%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2021}
\articlenumber{61}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc
%% note that the class file already loads {amsmath, amsthm, amssymb}

\title{The polytope algebra of generalized permutahedra}
\author{Jose Bastidas\thanks{\href{mailto:bastidas@math.cornell.edu}{bastidas@math.cornell.edu}.}\addressmark{1}}
\address{\addressmark{1}Department of Mathematics, Cornell University, Ithaca, NY 14853}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use~\cite, and instead write out your citation as something like (author year)
\abstract{We study the polytope algebra of McMullen relative to a fixed zonotope. We endow the corresponding subalgebra with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. In the case of Coxeter arrangements of type A and B, we find connections with statistics on (signed) permutations and with the Hopf monoid of generalized permutahedra of Aguiar and Ardila.}
\keywords{generalized permutahedra, valuation, hyperplane arrangements, Tits algebra, Eulerian idempotent, zonotope}

%% 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
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{61Bastidas.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\usepackage{bm,enumitem,tikz,tikz-cd}
\usetikzlibrary{shapes.multipart,positioning,decorations.pathreplacing,calc,intersections,folding}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem*{theorem*}{Theorem}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{example}[theorem]{Example}
\newtheorem{ex-thm}[theorem]{Example/Theorem}
\newtheorem{convention}[theorem]{Convention}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{plain}
\definecolor{cBLalt}{HTML}{007CFC}
\definecolor{cORalt}{HTML}{F59723}
\newcommand\blue[1]{{\color{cBLalt}{#1}}}
\newcommand\red[1]{{\color{red}{#1}}}
\usepackage{array} \newcolumntype{L}{>{\displaystyle}l} \newcolumntype{C}{>{\displaystyle}c} \newcolumntype{R}{>{\displaystyle}r}
\usepackage{nameref}

\newcommand{\RR}{\mathbb{R}} % Reals
\newcommand{\ZZ}{\mathbb{Z}} % Integers
\DeclareMathOperator{\supp}{s} % support
\DeclareMathOperator{\exc}{exc}
\newcommand{\opp}[1]{\overline{#1}} 
\newcommand{\p}{{\mathfrak{p}}}
\newcommand{\q}{{\mathfrak{q}}}
\newcommand{\f}{{\mathfrak{f}}}
\newcommand{\rr}{{\mathfrak{r}}}
\newcommand{\frakz}{{\mathfrak{z}}}  
\newcommand{\frakl}{{\mathfrak{l}}}
\newcommand{\frakS}{{\mathfrak{S}}}
\newcommand{\frakB}{{\mathfrak{B}}}
\newcommand{\frakC}{{\mathfrak{C}}}
\newcommand{\arr}{\mathcal A} %hyperplane arrangement 
\newcommand{\rmX}{\mathrm{X}} % Flat
\newcommand{\rmY}{\mathrm{Y}} % Flat
\newcommand{\rmH}{{\mathrm{H}}}
\newcommand{\vH}{{\mathbf{H}}}
\newcommand{\face}{\Sigma}
\DeclareMathOperator{\relint}{relint}
\renewcommand{\flat}{\mathcal{L}}
\newcommand{\B}[1]{\mathtt{#1}} % for basis elements
\newcommand{\qqand}{\qquad\text{and}\qquad}
\DeclareMathOperator{\Conv}{Conv}
\newcommand{\bdot}{\bm\cdot}
\DeclareMathOperator{\vol}{vol}
\newcommand{\cB}{{\mathcal{B}}}

%%tikz commands

\tikzset{
  anticlockwise arc centered at/.style={
    to path={
      let \p1=(\tikztostart), \p2=(\tikztotarget), \p3=(#1) in
      \pgfextra{
        \pgfmathsetmacro{\anglestart}{atan2(\y1-\y3,\x1-\x3)}
        \pgfmathsetmacro{\angletarget}{atan2(\y2-\y3,\x2-\x3)}
        \pgfmathsetmacro{\angletarget}%
        {\angletarget < \anglestart ? \angletarget+360 : \angletarget}
        \pgfmathsetmacro{\radius}{veclen(\y1-\y3,\x1-\x3)}
      }
      arc(\anglestart:\angletarget:\radius pt) -- (\tikztotarget)
    },
  },
  clockwise arc centered at/.style={
    to path={
      let \p1=(\tikztostart), \p2=(\tikztotarget), \p3=(#1) in
      \pgfextra{
        \pgfmathsetmacro{\anglestart}{atan2(\y1-\y3,\x1-\x3)}
        \pgfmathsetmacro{\angletarget}{atan2(\y2-\y3,\x2-\x3)}
        \pgfmathsetmacro{\angletarget}%
        {\angletarget > \anglestart ? \angletarget - 360 : \angletarget}
        \pgfmathsetmacro{\radius}{veclen(\y1-\y3,\x1-\x3)}
      }
      arc(\anglestart:\angletarget:\radius pt)  -- (\tikztotarget)
    },
  },
}

\newcommand{\phantomarcThroughThreePoints}[4][]{
\coordinate (middle1) at ($(#2)!.5!(#3)$);
\coordinate (middle2) at ($(#3)!.5!(#4)$);
\coordinate (aux1) at ($(middle1)!1!90:(#3)$);
\coordinate (aux2) at ($(middle2)!1!90:(#4)$);
\coordinate (center#2) at ($(intersection of middle1--aux1 and middle2--aux2)$);
\draw[#1,draw=none,name path=kamaan#2] %kamaan#2 is the name of the arc that is being drawn, and center of the arc is center#1 
%%here we have added draw=none to suppress the arc but use the coordinates later.
let \p1=($(#2)-(center#2)$),
      \p2=($(#4)-(center#2)$),
      \n0={veclen(\p1)},       % Radius
      \n1={atan2(\y1,\x1)}, % angles
      \n2={atan2(\y2,\x2)},
      \n3={\n2>\n1?\n2:\n2+360}
    in (#2) arc(\n1:\n3:\n0);
}

\newcommand{\arcThroughThreePoints}[3]{
\phantomarcThroughThreePoints{#1}{#2}{#3}; 
\draw (#1) to[anticlockwise arc centered at=center#1] (#3); %we could give different options to draw for drawing thick arc, colored arc, etc.
}

\begin{document}

\maketitle

\section*{Introduction}
	
Generalized permutahedra have been a central object of study for many combinatorialists in recent years.
They serve as a geometric model for many classical (type A) combinatorial objects.
We study generalized permutahedra modulo certain \emph{valuation} and \emph{translation invariance} relations, previously considered in the construction of McMullen's polytope algebra, and give the resulting space the structure of a module over the Tits algebra of the braid arrangement. This structure is compatible with the Hopf monoid structure of Aguiar and Ardila~\cite{aa17}.

We review McMullen's construction of the polytope algebra and its main properties in \cref{s:polytope}. In \cref{s:Tits}, we review the Tits algebra~$\RR\face[\arr]$ of a hyperplane arrangement and some generalities about modules over~$\RR\face[\arr]$. In particular, we focus on the action of certain elements of~$\RR\face[\arr]$ that we call \emph{characteristic}. Particular cases of the action of such elements are related to the antipode problem and to the computation of polynomial invariants in Hopf monoids, this is reviewed in \cref{ss:braid}. Our main construction, the module structure on \emph{generalized zonotopes} over the Tits algebra of the corresponding arrangement, is given in \cref{s:module}. Finally, in \cref{s:cox_eul} we specialize the previous construction to the case of generalized permutahedra and type B generalized permutahedra.
%, they are zonotopes of the Coxeter arrangements of type A and B respectively. 
For full details and proofs see~\cite{bastidas2020polytope}.

\subsubsection*{Preliminaries and notation}

Let~$V$ be a real vector space of dimension~$d$ endowed with an inner product~$\langle \,\cdot\, , \cdot\, \rangle$.
We let~$o \in V$ denote the zero vector of~$V$. Given a polytope~$\p \subseteq V$ and a vector~$v \in V$, we let~$\p_v$ denote the face of~$\p$ maximized in the direction~$v$. That is,~$\p_v := \{ p \in \p \,:\, \langle p , v \rangle \geq \langle q , v \rangle \text{ for all } q \in \p\}$. 
The \textbf{normal cone} of a face~$\f$ of~$\p$ is the polyhedral cone~$N(\f,\p) := \{ v \in V \,:\, \f \leq \p_v\}$. The \textbf{normal fan} of~$\p$ is the collection~$\face_\p := \{ N(\f,\p) \,:\, \f \leq \p\}$ of all normal cones of~$\p$.

Recall that a fan~$\face$ \textbf{refines}~$\face'$ if every cone in~$\face'$ is a union of cones in~$\face$. We say that a polytope~$\q$ is a \textbf{deformation} of~$\p$ if~$\face_\p$ refines~$\face_\q$.
The \textbf{Minkowski sum} of two polytopes~$\p,\q \subseteq V$ is the polytope~$\p + \q := \{ p + q \,:\, p \in \p,\, q \in \q \}$.
We say that~$\q$ is a \textbf{Minkowski summand} of~$\p$ if~$\p = \q + \q'$ for some polytope~$\q'$.
The normal fan of~$\p + \q$ is the common refinement of~$\face_\p$ and~$\face_\q$. 
Hence,~$\face_\p$ refines the normal fan of any of its Minkowski summands.

\section{McMullen's polytope algebra}\label{s:polytope}

The \textbf{polytope algebra}~$\Pi(V)$ is generated as a group by elements~$[\p]$, one for each polytope~$\p \subseteq V$.
These generators satisfy the following \emph{valuation} and \emph{translation invariance} relations:
\begin{equation}\label{eq:valuation_property}
[\p \cup \q] + [\p \cap \q] = [\p] + [\q]
\qqand
[\p + \{t\}] = [\p]
\end{equation}
whenever~$\p$,~$\q$ and~$\p \cup \q$ are polytopes; and for any translation vector~$t \in V$. 

The product of~$\Pi(V)$ is defined on generators by means of the Minkowski sum
\begin{equation}\label{eq:product_def}
[\p] \cdot [\q] := [ \p + \q ].
\end{equation}
It readily follows from~\eqref{eq:valuation_property} that the class of a point~$1 := [\{o\}]$ is the unit of~$\Pi(V)$.

\begin{lemma}[{\cite[Lemma 13]{mcmullen89}}]
Let~$\p$ be a~$k$-dimensional polytope.
Then,
$$
([\p] - 1)^k \neq 0 \qqand ([\p] - 1)^r = 0
\quad \text{for } r > k.
$$
\end{lemma}

Thus, we can define the \textbf{log-class} of a polytope~$\p$ by means of the usual power series of~$\log(x)$ centered at~$x=1$. If~$\p$ has dimension~$k$, then
$$
\log[\p] : = \sum_{r = 1}^k \tfrac{(-1)^{r-1}}{r}([\p]-1)^r.
$$
For~$r \geq 0$ let~$\Xi_r(V)$ be the subgroup of~$\Pi(V)$ generated by elements of the form~$(\log[\p])^r$.

\begin{theorem}[{\cite[Theorem 1]{mcmullen89}}]
$\Pi(V)$ is almost a graded~$\RR$-algebra, in the following sense:
\begin{enumerate}[label=\roman*.]
\item as an abelian group,~$\Pi(V)$ admits a direct sum decomposition~$\Pi(V) = \bigoplus_{r = 0}^d \Xi_r(V)$;
\item under multiplication,~$\Xi_r(V) \cdot \Xi_s(V) = \Xi_{r+s}(V)$, with~$\Xi_r(V) = 0$ for~$r > d$;
\item~$\Xi_0(V) \cong \ZZ$, and for~$r=1,	\dots,d$,~$\Xi_r(V)$ is a real vector space;
\item the product of elements in~$\bigoplus_{r \geq 1} \Xi_r(V)$ is bilinear.
\end{enumerate}
\end{theorem}

\begin{convention}
As in later work of McMullen~\cite{mcmullen93simple}, we replace~$\Xi_0(V) \cong \ZZ$ with the tensor product~$\Xi_0(V)_\RR := \Xi_0(V) \otimes_\ZZ \RR$ to get a genuine graded~$\RR$-algebra $\Pi(V)_\RR$. To simplify notation, we drop the subscript~$\RR$.
\end{convention}

For each scalar~$\lambda \in \RR$, define the \textbf{dilation}~$\delta_\lambda : \Pi(V) \rightarrow \Pi(V)$ on generators by~$\delta_\lambda [\p] := [\lambda \p]$, where~$\lambda \p := \{ \lambda p \,:\, p \in \p\}$.
One can easily verify that~$\delta_\lambda$ preserves relations~\eqref{eq:valuation_property} and defines an algebra morphism.
The following result characterizes the graded components of~$\Pi(V)$ as the eigenspaces of any positive nontrivial dilation.

\begin{lemma}[{\cite[Lemma 20]{mcmullen89}}]
Let~$x \in \Pi(V)$ and ~$\lambda > 0$, with~$\lambda \neq 1$.
Then,
\begin{equation}\label{eq:Xi_dilation}
x \in \Xi_r(V) 
\qquad\text{if and only if}\qquad 
\delta_\lambda x = \lambda^r x.
\end{equation}
\end{lemma}

\begin{definition}\label{d:subalgebra_polytope}
Fix a polytope~$\p \subseteq V$.
The \textbf{subalgebra relative to~$\p$}, denoted~$\Pi(\p)$, is the subalgebra of~$\Pi(V)$ generated by classes~$[\q]$ of deformations~$\q$ of~$\p$.
\end{definition}

\begin{remark}
McMullen~\cite{mcmullen93simple} defines~$\Pi(\p)$ in terms of Minkowski summands of~$\p$. A result of Shephard~\cite[Section 15.2.7]{grunbaum03convex} implies that both definitions are equivalent.
\end{remark}

The grading of~$\Pi(V)$ induces a grading of~$\Pi(\p)$.
We let~$\Xi_r(\p) = \Pi(\p) \cap \Xi_r(V)$.
The dimension of these spaces was described by McMullen in the case of simple polytopes.

\begin{theorem}[{\cite[Theorem 6.1]{mcmullen93simple}}]\label{t:simple_h_dimension}
If~$\p$ is a simple polytope, then~$\dim_\RR(\Xi_r(\p)) = h_r(\p)$ for all~$r$.
\end{theorem}

Fix a face~$\f$ of~$\p$ and a direction~$v \in V$ such that~$\p_v = \f$. If~$\q$ is a Minkowski summand of~$\p$, say~$\p = \q + \q'$, then~$\f = \q_v + \q'_v$. That is,~$\q_v$ is a Minkowski summand of~$\f$.
It then follows by~\cite[Theorem 7]{mcmullen89} that there is a well-defined algebra morphism
\begin{equation}\label{eq:morphism_to_face}
\psi_\f : \Pi(\p) \rightarrow \Pi(\f)
\end{equation}
sending a generator~$[\q]$ of~$\Pi(\p)$ to~$[\q_v] \in \Pi(\f)$. One easily checks that~$\psi_\f$ is independent on the particular choice of~$v \in \relint(N(\f,\p))$.

\begin{theorem}[{\cite[Theorem 2.4]{mcmullen93simple}}]\label{t:Mc_face_surj}
Let~$\p$ be a simple polytope and~$\f$ a face of~$\p$.
Then, the morphism~$\psi_\f : \Pi(\p) \rightarrow \Pi(\f)$ is surjective.
\end{theorem}

\section{The Tits algebra of a linear hyperplane arrangement}\label{s:Tits}

Let~$\arr$ be a linear hyperplane arrangement in~$V$. The hyperplanes in~$\arr$ split~$V$ into a collection~$\face[\arr]$ of convex cones called \textbf{faces} of~$\arr$. The set~$\face[\arr]$ has the structure of a monoid under the \emph{Tits product} illustrated in \cref{f:ex_arr_prod}.
The product of two faces~$F$ and~$G$, denoted~$FG$, is the first face you encounter after moving a small positive distance from an interior point of~$F$ to an interior point of~$G$.
The unit of this product is the \textbf{central face}~$O \in \face[\arr]$: the intersection of all the hyperplanes in~$\arr$.

\newpage
The \textbf{Tits algebra} of~$\arr$ is the monoid algebra~$\RR\face[\arr]$. See~\cite[Chapters 1 and 9]{am17} for more details. We let~$\B{H}_F$ denote the basis element of~$\RR\face[\arr]$ associated to the face~$F$ of~$\arr$.

\begin{figure}
$$
\begin{gathered}
\begin{tikzpicture}[scale=.8,<->,cBLalt]
\draw (0+180:2) -- (0,0) -- (0:2);
\draw (50+180:2) -- (0,0) -- (50:2);
\draw (130+180:2) -- (0,0) -- (130:2);
\node [circle, inner sep=1.5pt,fill,draw] at (0,0) {};
\node [black,below] at (-.075,.02) {\small~$O$};
\node [left,black] at (0+180:2) {\small~$F$};
\node [black,right] at (30:1) {\small~$C$};
\node [black,left] at (150:1) {\small~$FC$};
\end{tikzpicture}
\end{gathered}
\qquad\qquad\qquad\qquad
\begin{gathered}
\begin{tikzpicture}[scale=.57,cBLalt]
\newdimen\R
\R=2.5cm
\coordinate (madhya) at (0,0);
\draw (madhya) circle (\R);
\coordinate (P) at (-1,0);
\coordinate (Q) at (1,0);
\coordinate (A1) at (54.735:\R);
\coordinate (B1) at (180+54.735:\R);
\arcThroughThreePoints{A1}{P}{B1};
\arcThroughThreePoints{B1}{Q}{A1};
\coordinate (A2) at (180-54.735:\R);
\coordinate (B2) at (-54.735:\R);
\arcThroughThreePoints{A2}{P}{B2};
\arcThroughThreePoints{B2}{Q}{A2};
\coordinate (A3) at (0:\R);
\coordinate (B3) at (180:\R);
\draw (A3) -- (B3);
\path[name intersections={of=kamaanA1 and kamaanB2, by=a1b2}];
\path[name intersections={of=kamaanB1 and kamaanA2, by=b1a2}];
\node [circle, inner sep=1.5pt,fill,draw] at (a1b2) {};
\node [circle, inner sep=1.5pt,fill,draw] at (b1a2) {};
\node [circle, inner sep=1.5pt,fill,draw] at (P) {};
\node [circle, inner sep=1.5pt,fill,draw] at (Q) {};
\node [circle, inner sep=1.5pt,fill,draw] at (A1) {};
\node [circle, inner sep=1.5pt,fill,draw] at (B1) {};
\node [circle, inner sep=1.5pt,fill,draw] at (A2) {};
\node [circle, inner sep=1.5pt,fill,draw] at (B2) {};
\node [circle, inner sep=1.5pt,fill,draw] at (A3) {};
\node [circle, inner sep=1.5pt,fill,draw] at (B3) {};
\node [black] at (180+54.735:1.2\R) {\small~$F$};
\node [black] at (180+52:0.67\R) {\small~$FH$};
\node [black, circle, inner sep = 0pt, minimum size = 1pt, fill=white] at (0.47,.63) {\small~$H$};
\node [black, inner sep = 2pt, fill=white, right] at (27.367:.85\R) {\small~$G=GF$};
\node [black, inner sep = 1pt, fill=white] at (270:1\R) {\small~$FG$};
\end{tikzpicture}
\end{gathered}
$$
\setlength{\belowcaptionskip}{-20pt}
\caption{Product of faces in arrangements in dimension 2 and 3.
			The second arrangement has been intersected with a sphere around the origin.
%			$F$ is a ray (dimension 1),~$G$,~$H$ and~$FG$ are walls (codimension 1), and~$FH$ is a chamber (full-dimensional).
%The product of two faces~$F$ and~$G$, denoted~$FG$, is the first face you encounter after moving a small positive distance from an interior point of~$F$ to an interior point of~$G$.
}
\label{f:ex_arr_prod}
\end{figure}

An arbitrary intersection of hyperplanes in $\arr$ is a \textbf{flat} of the arrangement. The set~$\flat[\arr]$ of flats is a lattice with maximum~$\top:= V$ and minimum~$\perp$, the intersection of all hyperplanes in~$\arr$. The \textbf{support} of a face~$F$ is the smallest flat~$\supp(F)$ containing it.

We view~$\flat[\arr]$ as a commutative monoid with the join operation for the product. Then, the support map
$
\supp:\face[\arr]\to\flat[\arr]
$
is a morphism of monoids.

The monoid algebra~$\RR \flat[\arr]$ is the maximal (split-)semisimple quotient of~$\RR\face[\arr]$ via the support map.
We let~$\B{H}_\rmX$ denote the basis element of~$\RR\flat[\arr]$ associated to the flat~$\rmX$ of~$\arr$, so that~$\B{H}_\rmX \cdot \B{H}_\rmY = \B{H}_{\rmX \vee \rmY}$. It follows that the simple modules over the Tits algebra are one-dimensional and are indexed by the flats of~$\arr$. This rests on the fact that~$\RR \flat[\arr]$ has a unique complete system of orthogonal idempotents~$\{\B{Q}_\rmX\}_{\rmX \in \flat[\arr]}$ determined by
\begin{equation}\label{eq:H_and_Q}
\B{H}_\rmX = \sum_{\rmY:\,\rmY\geq\rmX}\B{Q}_\rmY
\quad\text{or equivalently}\quad
\B{Q}_\rmX = \sum_{\rmY:\,\rmY \geq \rmX} \mu(\rmX,\rmY) \B{H}_\rmY,
\end{equation}
where~$\mu$ denotes the Möbius function of the lattice~$\flat[\arr]$. This is a result of Solomon.

For every flat~$\rmX$, let~$\chi_\rmX$ denote the character of the simple module of~$\RR\face[\arr]$ indexed by~$\rmX$.
%An element~$w \in \RR \face[\arr]$ is \textbf{characteristic of parameter~$t \in \RR$} if for every flat $\rmX \in \flat[\arr]$,~$\chi_\rmX(w) = t^{\dim(\rmX)}$.
An element~$w \in \RR \face[\arr]$ is \textbf{characteristic of parameter~$t \in \RR$} if $\chi_\rmX(w) = t^{\dim(\rmX)}$ for every flat $\rmX \in \flat[\arr]$.
Characteristic elements determine the characteristic polynomial of~$\arr$ and of its contractions. See~\cite{abm19} and~\cite[Section 12.4]{am17} for more information.

A main motivation to study characteristic elements and their action on modules over the Tits algebra comes from the theory of Hopf monoids. We expand on this next.

\subsection{Hopf monoids and the braid arrangement}\label{ss:braid}

The theory of Hopf monoids in the category of species was developed by Aguiar and Mahajan~\cite{am10,am13}, and has received significant attention in recent years. It provides a unified framework to study families of combinatorial objects that have a natural way to \emph{merge} and \emph{break} structures.

A \textbf{species}~$\vH$ consists of a vector space~$\vH[I]$ for each finite set~$I$ and \emph{relabeling maps} $\vH[I] \rightarrow \vH[J]$ for each bijection~$I \rightarrow J$. A \textbf{Hopf monoid} is a species~$\vH$ with \emph{compatible} product and coproduct maps
$$
\mu_{S_1,\dots,S_k} : \vH[S_1] \otimes \dots \otimes \vH[S_k] \rightarrow \vH[I]
\qqand
\Delta_{S_1,\dots,S_k} : \vH[I] \rightarrow \vH[S_1] \otimes \dots \otimes \vH[S_k]
$$
indexed by set compositions~$(S_1,\dots,S_k)$ of~$I$, and an \textbf{antipode} map~${\bf s}_I : \vH[I] \rightarrow \vH[I]$, which generalizes the notion of inversion in a group. A fundamental problem in Hopf monoid theory is to find a reduced formula for the antipode map of a given Hopf monoid~$\vH$.

Let~$\arr_d$ denote the \textbf{braid arrangement} in~$\RR^d$. It consists of the hyperplanes~$x_i = x_j$ for $1 \leq i < j \leq d$.
We identify faces and flats of~$\arr_d$ with set compositions $F \vDash [d]$ and set partitions $\rmX \vdash [d]$ of~$[d]$ in the usual way. For instance, the set composition~$F = (\{1,3\},\{2\},\{4,5\})$ represents the face with equations~$x_1 = x_3 > x_2 > x_4 = x_5$ of~$\arr_5$. Similarly, the set partition $\rmX = \big\{\{1,3\},\{2\},\{4,5\}\big\}$ represents the flat satisfying~$x_1 = x_3$ and~$x_4 = x_5$.
Note that $\supp(F) = \rmX$.

The \textbf{Takeuchi element}~$\tau \in \RR\face[\arr_d]$ and the \textbf{Adams element}~$\alpha_t \in \RR\face[\arr_d]$ are defined by
$$
\tau = \sum_F (-1)^{\dim(F)} \B{H}_F
\qqand
\alpha_t = \sum_F {t \choose \dim(F)} \B{H}_F.
$$
They are characteristic elements of parameter~$-1$ and~$t$, respectively.

\begin{ex-thm}[{\cite[Section 13.1]{am13}}]\label{ex:hopf_to_module}
Let~$\vH$ be a commutative Hopf monoid. Then, the space~$\vH[[d]]$ is a right~$\RR\face[\arr_d]$-module. The action of the basis element~$\B{H}_F$ corresponding to a face~$F = (S_1,\dots,S_k)$ is given by the composition~$\mu_{S_1,\dots,S_k} \circ \Delta_{S_1,\dots,S_k}$.

Then, the antipode map~${\bf s}$ is determined by the action of the Takeuchi element. Explicitly,~${\bf s}_{[d]}(x) = x \cdot \tau$.
On the other hand, when $t = n$ is an integer, the action of the Adams element~$\alpha_n$ agrees with the~$n$-th \emph{convolution power of the identity}.
If~$\zeta$ is a \emph{character} of~$\vH$ and~$P(x,t)$ denotes the corresponding polynomial invariant on an element~$x \in \vH[[d]]$ (see~\cite[Section 16]{aa17}), then~$P(x,t) = \zeta(x \cdot \alpha_t)$.
\end{ex-thm}

\subsection{Eulerian idempotents and diagonalization}

An \textbf{Eulerian family} of~$\arr$ is a collection~$\{\B{E}_\rmX\}_{\rmX \in \flat[\arr]}$ of idempotent and mutually orthogonal elements of~$\RR\face[\arr]$ of the form
$$
\B{E}_\rmX = \sum_{F \,:\, \supp(F) \geq \rmX} a^F \B{H}_F,
$$
with~$a^F \neq 0$ for at least one~$F$ with~$\supp(F) = \rmX$. It follows~\cite[Theorem 11.20]{am17} that~$\{\B{E}_\rmX\}_\rmX$ is a complete system of orthogonal idempotents and that 
$$
\supp(\B{E}_\rmX) = \B{Q}_\rmX.
$$

A characteristic element~$w$ of \emph{non-critical}\footnote{$t \in \RR$ is non-critical if it is not a root of the characteristic polynomial of~$\arr$ nor any of its contractions.} parameter~$t$ uniquely determines an Eulerian family~$\B{E} = \{\B{E}_\rmX\}_\rmX$, which satisfies
\begin{equation}\label{eq:char_eul_idempotent}
w = \sum_\rmX t^{\dim(\rmX)} \rm E_\rmX.
\end{equation}
It follows that the action of~$w$ on any~$\RR\face[\arr]$-module~$M$ is diagonalizable.

Let~$M$ be a (right)~$\RR\face[\arr]$-module,~$w \in \RR\face[\arr]$ be a characteristic element of non-critical parameter~$t$ and~$\{\B{E}_\rmX\}_\rmX$ be the corresponding Eulerian Family. Then, we have a decomposition
$$
M = \bigoplus_{\rmX} M \cdot \B{E}_\rmX
$$
of vector spaces. Expression~\eqref{eq:char_eul_idempotent} shows that~$w$ acts on~$M \cdot \B{E}_\rmX$ by multiplication by~$t^{\dim(\rmX)}$. We define
\begin{equation}\label{eq:def_eta}
\eta_\rmX(M) := \dim_\RR(M \cdot \B{E}_\rmX).
\end{equation}
The character~$\chi_M: \RR \face[\arr] \rightarrow \RR$ of~$M$ factors through~$\RR \flat[\arr]$:

$$
\begin{tikzcd}[row sep=small]
{\RR\face[\arr]} \arrow[rd, "\chi_M"] \arrow[d, "\supp"'] & \\
{\RR\flat[\arr]} \arrow[r, "\overline{\chi}_M"', dashed] & \RR
\end{tikzcd}
$$
Thus,~$\eta_\rmX(M) = \dim_\RR(M \cdot \B{E}_\rmX) = \chi_M(\B{E}_\rmX) = \opp{\chi}_M(\B{Q}_\rmX)$ is independent of the characteristic element~$w$. Furthermore, using relations~\eqref{eq:H_and_Q} and the linearity of $\opp{\chi}_M$ we deduce
\begin{multline}\label{eq:compute_eta}
\eta_\rmX(M) 
= \opp{\chi}_M(\B{Q}_\rmX) 
= \sum_{\rmY \geq \rmX} \mu(\rmX,\rmY) \opp{\chi}_M(\B{H}_\rmY)\\
= \sum_{\rmY \geq \rmX} \mu(\rmX,\rmY) \chi_M(\B{H}_{F_\rmY})
= \sum_{\rmY \geq \rmX} \mu(\rmX,\rmY) \dim_\RR(M \cdot \B{H}_{F_\rmY})
\end{multline} 
where~$F_\rmY \in \face[\arr]$ is such that~$\supp(F_\rmY) = \rmY$. The last equality follows since~$\B{H}_{F_\rmY}$ is an idempotent element, and thus~$\chi_M(\B{H}_{F_\rmY}) = \dim_\RR(M \cdot \B{H}_{F_\rmY})$. 

Moreover, the number of composition factors~$M_{i+1}/M_i$ isomorphic to the simple module indexed by~$\rmX$ in a composition series
$
0 \subset M_1 \subset M_2 \subset \dots \subset M_k = M
$
of~$M$ is precisely~$\eta_\rmX(M)$.

\section{The McMullen module of generalized zonotopes}\label{s:module}

Fix a hyperplane arrangement~$\arr$ in~$V$.
Take a normal vector~$v_\rmH$ for each hyperplane $\rmH \in \arr$, and consider the zonotope~$\frakz := \sum_\rmH \Conv\{o,v_\rmH\}$. Its normal fan~$\face_\frakz$ coincides with the collection of faces~$\face[\arr]$ of the arrangement. We say that a polytope~$\p$ is a \textbf{generalized zonotope} of~$\arr$ if it is a deformation of~$\frakz$.

\newpage
We now consider the algebra~$\Pi(\frakz)$ introduced in \cref{d:subalgebra_polytope}.
It is generated by the classes of generalized zonotopes of~$\arr$. It only depends on the arrangement~$\arr$ and not on the particular choice normal vectors~$v_\rmH$.

Faces of a zonotope~$\frakz$ are Minkowski summands of~$\frakz$. Therefore, for each face~$\f$ of~$\frakz$, the algebra~$\Pi(\f)$ is a subalgebra of~$\Pi(\frakz)$. Moreover, if~$\f = \frakz_v$ for some~$v \in V$ and~$\q$ is a Minkowski summand of~$\f$, then~$\q_v = \q$.
Hence, the composition
$
\Pi(\f) \hookrightarrow \Pi(\frakz) \xrightarrow{\psi_\f} \Pi(\f),
$
where~$\psi_\f$ is the morphism~\eqref{eq:morphism_to_face}, is the identity map.
We have proved the following.
	
\begin{proposition}\label{p:zonotope_surjective}
Let~$\frakz$ be a zonotope and~$\f$ a face of~$\frakz$.
Then, the morphism~$\psi_\f$ is surjective.
\end{proposition}

\begin{remark}
Compare this to \cref{t:Mc_face_surj} and note that we do not assume that~$\frakz$ is simple. For an arbitrary polytope~$\p$ and a face~$\f$ of~$\p$, there is no natural morphism~$\Pi(\f) \rightarrow \Pi(\p)$, unlike in the case of zonotopes.
\end{remark}

\begin{theorem}\label{t:mod_str}
The algebra~$\Pi(\frakz)$ is a right~$\RR\face[\arr]$-module under the following action.
For a generator~$[\q]$ of~$\Pi(\frakz)$ and a basis element~$\B{H}_F$ of~$\RR\face[\arr]$,
\begin{equation}\label{eq:def_action}
[\q] \cdot \B{H}_F := [\q_v],
\end{equation}
where~$v \in \relint(F)$.
Moreover, each graded component~$\Xi_r(\frakz)$ is a~$\RR\face[\arr]$-submodule and the action of basis elements~$\B{H}_F$ on~$\Pi(\frakz)$ is by (graded) algebra endomorphisms.
\end{theorem}

\begin{proof}[Sketch of proof]
It follows from~\cite[Section 3.1.5]{grunbaum03convex} that for a polytope~$\q \subseteq V$ and vectors~$v,w \in V$, 
$
(\q_v)_w = \q_{v + \lambda w}
$
for any small enough~$\lambda > 0$.
Similarly, if~$v \in \relint(F)$ and~$w \in \relint(G)$ for~$F,G \in \face[\arr]$, then~$v + \lambda w \in \relint(FG)$ for any small enough~$\lambda > 0$.


The last statement follows form~\cite[Theorem 7]{mcmullen89} and the characterization of the graded components~$\Xi_r$ in~\eqref{eq:Xi_dilation}.
\end{proof}

Let~$w \in \RR\face[\arr]$ be a characteristic element of parameter~$t$ and~$\{\B{E}_\rmX\}_\rmX$ the corresponding Eulerian family. It follows from \eqref{eq:Xi_dilation} and \eqref{eq:char_eul_idempotent} that for any~$\lambda > 0$,
$$
\text{If } x \in \Xi_r(\frakz) \cdot \B{E}_\rmX,
\qquad \text{then} \qquad
\delta_\lambda x = \lambda^r x
\qqand
x \cdot w = t^{\dim(\rmX)} x.
$$
We say that elements in~$\Xi_r(\frakz) \cdot \B{E}_\rmX$ are \textbf{double-eigenvectors}.
%We proceed to compute the numbers~$\eta_\rmX(\Xi_r(\frakz)) = \dim_\RR(\Xi_r(\frakz) \cdot \B{E}_\rmX)$ defined in \eqref{eq:def_eta}.
We proceed to compute the dimension of these \emph{double-eigenspaces}~$\eta_\rmX(\Xi_r(\frakz)) = \dim_\RR(\Xi_r(\frakz) \cdot \B{E}_\rmX)$ defined in \eqref{eq:def_eta}.

Let~$F \in \face[\arr]$ and~$\f$ be the corresponding face of~$\frakz$. Observe that for any $x \in \Pi(\frakz)$, $x \cdot \B{H}_F = \psi_\f(x)$. Applying \cref{p:zonotope_surjective} to \eqref{eq:compute_eta}, we have
$$
\eta_\rmX(\Xi_r(\frakz))
= \sum_{\rmY \geq \rmX} \mu(\rmX,\rmY) \dim_\RR(\Xi_r(\frakz)\cdot \B{H}_{F_\rmY})
= \sum_{\rmY \geq \rmX} \mu(\rmX,\rmY) \dim_\RR(\Xi_r(\frakz_\rmY)),
$$
where~$\frakz_\rmY$ is any face of~$\frakz$ perpendicular to~$\rmY$. That is, such that $N(\frakz_\rmY,\frakz_\rmY)= \rmY$.
If in addition~$\arr$ is a simplicial arrangement (i.e.~$\frakz$ is simple), \cref{t:simple_h_dimension} implies that
\begin{equation}\label{eq:eta_sum_h}
\sum_r \eta_\rmX(\Xi_r(\frakz)) z^r = \sum_{\rmY \geq \rmX} \mu(\rmX,\rmY) h(\frakz_\rmY,z),
\end{equation}
where~$h(\frakz_\rmY,z)$ denotes the~$h$-polynomial of~$\frakz_\rmY$.

\section{Generalized permutahedra and Eulerian polynomials}\label{s:cox_eul}

\subsection{Type A}

Let~$\arr_d$ denote the braid arrangement in~$\RR^d$, see \cref{ss:braid}. It is the Coxeter arrangement of type A, corresponding to the symmetric group~$\frakS_d$. This group acts on~$\RR^d$ by permuting coordinates~$\sigma(x_1,x_2,\dots,x_d) = (x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(d)})$.
The \textbf{permutahedron}~$\pi_d \subseteq \RR^d$ is the convex hull of the points obtained by permuting the coordinates of~$(1,2,\dots,d)$. It is a zonotope of~$\arr_d$ and has dimension~$d-1$. Deformations of~$\pi_d$ are called \textbf{generalized permutahedra}. In this section we study the algebra/$\RR\face[\arr_d]$-module~$\Pi(\pi_d)$. 

\subsubsection{Hopf monoid structure}

Aguiar and Ardila introduced in~\cite{aa17} the Hopf monoid of generalized permutahedra~$\bf GP$.
For any finite set~$I$,~${\bf GP}[I] = \RR \{ \p \subseteq \RR^I \,:\, \p$ is a generalized permutahedron$\}$. The product and coproduct maps of~$\bf GP$ are completely determined by
$$
\mu_{S,T}(\p \otimes \q) = \p \times \q
\qqand
(\mu_F \circ \Delta_F) (\p) = \p_v
$$
where~$v \in \relint(F)$. Compare this to \eqref{eq:def_action}. Let~$\bm \Pi$ be the species defined by setting~${\bm \Pi}[I] := \Pi(\pi_I)$, where~$\pi_I$ denotes the permutahedron in~$\RR^I$.

\begin{theorem}
The species $\bm \Pi$ is the Hopf monoid quotient of~$\bf GP$ induced by the map~$\p \mapsto [\p]$.
Moreover, $\bm \Pi$ has the structure of a (2,1)-monoid in the category of species with the Cauchy and Hadamard product.
\end{theorem}

A result similar to the first statement was recently obtained by Ardila and Sanchez in~\cite{as20valGP}. The \emph{higher monoidal structure} is a consequence of the last statement in \cref{t:mod_str}. For more information on higher monoidal categories see~\cite[Chapter 7]{am10}.

\subsubsection{Module structure}

We let~$\supp(\sigma)$ denote the subspace of fixed points by the action of~$\sigma$ on~$\RR^d$; it is a flat of~$\arr_d$.
Recall that~$i \in [d]$ is an \textbf{excedance} of a permutation~$\sigma \in \frakS_d$ if~$\sigma(i) > i$. Let~$\exc(\sigma)$ denote the number of excedances of~$\sigma$. We completely determine the numbers~$\eta_\rmX(\Xi_r(\pi_d))$.

\begin{theorem}\label{t:dims_double_e-spaces_A}
For any flat~$\rmX \in \flat[\arr_d]$ and~$r=0,1,\dots,d-1$,
$$
\eta_\rmX(\Xi_r(\pi_d)) = \big| \{ \sigma \in \frakS_d \,:\, \supp(\sigma) = \rmX,\ \exc(\sigma) = r \}\big|.
$$
\end{theorem}

Building on top of work by Björner, Brenti showed that for any Coxeter group~$W$, the~$h$-polynomial of the~$W$-permutahedron is the corresponding~$W$-Eulerian polynomial~\cite[Theorem 2.3]{brenti94cox-eul}. The \emph{classical} \textbf{Eulerian polynomial}~$A_d(z)$ is
$$
A_d(z) = \sum_{k=0}^{d-1} A_{d,k} z^k = \sum_{\sigma \in \frakS_d} z^{\exc(\sigma)}.
$$
That is, the coefficient~$A_{d,k}$ counts the number of permutations of~$[d]$ with exactly~$k$ excedances.
These coefficients are called \emph{Eulerian numbers} (OEIS: \href{https://oeis.org/A008292}{A008292}).
It is customary to define~$A_d(z)$ in terms of the \emph{descents} of permutations in~$\frakS_d$. However, Foata's \emph{fundamental transformation} shows that the distribution of these two statistics are equal.

The exponential generating function for the classical Eulerian polynomials was first described by Euler:
\begin{equation}\label{eq:generating_eulerian}
A(z,x) = \sum_{d \geq 0} A_d(z) \dfrac{x^d}{d!} = \dfrac{z-1}{z-e^{x(z-1)}}.
\end{equation}

Under the identification between flats of~$\arr_d$ and set partitions of~$[d]$,~$\supp(\sigma)$ is the partition of~$[d]$ underlying the cycle decomposition of~$\sigma$.
Given a finite set~$S$, we let~$\frakC(S)$ denote the collection of cyclic permutations on~$S$.
For a block~$S \in \supp(\sigma)$, we let~$\sigma|_S \in \frakC(S)$ be the restriction of~$\sigma$ to~$S$.

\begin{lemma}\label{l:cyc_exc}
$
\displaystyle\sum_{\rmX = \{S_1,\dots,S_k\} \vdash [d]} \mu(\perp,\rmX) A_{|S_1|}(z) \cdot \ldots \cdot A_{|S_k|}(z) = \sum_{\sigma \in \frakC(d)} z^{\exc(\sigma)}.
$
\end{lemma}

\begin{proof}[Sketch of proof]
We verify that the exponential generating functions of both sides of the equation above is~$\log(A(z,x))$.
Recall that ~$\mu(\perp,\rmX) = (-1)^{k-1}(k-1)!$, where~$k = |\rmX|$, and that~$\log(1+x) = \sum_{d \geq 1} (-1)^{d-1}(d-1)! \tfrac{x^d}{d!}$.
An application of the Compositional Formula~\cite[Theorem 5.1.4]{stanley99EC2} yields the result for the terms on the left.

For the terms on the right, observe that the number of excedances of~$\sigma$ is the sum of the excedances on each of its cycles. That is,
\begin{equation}\label{eq:exc_sum}
\exc(\sigma) = \sum_{S \in \supp(\sigma)} \exc(\sigma|_S)
\end{equation}
An application of the Exponential formula then shows that~$A(z,x) = \exp(\widetilde{A}(z,x))$ where~$\widetilde{A}(z,x)$ is the exponential generating function of the terms on the right.
\end{proof}

\begin{proof}[Proof of \cref{t:dims_double_e-spaces_A}. Sketch]
Faces of~$\pi_d$ are products of permutahedra of lower dimension. Explicitly, if~$\rmX =\{S_1,\dots,S_k\}$, then~$
(\pi_d)_\rmX \cong \pi_{|S_1|} \times \dots \times \pi_{|S_k|}$.
On the other hand, flats $\rmY$ containing $\rmX = \{S_1,\dots,S_k\}$ correspond to a choice of partitions $\rmY|_{S_i}$ for each $S_i$, and in this case $\mu(\rmX,\rmY) = \mu(\{S_1\},\rmY|_{S_1}) \cdots \mu(\{S_k\},\rmY|_{S_k})$.

Using that~$h(\p \times \q,z) = h(\p,z)h(\q,z)$, formula \eqref{eq:eta_sum_h} then yields
$$
\sum_r \eta_\rmX(\Xi_r(\pi)) z^r
= \prod_{S \in \rmX} \Big( \sum_{\rmY=\{T_1,\dots,T_\ell\} \vdash S} \mu(\perp,\rmY) A_{|T_1|}(z) \cdot \ldots \cdot A_{|T_\ell|}(z) \Big).
$$
Applying \cref{l:cyc_exc}, we get
$$
\sum_r \eta_\rmX(\Xi_r(\pi)) z^r
= \prod_{S \in \rmX} \Big( \sum_{\sigma \in \frakC(S)} z^{\exc(\sigma)} \Big)
= \sum_{\substack{\sigma \in \frakS_d \\ \supp(\sigma) = \rmX}} z^{\exc(\sigma)}.
$$
Finally, taking the coefficient of~$z^r$ on both sides yields the result.
\end{proof}

Let~$\{\B{E}_\rmX\}_\rmX$ be the Eulerian family corresponding to the Adams element~$\alpha_t$ from \cref{ss:braid}.
The particular case~$r=1$ of \cref{t:dims_double_e-spaces_A} reads
\begin{equation}\label{eq:A_grade1}
\eta_\rmX(\Xi_1(\pi_d)) = \begin{cases}
1 & \text{if~$\rmX$ has exactly 1 non-singleton singleton block}, \\
0 & \text{otherwise}.
\end{cases}
\end{equation}
Let~$\rmX$ be such a flat and let~$J \subseteq [d]$ be the corresponding non-singleton block.

\begin{proposition}\label{p:diag_A-1}
The space~$\Xi_1(\pi_d) \cdot \B{E}_\rmX$ is spanned by the element~$\log[\Delta_J] \cdot \B{E}_\rmX$, where~$\Delta_J$ denotes the simplex with vertex set~$\{e_j \,:\, j \in J\}$.
\end{proposition}

In view of \eqref{eq:A_grade1}, the proof of this boils down to showing that~$\log[\Delta_J] \cdot \B{E}_\rmX \neq 0$.
This rests on a result of Ardila, Benedetti and Doker~\cite[Proposition 2.4]{abd10volume} that implies that~$\{\log[\Delta_J] \,:\, |J| \geq 2\}$ is a linear basis of~$\Xi_1(\pi_d)$.

Their result shows that every generalized permutahedron~$\p$ can be written uniquely as a \textbf{signed Minkowski sum} of the simplices~$\{\Delta_J, \emptyset \neq J \subseteq [d]\}$. That is, there are unique coefficients~$\{y_J\}_J \subseteq \RR$ such that~$\p + \sum_{y_J < 0}|y_J|\Delta_J = \sum_{y_J > 0}y_J\Delta_J$. Taking log-classes, we get that
$$
\log[\p] = \sum_J y_J\log[\Delta_J].
$$
Since~$\log[\Delta_J] = \log(1) = 0$ for~$|J| = 1$, we deduce that every log-class~$\log[\p]$ can be written \textbf{uniquely} as a linear combination of~$\{\log[\Delta_J], |J| \geq 2\}$.

\subsection{Type B}

Let~$\cB_d$ denote the Coxeter arrangement of type B in~$\RR^d$. It consists of hyperplanes $x_i = x_j$,~$x_i = - x_j$ and~$x_i = 0$. Flats of~$\cB_d$ are in correspondence with signed set partitions $\rmX \vdash^B [d]$ of~$[d]$. A \textbf{signed set partitions} of~$[d]$ is a weak partition~$\{S_0,S_1,\opp{S_1},\dots,S_k,\opp{S_k}\}$ of $\pm[d] := \{i , \opp{i} \,:\, i \in [d]\}$ such that~$S_0 = \opp{S_0}$,~$S_i \neq \emptyset$ and~$S_i \cap \opp{S_i} = \emptyset$ for~$i \neq 0$. For instance, the type B partition~$\big\{\{1,\opp{1},4,\opp{4},5,\opp{5}\},\{2,\opp{3}\},\{\opp{2},3\}\big\}$ represents the flat of~$\cB_5$ with equations~$x_1 = x_4 = x_5 = 0$,~$x_2 = -x_3$. For~$i \in [d]$, write~$i = |\opp{i}| = |i|$.

The \textbf{hyperoctahedral group}~$\frakB_d$ consists of \textbf{signed permutations} $\tau: \pm[d] \rightarrow \pm[d]$ satisfying~$\tau(i) = j$ if and only if~$\tau(\opp{i}) = \opp{j}$. Signed permutations act on~$\RR^d$ by permuting coordinates and changing signs.
For example, let~$\tau = (1\opp{4}\opp{1}4)(5\opp{5})(2\opp{3})(\opp{2}3)$.
Then,~$\tau(x_1,x_2,x_3,x_4,x_5) = (-x_4,-x_3,-x_2,x_1,-x_5)$. Note that~$\supp(\tau)$ is the flat in the previous paragraph.

The \textbf{type B permutahedron}~$\pi^B_d \subseteq \RR^d$ is the convex hull of all the signed permutations of the point~$(1,2,\dots,d)$.
It is full-dimensional and a zonotope of~$\cB_d$, its $h$-polynomial is the \textbf{type B Eulerian polynomial} $B_d(z)$, it keeps track of the statistic $\exc_B(\tau)$ on $\frakB_d$.
Let $B(z,x)$ denote the type B generating function of the polynomials $B_d(z)$.
Deformations of~$\pi^B_d$ are called \textbf{type B generalized permutahedra}.
We now consider the module~$\Pi(\pi^B_d)$.

\begin{theorem}\label{t:dims_double_e-spaces_B}
For any flat~$\rmX \in \flat[\cB_d]$ and~$r=0,1,\dots, d$,
$$
\eta_\rmX(\Xi_r(\pi^B_d)) = \big|\{ \tau \in \frakB_d \,:\, \supp(\tau) = \rmX,\ \exc_B(\tau) = r \}\big|.
$$
\end{theorem}

The statistic $\exc_B$ is closely related to the \emph{flag-excedance} of a signed permutation, see~\cite{fh09signedV}. We do not go into the details of its definition, but we highlight a key property of $\exc_B$ analogous to~\eqref{eq:exc_sum}. If $\supp(\tau) = \{S_0,S_1,\opp{S_1},\dots,S_k,\opp{S_k}\}$, then $\tau|_{S_0}$ is a signed permutation with $\supp(\tau|_{S_0}) = \perp$ and each $\tau|_{S_i} \in \frakC(S_i)$ for $i \geq 1$ is a cyclic permutation. Moreover, $\exc_B(\tau) = \exc_B(\tau|_{S_0}) + \sum_{i \geq 1} \exc_\prec(\tau|_{S_i})$, where $\exc_\prec$ denotes the usual excedance but computed with respect to a particular order $\prec$ on $S_i$.

\begin{proof}[Sketch of proof]
The analogous result to \cref{l:cyc_exc} is
\begin{equation}\label{eq:cyc_exc_B}
\sum_{\{S_0,\dots,S_k,\opp{S_k}\} \vdash^B [d]} \mu(\perp,\rmX) B_{|S_0|/2} A_{|S_1|} \dots A_{|S_k|}
=
\sum_{\substack{\tau \in \frakB_d \\ \supp(\tau) = \perp}} z^{\exc_B(\tau)}.
\end{equation}
We again proceed by comparing the \emph{type B} exponential generating function of both sides and verify it equals~$\tfrac{B(z,x)}{\sqrt{A(z,x)}}$. The proof involves a type B analogous of the Compositional and Exponential formula. With~$\rmX$ as above,~$\mu(\perp,\rmX) = (-1)^k(2k-1)!!$ (compare with the generating function of~$(1+x)^{-1/2}$). On the other hand, the type B exponential formula together with the observations following the statement of \cref{t:dims_double_e-spaces_B} yield~$B(z,x) = \widetilde{B}(z,x)\exp(\tfrac{\log(A(z,x)}{2})$, where~$\widetilde{B}(z,x)$ is the type B exponential generating function of the right side of \eqref{eq:cyc_exc_B}. See~\cite[Section 6.2]{bastidas2020polytope} for all the details.
\end{proof}

As a byproduct of the proof, we obtain the following formula. To the best of our knowledge, this is a new formula. The analogous result for the symmetric group was described by Brenti in~\cite[Proposition 7.3]{brenti00qeulerian}.

\begin{corollary}
$\displaystyle\sum_{d \geq 0} \Big( \sum_{\tau \in \frakB_d} t^{\dim(\supp(\tau))} z^{\exc_B(\tau)} \Big) \dfrac{x^d}{2^d\,d!} = B(z,x)\,A(z,x)^{\tfrac{t-1}{2}}$.
\end{corollary}

In~\cite{acep20submodular}, the authors ask for a nice set of generators for the family of type B generalized permutahedra, in the sense of Ardila, Benedetti and Doker in the previous section. The special case~$r = 1$ of \cref{t:dims_double_e-spaces_B} implies the following.

\begin{corollary}
Any set of generators for the family of type B generalized permutahedra must contain at least~$2^{d-1}$ full-dimensional polytopes.
\end{corollary}

\begin{proof}
As discussed after \cref{p:diag_A-1}, the log-classes of any such set would linearly generate~$\Xi_1(\pi^B_d)$.
Let~$\{\B{E}_\rmX\}_\rmX$ be an Eulerian family of~$\cB_d$.

Let~$\p$ be a type B generalized permutahedron that is not full-dimensional. Let~$F$ be a maximal face in the flat~$N(\p,\p)$. Then,~$\supp(F) \neq \perp$. It follows from~\cite[Lemma 11.12]{am17} that~$\B{H}_F \cdot \B{E}_\perp = 0$. Since the action of basis elements~$\B{H}_F$ on~$\Pi(\pi^B_d)$ is by endomorphisms, we get
$
\log[\p] \cdot \B{H}_F = \log([\p] \cdot \B{H}_F) = \log[\p]
$
and consequently
$
\log[\p] \cdot \B{E}_\perp = (\log[\p] \cdot \B{H}_F) \cdot \B{E}_\perp = 0.
$
The result follows since, by \cref{t:dims_double_e-spaces_B},~$\dim_\RR(\Xi_1(\pi^B_d) \cdot \B{E}_\perp) = \eta_\perp(\Xi_1(\pi^B_d)) = 2^{d-1}$.
\end{proof}

This is in sharp contrast with the type A case, where the set of generators is formed by the collection of faces of the standard simplex.

\printbibliography
\end{document}