiDoker}]
\label{prop:faceSimplexBasis}
Any deformed permutahedron has a unique decomposition as a Minkowski combination~$\deformedPermutahedronFS \eqdef \sum_{J \subseteq [n]} \coeffFS_J \, \triangle_J$ of faces of the standard simplex, with~$\coeffFS_J \in \R$ for~$J \subseteq [n]$.
\end{proposition}

\begin{proposition}[\cite{Postnikov,ArdilaBenedettiDoker}]
\label{prop:FStoRHS}
The parameters~$\coeffFS$ and~$\coeffRHS$ in \cref{prop:deformedPermutahedraZ,prop:faceSimplexBasis} are related by
\[
\coeffRHS_R = \sum_{J \subseteq R} \coeffFS_J
\qquad\text{and}\qquad
\coeffFS_J = \sum_{R \subseteq J} (-1)^{|J \ssm R|} \, \coeffRHS_R.
\]
\end{proposition}


\begin{example}
Type cones are high dimensional objects difficult to visualize.
We can however see the type cone of the $2$-dimensional permutahedron~$\Perm[3]$ by intersecting it with a hyperplane.
The resulting polytope is a triangular bipyramid illustrated in \cref{fig:submodularFunctions}.
We have located in the type polytope the shard polytopes of the four arcs of~$\arcs_3$ together with different polytopes considered along this extended abstract.
Note that the four shard polytopes are all vertices of the type polytope (see \cref{prop:SPindecomposable}) and form an affine basis of the space (see \cref{prop:shardPolytopeBasis}).

\begin{figure}[t]
	\capstart
	\centerline{\includegraphics[width=.9\textwidth]{submodularFunctions}}
	\caption{The type polytope of the permutahedron~$\Perm[3]$.}
	\label{fig:submodularFunctions}
\end{figure}
\end{example}

The main point of this section is the following statement.
It is shown in~\cite{PadrolPilaudRitter} either from a direct Minkowski indecomposability criterion, or from the description of shard polytopes as matroid polytopes of $2$-connected series-parallel graphs (see \cref{subsec:matroidPolytopes}).

\begin{proposition}
\label{prop:SPindecomposable}
For any arc~$\arc$, the shard polytope~$\shardPolytope$ is Minkowski indecomposable.
\end{proposition}

Thus, shard polytopes correspond to certain rays of the submodular cone.
However, not all indecomposable deformed permutahedra are shard polytopes. 

\begin{theorem}
\label{thm:shardPolytopesRaysTypeCone}
For any arc~$\arc \in \arcs_n$, the shard polytopes of the arcs forcing~$\arc$ are precisely (representatives of) the rays of the type cone of the $\arc$-associahedron.
\end{theorem}

We get the following result as a direct consequence of the simpliciality of the type cones of associahedra and the description of its rays~\cite{PadrolPaluPilaudPlamondon}.

\begin{corollary}
\label{coro:uniqueDecompositionCambrian}
Any polytope whose normal fan is the $\arc$-Cambrian fan~$\Fan_\arc$ has a unique decomposition (up to translation) as a Minkowski sum of dilated shard polytopes~$\shardPolytope[\arc']$ for~$\arc'$ forcing~$\arc$.
\end{corollary}

\begin{remark}
\label{rem:shardPolytopesAlreadyExisted}
\cref{thm:shardPolytopesRaysTypeCone} connects shard polytopes to other interpretations of the rays of the type cone of the Cambrian fans: as Newton polytopes of $F$-polynomials of cluster variables of acyclic type~$A$ cluster algebras~\cite{BazierMatteDouvilleMousavandThomasYildirim}, and as brick polytope summands of certain sorting networks~\cite{JahnLoweStump}.
We skip all precise definitions here as these interpretations are not needed in the rest of our discussion.
We are not aware that our vertex and facet descriptions from \cref{prop:shardPolytope} have been observed earlier for these polytopes.
\end{remark}

%%%%%%%

\subsection{Minkowski basis of shard polytopes}
\label{subsec:MinkowskiBasis}

It turns out that shard polytope also provide a relevant Minkowski basis for the space of deformed permutahedra, similar to faces of the standard simplex (see \cref{prop:faceSimplexBasis}).

\begin{proposition}
\label{prop:shardPolytopeBasis}
Up to translations, any deformed permutahedron has a unique decomposition as a Minkowski combination~$\deformedPermutahedronSP \eqdef \sum_{\arc \in \arcs_n} \coeffSP_\arc \, \shardPolytope[\arc]$ of shard polytopes, with~$\coeffSP_\arc \! \in \! \R$~for~$\arc \! \in \! \arcs_n$.
\end{proposition}

We thus have three parametrizations of the space of deformed permutahedra: as Min\-kowski combinations of shard polytopes~$\deformedPermutahedronSP \eqdef \sum_{\arc \in \arcs} \coeffSP_\arc \, \translatedShardPolytope[\arc]$ (see \cref{prop:shardPolytopeBasis}), as Minkowski combinations of faces of the standard simplex~$\deformedPermutahedronFS \eqdef \sum_{J \subseteq [n]} \coeffFS_J \, \triangle_J$ (see \cref{prop:faceSimplexBasis}), or from their right hand sides as~$\deformedPermutahedronRHS$ (see \cref{prop:deformedPermutahedraZ}).
The exchange matrices between the parameters~$\coeffSP$, $\coeffFS$ and~$\coeffRHS$ are given by explicit combinatorial formulas.
Next, we describe the connection between the parameters~$\coeffSP$ and $\coeffFS$, which can be combined with \cref{prop:FStoRHS} to get the connection between the parameters~$\coeffSP$~and~$\coeffRHS$. 
Note that we only consider simplices~$\simplex_J$ with~$|J| \ge 2$, as we work up to translations.

\begin{proposition}
\label{prop:SPtoFS}
The parameters~$\coeffSP$ and~$\coeffFS$ in \cref{prop:faceSimplexBasis,prop:shardPolytopeBasis} are related by
\[
\coeffSP_\arc = \!\!\! \sum_{J \conn (A \cup \{a, b\})} \!\!\! (-1)^{|\{a,b\} \cap \{\min J, \max J\}|} \, \coeffRHS_R
\qquad\text{and}\qquad
\coeffFS_J = \!\!\!\!\!\! \sum_{\substack{\arc = (a, b, A, B) \\ (A \cup \{a,b\}) \conn J}} \!\!\!\!\!\! (-1)^{|J\cap (B\cup\{a,b\})|} \, \coeffSP_\arc,
\]
where we write~$I \conn J$ when~$\{\min J, \max J\} \subseteq {]\min I, \max I[} \symdif I$ and ${]\min J, \max J[} \cap I \subseteq J$.
\end{proposition}

%%%%%%%

\subsection{Mixed volumes of shard polytopes}
\label{subsec:mixedVolumes}

The \defn{mixed volume} is the unique function~$\Vol(-,\cdots,-)$ on $n$-tuples of polytopes such that
\(
\Vol(y_1 \polytope{P}_1 + \dots + y_m \polytope{P}_m) = \sum_{(i_1, \dots, i_n) \in \binom{[m]}{n}} \Vol(\polytope{P}_{i_1}, \dots, \polytope{P}_{i_n}) \, y_{i_1} \dots y_{i_n}
\)
for any collection of $m \ge n$ polytopes $\polytope{P}_1, \dots, \polytope{P}_m$ and any real numbers $y_1, \dots, y_m$ such that~${y_1 \polytope{P}_1 + \dots + y_m \polytope{P}_m}$ is a polytope.
Note that $\Vol(\polytope{P}, \dots, \polytope{P}) = \Vol(\polytope{P})$ and that mixed volumes are multilinear.
Via \cref{prop:SPtoFS}, we can thus compute (mixed) volumes of shard polytopes using mixed volumes of simplices already computed by A.~Postnikov in \cite{Postnikov}.

\begin{lemma}[\cite{Postnikov}]
The mixed volume of the faces~$\triangle_{J_1}, \dots, \triangle_{J_{n-1}}$ of the standard simplex is
\[
\Vol(\triangle_{J_1}, \dots, \triangle_{J_{n-1}}) \! = \!
\begin{cases}
	\displaystyle \! \frac{1}{(n-1)!} \!\!\! & \begin{array}{@{}l@{}} \text{if } J_1, \dots, J_{n-1} \text{ satisfy the \defn{dragon marriage condition} of \cite{Postnikov}:} \\[-.1cm] |J_{i_1} \cup \cdots \cup J_{i_k}| \ge k+1 \text{ for any distinct } i_1, \dots, i_k \in [n-1]\end{array} \\
	0 & \text{otherwise.}
\end{cases}
\]
\end{lemma}

\begin{theorem}
For any arcs~$\arc_1, \dots, \arc_{n-1} \in \arcs_n$, the mixed volume of~$\shardPolytope[\arc_1], \dots, \shardPolytope[\arc_{n-1}]$ is
\[
\Vol(\shardPolytope[\arc_1], \dots, \shardPolytope[\arc_{n-1}]) = \frac{1}{(n-1)!}\sum_{J_1, \dots, J_{n-1}} (-1)^{\sum_{i \in [n-1]} |J_i \cap (B_i \cup \{a_i, b_i\})|}
\]
summing over all~$J_1, \dots, J_{n-1}$ with~$|J_i| \ge 2$ and $(A_i \cup \{a_i,b_i\}) \conn J_i$ for all $i \in [n-1]$, and such that $J_1, \dots ,J_{n-1}$ satisfy the dragon marriage condition.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Type~$B$ shard polytopes}
\label{sec:typeB}

\begin{figure}[b]
	\capstart
	\centerline{\includegraphics[scale=.85]{B2CoxeterArrangement} \includegraphics[scale=.85]{B2shards}}
	\caption{The type~$B_2$ Coxeter fan~$\BFan_2$ (left) and the corresponding $B$-shards (right).}
	\label{fig:B2shards}
\end{figure}

Based on \cref{prop:symmetriesShardPolytope,coro:symmetriesMinkowskiSumShardPolytopes}, we extend shard polytopes to the type~$B_n$ Coxeter group.
It is the group of permutations~$\sigma$ of~$[\pm n] = \{-n, \dots, -1\} \cup \{1, \dots, n\}$ such that $\sigma(-i) = -\sigma(i)$ for all~$i \in [\pm n]$.
The following are the analogues~of~arcs.

\begin{definition}
A \defn{$B$-arc} on $[\pm n]$ is either a centrally symmetric $A$-arc on~$[\pm n]$ or a centrally symmetric and noncrossing pair of $A$-arcs on~$[\pm n]$ with disjoint endpoints.
\end{definition}

As in type~$A$, there is a forcing order on $B$-arcs, and the lattice congruences of the type~$B_n$ weak order correspond to the upper ideals in forcing order.
Geometrically, the type~$B_n$ arrangement is defined by the hyperplanes~$\set{\b{x} \in \R^n}{\b{x}_a = \b{x}_b}$ for~${a < b \in [\pm n]}$, with the convention that~$\b{x}_{-i} \eqdef -\b{x}_i$.
Each $B$-arc~$\Barc \eqdef (-\arc, \arc)$ with~$\arc \eqdef (a, b, A, B)$ corresponds to a shard~$\shard(\Barc)$ defined as the piece of the hyperplane~$\b{x}_a = \b{x}_b$ satisfying the inequalities~${\b{x}_{a'} \le \b{x}_a = \b{x}_b \le \b{x}_{b'}}$ for all~$a' \in A$ and~$b' \in B$, again with the convention that~$\b{x}_{-i} = -\b{x}_i$.
See \cref{fig:B2shards}.
The following are the analogues of shard polytopes.

\begin{definition}
\label{def:BshardPolytope}
The \defn{shard polytope}~$\shardPolytope[\Barc]$ of a $B$-arc~$\Barc \eqdef (-\arc, \arc)$ is the convex hull of the characteristic vectors of all $\arc$-alternating matchings, with the convention that~$\b{e}_{-i} = -\b{e}_i$.
\end{definition}

Again, these polytopes are designed to fulfill the following normal fan property.

\begin{proposition}
\label{prop:shardPolytopeFanB}
For any $B$-arc~$\Barc$, the union of the walls of the normal fan of the shard polytope $\shardPolytope[\Barc]$ contains the shard~$\shard(\Barc)$ and is contained in the union of the shards~$\shard(\Barc')$ for~$\Barc'$ forcing~$\Barc$.
\end{proposition}

\begin{corollary}
\label{coro:MinkowskiSumShardPolytopesB}
For any $B$-arc ideal~$\Barcs \subseteq \Barcs_n$, the quotient fan~$\BFan_{\Barcs}$ is the normal fan of the Minkowski sum~$\shardPolytope[\Barcs] \eqdef \sum_{\Barc \in \Barcs} \shardPolytope[\Barc]$ of the shard polytopes~$\shardPolytope[\Barc]$ of all $B$-arcs~${\Barc \in \Barcs}$.
\end{corollary}

Finally, we conjecture that type~$B$ shard polytopes are Minkowski indecomposable. In any case, we prove that they form a Minkowski basis for type~$B$ deformed permutahedra.

\begin{theorem}
Up to translation, any type~$B$ deformed permutahedron has a unique decomposition as a Minkowski sum and difference of dilated type~$B$ shard polytopes.
\end{theorem}

%\acknowledgements{}

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