\section{Introduction}\label{intro}

By definition, a \emph{matroid} on a finite set $E$ is given by a rank function
$ \rk: 2^E \to \Z_{\geq 0} $
satisfying the following:
\begin{enumerate}[]\itemsep 5pt
\item[] (Submodularity) For any $A_1,A_2 \subseteq E$, we have $\rk(A_1 \cup A_2) + \rk(A_1 \cap A_2) \leq \rk(A_1) + \rk(A_2)$.
\item[] (Monotonicity) For any $A_1 \subseteq A_2 \subseteq E$, we have $\rk(A_1) \leq \rk(A_2)$.
\item[] (Boundedness) For any $A \subseteq E$, we have $\rk(A) \leq |A|$.
\item[] (Normalization) The rank of the empty subset is zero.
\end{enumerate}
Sans ``boundedness'', the axioms above define a \textit{polymatroid}.
Throughout this paper, we assume that the polymatroid is \emph{loopless}:
\begin{enumerate}[]\itemsep 5pt
\item[] (Looplessness) The rank of any nonempty subset is nonzero.
\end{enumerate}
If $P$ is a polymatroid on $E$, then its \emph{rank} is $\rk(P) \coloneqq \rk(E)$.
A \textit{flat} of $P$ is a subset $F \subseteq E$ that is maximal among sets of its rank.
Ordered by inclusion, the flats of $P$ form a lattice $\L_P$.\footnote{Unlike in the case of matroids, the lattice of flats of a polymatroid can fail to be graded or atomic.} 
The intersection of two flats is a flat, so any subset $A$ of $E$ is contained in a unique minimal flat $\cl_P(A)$, called the \emph{closure} of $A$ in $P$, which is obtained by intersecting all flats that contain $A$.

Matroids can be viewed as combinatorial abstractions of hyperplane arrangements.
More generally, polymatroids can be viewed as combinatorial abstractions of subspace arrangements.

\begin{example}\label{subspace}
  Let $V_1, \ldots, V_n$ be linear subspaces of a vector space $V$ over a field $\mathbb{F}$.
  The rank function
  \[ \rk(A) \coloneqq \codim_V(\cap_{i \in A} V_i) \]
   defines a polymatroid $P$ on the set of indices $E=\{1,\ldots,n\}$, which is a matroid if and only if every $V_i$ is a hyperplane.
   The map
   $F \mapsto \cap_{i \in F} V_i$
   is a bijection between the flats of $P$ and the subspaces of $V$ obtained by intersecting some of the $V_i$'s.
   A polymatroid arising in this way is said to be \word{realizable} over $\mathbb{F}$, and the subspace arrangement is called a \emph{realization} of $P$ over $\mathbb{F}$. 
\end{example}

Much of a hyperplane arrangement's combinatorial data is captured by intersection theory on its wonderful compactification \cite{dCP95}.
To extend this from hyperplane arrangements to non-realizable matroids, one must replace the wonderful compactification with a combinatorial object, the Bergman fan of a matroid.
The purpose of the present paper is to introduce the \emph{Bergman fan of a polymatroid}, a combinatorial model for the wonderful compactification of a subspace arrangement.
As in the case of matroids, the Bergman fan of a polymatroid is a tropical variety of degree one.
In Section \ref{sec:chow}, we show that the Chow ring of a polymatroid satisfies the K\"{a}hler package with respect to any strictly convex piecewise linear function on its Bergman fan, 
 recovering and strengthening a result of Pagaria and Pezzoli \cite[Theorems 4.7 and 4.21]{PP21}.

Our construction of the Bergman fan is inspired by a geometric observation: over an infinite field, the wonderful compactification of any subspace arrangement can be realized as the wonderful compactification of a hyperplane arrangement, taken with respect to an appropriate building set (\cref{rmk:geomOfBS}).
The construction immediately reveals that the Bergman fan of a polymatroid and the Bergman fan of the associated matroid have the same support.
Thus, the K\"{a}hler package for the polymatroid follows from that of the associated matroid \cite{AHK18}
and the general fact that the validity of the K\"{a}hler package for the Chow ring of a fan depends only on the support of the fan \cite{ADH20}.

\subsection{The Bergman fan of a Boolean polymatroid}\label{intro:boolean}
An important special case is that of Boolean polymatroids.
Let $\pi:\widetilde E \to E$ be a surjective map between finite sets.
The \textit{Boolean polymatroid} $B(\pi)$ is the polymatroid on $E$ defined by the rank function 
\[
\rk_{B(\pi)}(A) = |\pi^{-1}(A)| \ \ \text{for $A \subseteq E$.}
\]
We write $N_{\widetilde E}$ for $\mathbb{Z}^{\widetilde E} / \mathbb{Z}(1,1,\ldots,1)$,
and, for a subset $S$ of $\widetilde E$, write $\e_S$ for the vector
$\sum_{i \in S} \e_i$ in $N_{\widetilde E} \otimes \R$.

\begin{dfn}\label{dfn:booleanBergman}
 The \word{Bergman fan} $\Sigma_{B(\pi)}$ of the Boolean polymatroid $B(\pi)$ is the fan in $N_{\widetilde E}\otimes \R$ with cones
  \[
    \sigma_{\mathscr F, A} \coloneqq \cone( \e_{\pi^{-1}(F_1)}, \ldots, \e_{\pi^{-1}(F_k)}) + \cone(\e_i)_{i \in S},
  \]
  for every chain $\mathscr{F} = \{\emptyset \subsetneq F_1 \subsetneq F_2 \subsetneq \cdots \subsetneq F_k \subsetneq E \}$
  and subset $S$ of $\widetilde E$ not containing any fiber of $\pi$.
\end{dfn}

Throughout the paper, we write $n$ for the cardinality of $E$.

\begin{dfn}
  An \emph{ordered transversal} of $\pi$ is a sequence $s_1,\ldots,s_n$ of elements of $\tilde{E}$ such that each fiber of $\pi$ contains exactly one element of the sequence.
  The \textit{polypermutohedron}  $Q(\pi)$ is the convex hull of the vectors
  $\sum_{i =1}^n i \e_{s_i}$ in $\R^{\widetilde E}$, 
    where $s_1, \ldots, s_n$ range over all ordered transversals of $\pi$.
\end{dfn}

In \cref{appendix:polytope}, we show $\Sigma_{B(\pi)}$ is the inner normal fan of the polypermutohedron $Q(\pi)$. It follows immediately from Definition \ref{dfn:booleanBergman} that $\Sigma_{B(\pi)}$ is a complete unimodular fan in $N_{\widetilde E}$.


\begin{example}
When $\pi$ is a bijection, an ordered traversal of $\pi$ is a permutation of $\tilde{E}$, and $Q(\pi)$ is the standard permutohedron in $\R^{\widetilde E}$. This recovers a familiar statement: the Bergman fan of the Boolean matroid is the normal fan of the standard permutohedron.
\end{example}
\begin{example}
When $E$ is a singleton, an ordered transversal of $\pi$ is an element $\tilde{E}$, and $Q(\pi)$ is the standard simplex in $\R^{\widetilde E}$.
Thus, the Bergman fan of a Boolean polymatroid on a singleton is the normal fan of the standard simplex.
\end{example}

\subsection{The Bergman fan of a polymatroid}

Let $P$ be a polymatroid on $E$, and let $\pi:\widetilde E \to E$ be a surjective map satisfying $\rk_P(i)=|\pi^{-1}(i)|$ for every $i$ in $E$.

\begin{dfn}\label{dfn:maximalBergman}
   The \word{Bergman fan} $\Sigma_P$ of the polymatroid $P$ is the subfan of $\Sigma_{B(\pi)}$ with cones
  \[
    \sigma_{\mathscr F, S} \coloneqq \cone( \e_{\pi^{-1}(F_1)}, \ldots, \e_{\pi^{-1}(F_k)}) + \cone(\e_i)_{i \in S},
  \]
  one for every chain of flats $\mathscr F = \{\emptyset = F_0 \subsetneq F_1 \subsetneq F_2 \subsetneq \cdots \subsetneq F_k \subsetneq E \}$ of $P$ and a subset $S$ of $\widetilde E$ such that 
\[
\rk(F \cup \pi(T)) > \rk(F) + |T| \ \ \text{for every proper flat $F$ in $\mathscr{F}$ and every nonempty subset $T$ of $S\setminus \pi^{-1}(F)$.}
\]
\end{dfn}

The Bergman fan $\Sigma_P$ is unimodular with respect to $N_{\widetilde E}$, defining a smooth toric variety $X_P$ over $\mathbb{C}$. We write $A(\Sigma_P)$ for the Chow ring of $X_P$. 
We relate $A(\Sigma_P)$ to the \emph{Chow ring} of the polymatroid $P$ (Definition \ref{dfn:ppchow}), denoted $\DP(P)$, introduced in \cite[Section 4]{PP21}. Our main result states the following.
 
\begin{manual}{\cref{thm:isomorphism}}
 There is a natural isomorphism of graded rings $\DP(P) \cong A(\Sigma_P)$.
\end{manual}

In \cref{cor:gb}, we use \cref{thm:isomorphism} to recover a Gr\"obner basis for $\DP(P)$ found in \cite{PP21}. 
In \cref{cor:hodge}, we prove the K\"{a}hler package for $A(\Sigma_P)$ with respect to the cone of strictly convex piecewise linear function on $\Sigma_P$, extending the K\"{a}hler package for $\DP(P)$ with respect to the $\sigma$-cone in  \cite{PP21}. See Remark \ref{ConeRemark} for a comparison of the two cones.

\subsection{Building sets}
In fact, our results hold for polymatroids $P$ equipped with a \word{geometric building set} $\G$ (\cref{sec:buildingsets}). The statements above are specializations of our results to the case when $\G$ consists of all nonempty flats of $P$.
In maximal generality, we define the \word{Bergman fan} of $(P, \G)$, denoted $\Sigma_{P,\G}$ (\cref{dfn:bergmanfan}).
The Chow ring $\DP(P, \G)$ associated to $(P, \G)$ was introduced in \cite{PP21}, and it is isomorphic to 
$A(\Sigma_{P, \G})$ (\cref{thm:isomorphism}).
All corollaries continue to hold, including the K\"{a}hler package for $\DP(P, \G)$.

\subsection*{Organization}
In \cref{sec:multisymmetric}, we develop the combinatorics of multisymmetric matroids and lifts, a key tool throughout this paper.
We use lifts to define the Bergman fan of a polymatroid (with respect to a geometric building set) in \cref{sec:buildingsets}.
Finally, in \cref{sec:chow}, we show that the Chow ring of the Bergman fan agrees with the polymatroid Chow ring of \cite{PP21} and derive consequences.
Examples \ref{subspace} and \ref{rmk:why-geometric}, and Remarks \ref{rmk:geomOfBS}, \ref{tropicalization}, and \ref{rmk:geoProof} explain the geometry underlying this work.

\subsection*{Acknowledgements}
We thank Spencer Backman for comments and conversations. The second author is partially supported by a Simons Investigator Grant and NSF Grant DMS-2053308, the third is supported by an NDSEG fellowship, and the last is supported by a Sloan fellowship.


\iffalse
\textcolor{blue}{(Please move this explicit description to the main body.) The Bergman fan has the associated \word{Chow ring}
\[
  A(\Sigma_P) \coloneqq \Z[z_F : F \text{ nonempty proper flat of $P$}] \otimes \R[z_i : i \in \widetilde E] / I,
\]
where $I$ is an ideal generated by the following polynomials:
\begin{align*}
  z_{F_1} &z_{F_2},  \qquad \text{$F_1$ and $F_2$ are incomparable nonempty proper flats of $P$,} \\
  z_F &\prod_{i \in S} z_i,  \quad \text{$S$ is a subset of $\widetilde E$ satisfying $\rk(F \cup \pi(S)) \leq \rk(F) + |S \setminus \pi^{-1}(F)|$,} \\
  \sum_{i \in F} z_F &- \sum_{j \in G} z_G,  \quad \text{$i$ and $j$ are elements of $\widetilde E$.}
\end{align*}}
\fi


\section{Multisymmetric matroids}\label{sec:multisymmetric}
Many proofs in \cref{sec:chow} reduce statements about polymatroids to known statements about matroids.
The key tool for this reduction is multisymmetric matroids, a new cryptomorphic formulation of polymatroids. 
Let $\sym_{\widetilde{E}}$ denote the symmetric group on a finite set $\widetilde{E}$. We continue to assume that all  (poly)matroids are loopless.
\begin{dfn}\label{dfn:multisymmetric}
  A \emph{multisymmetric matroid} is a matroid $M$ on $\widetilde{E}$ equipped with a partition
  $\widetilde{E} = \widetilde{E}_1 \sqcup \cdots \sqcup \widetilde{E}_n$
  such that the action of $\Gamma = \sym_{\widetilde{E}_1} \times \cdots \times \sym_{\widetilde{E}_n}$ on $\widetilde{E}$ takes flats to flats.
The \word{geometric part} of a subset $S \subseteq \widetilde{E}$ is $S^{\hspace{0.05ex}\mathrm{geo}} \coloneqq \cap_{\gamma \in \Gamma} (\gamma \cdot S)$. We call a subset $S\subseteq \widetilde{E}$ \word{geometric} if $S = S^{\hspace{0.05ex}\mathrm{geo}}$.
\end{dfn}

For a multisymmetric matroid $M$, we write $\L_M^\Gamma$ for its poset of geometric flats.

\begin{example}
  Any matroid $M$ on $\widetilde{E}$ can be given the \word{trivial} multisymmetric structure by setting $\Gamma = \prod_{e \in \widetilde{E}} \sym_{\{e\}}$. In this case, $\L_M^\Gamma = \L_M$.
\end{example}
\begin{example}\label{ex:uniform}
  If $M$ is multisymmetric on $\widetilde{E} = \widetilde{E}_1$, then $M$ is a uniform matroid, and $\L_M^\Gamma$ is $\{\emptyset, \widetilde{E}\}$.
\end{example}
\begin{example}
  Let $\widetilde{E}$ be the set of edges of the complete graph $K_4$, and let $M$ be the graphic matroid of $K_4$.
  No transposition of $\sym_{\widetilde{E}}$ preserves the flats of $M$, so $M$ has no non-trivial multisymmetric structures.
\end{example}

Closure in a multisymmetric matroid is restricted by the group action.
\begin{lem}\label{cor:union}
  If $M$ is multisymmetric and $S \subseteq \widetilde{E}$ is geometric, then $\cl_M(S)$ is also geometric.
\end{lem}
\begin{proof}
For any $S \subseteq \widetilde{E}$ geometric and $\gamma \in \Gamma$, we have $\gamma \cdot \cl_M(S) = \cl_M(\gamma\cdot S) = \cl_M(S)$.
\end{proof}

\begin{cor}\label{cor:sublattice}
  If $M$ is multisymmetric, then the geometric flats of $M$ form a sublattice of $\L_M$.
\end{cor}
\begin{proof}
  If $F$ and $G$ are two geometric flats, then $\cl_M(F \cup G)$ is geometric by \cref{cor:union}.
  The intersection of two geometric flats is also geometric.
  In other words, the set of geometric flats is closed under both join and meet, and therefore forms a sublattice of $\L_M$.
\end{proof}
\begin{cor}\label{cor:polymat}
  If $M$ is multisymmetric on $\widetilde{E} = \widetilde{E}_1 \sqcup \cdots \sqcup \widetilde{E}_n$, then $\L_M^\Gamma$ is the lattice of flats of the polymatroid $P$ on the set of indices $E = \{1, \ldots, n\}$ defined by the rank function
    $\rk_P(A) \coloneqq \rk_M( \cup_{i \in A} \widetilde{E}_i).$ 

\end{cor}
\begin{proof}
  Define $\pi: \widetilde{E} \to E$ by setting $\pi^{-1}(i) = \widetilde{E}_i$.
If $F$ is a flat of $P$, then for all $F \subsetneq A \subseteq \{1, \dotsc, n\}$,
\[ \rk_M(\pi^{-1}(F)) = \rk_P(F) < \rk_P(A) = \rk_M(\pi^{-1}(A)). \]
By \cref{cor:union}, $\cl_M(\pi^{-1}(F))$ is geometric, so we conclude that $\cl_M(\pi^{-1}(F)) = \pi^{-1}(F)$. 
In other words, $\pi^{-1}(F)$ is a flat of $M$.
Conversely, if $F$ is not a flat of $P$, then $\rk_P(F) = \rk_P(F \cup i)$ for some $i$ not in $F$.
This implies $\cl_M(\pi^{-1}(F)) \supseteq \pi^{-1}(F \cup i)$, so $\pi^{-1}(F)$ is not a flat of $M$.
Therefore, $F$ is a flat of $P$ if and only if $\pi^{-1}(F)$ is a geometric flat of $M$, and we have an isomorphism of lattices
\[ \L_P \longrightarrow \L_M^\Gamma, \qquad F \longmapsto \pi^{-1}(F). \qedhere\]
\end{proof}



\begin{lem}\label{lem:closure}
  Let $M$ be a multisymmetric matroid on $\widetilde{E} = \widetilde{E}_1 \sqcup \cdots \sqcup \widetilde{E}_n$. 
\begin{enumerate}\itemsep 5pt
  \item If $S \subseteq \widetilde{E}$, then either $\cl_M(S) \cap \widetilde{E}_i = \widetilde{E}_i$ or $\cl_M(S) \cap \widetilde{E}_i = S \cap \widetilde{E}_i$.\label{lem:extraclosure}
  \item If $F$ is a flat of $M$, then $\rk_M(F) = \rk_M(F^{\hspace{0.05ex}\mathrm{geo}}) + |F \setminus F^{\hspace{0.05ex}\mathrm{geo}}|$. \label{lem:closurerank}
\end{enumerate}
\end{lem}

\begin{proof}
    A permutation of $\widetilde{E}_i \setminus S$ induces an automorphism of $M$ that fixes $S$. Any such automorphism also fixes $\cl_M(S)$ because automorphisms commute with closure.
    Hence, if $(\widetilde{E}_i \setminus S) \cap \cl_M(S)$ is nonempty, then $\widetilde{E}_i \subseteq \cl_M(S)$. This proves the first part.
     
      For the second part, let $\{s_1, \ldots, s_k\} = F \setminus F^{\hspace{0.05ex}\mathrm{geo}}$. If \cref{lem:closurerank} fails then for some $1 \leq i \leq k,$
       \[
         \rk_M(F^{\hspace{0.05ex}\mathrm{geo}}\cup \{s_1, \ldots, s_i\}) = \rk_M(F^{\hspace{0.05ex}\mathrm{geo}}\cup \{s_1, \ldots, s_{i+1}\}).
       \]
       Therefore if $s_{i+1} \in \widetilde{E}_j$ then $\widetilde{E}_j \subseteq \cl_M(F^{\hspace{0.05ex}\mathrm{geo}}\cup \{s_1, \ldots, s_i\}) \subseteq F$ by the first part.
        Consequently, $s_{i+1} \in F^{\hspace{0.05ex}\mathrm{geo}}$, a contradiction. 
\end{proof}





\subsection{Lifts}
  Let $P$ be a polymatroid on $E = \{1, \ldots, n\}$ and
   $M$ a multisymmetric matroid on $\widetilde{E}~=~\widetilde{E}_1~\sqcup~\cdots~\sqcup~\widetilde{E_n}$. If $P$ is the polymatroid given by Corollary \ref{cor:polymat}, then we say that $M$ is a \word{multisymmetric lift} of $P$.
  If $\rk_M(\widetilde{E}_i) = |\widetilde{E_i}|$ for all $1 \leq i \leq n$, then we say that $M$ is a \word{minimal} multisymmetric lift of $P$.
\begin{thm}\label{lift}
A polymatroid $P$ has a unique minimal multisymmetric lift $\widetilde P$ constructed as follows.
  For $1 \leq i \leq n$, let $\widetilde E_i = \{1, \ldots, \rk_P(i)\}$, and 
$ \widetilde E = \widetilde E_1 \sqcup \cdots \sqcup \widetilde E_n$. Define the projection $\pi: \widetilde E \to E$ by $\pi^{-1}(i) = \widetilde E_i$.
  
 
  The minimal multisymmetric lift of $P$ is the matroid $\widetilde P$ on $\widetilde E$ with rank function
  \[
    \rk_{\widetilde P}(S) = \min \{ \rk_P(A) + |S \setminus \pi^{-1}(A)| : A \subseteq E \}.
\]
\end{thm}
\begin{example}\label{rmk:why-geometric}
  Suppose $P$ is the polymatroid realized by an arrangement of subspaces $V_1,\ldots, V_n$ in $V$ as in \cref{subspace}.
  The minimal multisymmetric lift $\widetilde P$ is realized by any hyperplane arrangement
  \[ \{V_{i,j} : 1 \leq i \leq n, \; 1 \leq j \leq \rk_P(i)\}, \]
  where $V_{i,1}, \ldots, V_{i, \rk_P(i)}$ are generic hyperplanes containing $V_i$.
  Under the correspondence between flats of $\widetilde P$ and intersections of the $V_{i,j}$'s, the geometric flats correspond to the subspaces of $V$ arise in for any choice of $\{V_{i,j}\}_{i,j}$.
\end{example}
  Versions of the construction in \cref{lift} make many independent appearances in the literature, for example, in \cite[\S2]{H72} and \cite[Propositions 3.1 and 3.2]{L77}. The most complete treatment we are aware of is \cite[\S2]{N86}, whose terminology differs from ours.
\subsubsection*{Notation.} We continue to use the notations of \cref{lift} in the remainder of this section. For visual clarity, we often write $M = \widetilde P$.
As usual, $\Gamma$ stands for the product of symmetric groups that acts on $M$.

\medskip
\begin{prop}\label{liftwelldefined}
The minimal lift $M = \widetilde P$ is a multisymmetric matroid. Explicitly,
\begin{enumerate}\itemsep 5pt
\item $\rk_{M}$ is a matroid rank function, and
\item the action of $\Gamma$  preserves flats.
\end{enumerate}
\end{prop}
\begin{proof}
 The proof of the first part is reproduced from \cite[Proof of Theorem 11.1.9]{O11}. We need to check that $\rk_M$ is non-negative, increasing, submodular, and satisfies $\rk_M(S)  \leq |S|$ for $S \subseteq \widetilde E$. 

It is clear that values of $\rk_M$ are non-negative.
Let $S \subseteq \widetilde E$ and $s \in \widetilde E \setminus S$. For all $A \subseteq E$,
\[
\rk_P(A) + |S \setminus \pi^{-1}( A)| \leq \rk_P(A) + |S\cup a \setminus \pi^{-1}( A)|,
\]
so $\rk_M$ is increasing. Moreover, by induction on $|S|$,
\begin{align*}
  \rk_M(S \cup s)
  =& \min\{\rk_P(A) + |S \cup s \setminus \pi^{-1}(A)| : A \subseteq E\}\\
  \leq& \min\{\rk_P(A) + |S \setminus \pi^{-1}(A)| : A \subseteq E\} + 1 = \rk_M(S) + 1\leq |S|+1 = |S \cup s|,
\end{align*}
so it only remains to check that $\rk_M$ is submodular.
Let $S_1, S_2 \subseteq \widetilde E$ and 
  $A_1, A_2 \subseteq E$ such that $\rk_M(S_i) = \rk_{P}(A_i) + 
  |S_i \setminus \pi^{-1}(A_i)|$. Then
\begin{align*}
\rk_M&(S_1) + \rk_M(S_2)
= \rk_{P}(A_1) + |S_1 \setminus \pi^{-1}(A_1)| + \rk_{P}(A_2) + |S_2 \setminus \pi^{-1}(A_2)|  \\
\geq& \rk_{P}(A_1 \cup A_2) + \rk_{P}(A_1 \cap A_2) + |S_1 \setminus \pi^{-1}(A_1)| + |S_2 \setminus \pi^{-1}(A_2)| \quad \text{by submodularity of $\rk_{P}$} \\
\geq& \rk_{P}(A_1 \cup A_2) + \rk_{P}(A_1 \cap A_2) + |(S_1 \cup S_2) \setminus \pi^{-1}(A_1 \cup A_2)| + |(S_1 \cap S_2) \setminus \pi^{-1}(A_1 \cap A_2)| \\
\geq& \rk_M(S_1 \cup S_2)  + \rk_M(S_1 \cap S_2).
\end{align*}
For the second part, let $\gamma \in \Gamma$ and $S \subseteq \widetilde E$.
  For any $A \subseteq E$, we have
   $|S \setminus \pi^{-1}(A)| = |(\gamma \cdot S) \setminus \pi^{-1}(A)|$, and hence 
   $\rk_M(S) = \rk_M(\gamma \cdot S)$. 
\end{proof}
The following lemma implies $\widetilde P$ is a multisymmetric lift of $P$.
\begin{lem}\label{Gfixed}
  Fix notation as in \cref{lift}. If $S \subseteq \widetilde E$ is stable under $\Gamma$, then
    $\rk_M(S) = \rk_P(\pi(S)).$
\end{lem}
\begin{proof}
  Since $S$ is stable under the action of $\Gamma$, $S$ is a union of 
  fibers of $\pi$. Hence,
    \begin{align*}
      \rk_{M}(S) =& \min \{ \rk_P(A) + |S \setminus \pi^{-1}(A)| : A \subseteq E \}\quad \text{by the definition of $\rk_{M}$}\\
      =& \min \{ \rk_P(A) + |S \setminus \pi^{-1}(A)| : A \subseteq \pi(S) \} \quad \text{because $\rk_P$ is increasing}\\
      =& \min \bigg\{ \rk_P(A) + \sum_{i\in \pi(S)\setminus 
      A}|\pi^{-1}(i)| : A \subseteq \pi(S) \bigg\}\quad \text{because 
      $S$ is a union of fibers of $\pi$}\\
      = & \min \bigg\{ \rk_P(A) + \sum_{i\in \pi(S)\setminus A}\rk_P(i) 
      : A \subseteq \pi(S) \bigg\} \quad\text{by the definition of $\pi$ 
      in \cref{lift}}\\
      \geq& \rk_P(\pi(S))\quad \text{by submodularity of $\rk_P$.}
    \end{align*}
    On the other hand, taking $A=\pi(S)$ in the definition of $\rk_M(S)$, we have $\rk_{M}(S)\leq \rk_P(\pi(S))$.
\end{proof}
\begin{proof}[Proof of \cref{lift}]
  Let $P$ be a polymatroid, and let $M = \widetilde P$ on $\widetilde E = \widetilde E_1 \sqcup \cdots \sqcup \widetilde E_n$, acted upon by $\Gamma$, be as in the statement of \cref{lift}.
  By \cref{liftwelldefined} and \cref{Gfixed}, $M$ is a multisymmetric lift of $P$. It is minimal because
  \[
    \rk_M(\widetilde E_i) = \rk_P(\pi(\widetilde E_i)) = \rk_P(i) = |\widetilde E_i|.
   \]
   It remains to check the uniqueness statement.

   Suppose $M'$ is a second minimal multisymmetric lift of $P$. Then there is an equivariant bijection between their ground sets; hence, we may think of both $M$ and $M'$ as matroids on $\widetilde E$.
   By \cref{Gfixed}, $\rk_M(A) = \rk_{M'}(A)$ for any $A \subseteq \widetilde E$ stable under the $\Gamma$-action.
   By \cref{lem:closure}\cref{lem:closurerank}, a multisymmetric matroid is determined by the ranks of its $\Gamma$-stable sets, so $M \cong M'$.
\end{proof}

\subsection{Operations}
The formation of minimal multisymmetric lifts commutes with some polymatroid operations.
Let $F$ be a flat of a polymatroid $P$ on $E$. The \word{restriction} of $P$ to $F$, denoted $P|_F$, is the polymatroid on $F$ with rank function
\[
  \rk_{P|_F}(A) \coloneqq \rk_P(A), \quad A \subseteq F.
\]
There is a lattice isomorphism
\[ \L_{P|_F} \to \{G \in \L_P: G \leq F\},\quad H \mapsto H. \]

If $P_1$ and $P_2$ are polymatroids on $E^1$ and $E^2$, respectively, then their \textit{direct sum} is the polymatroid $P_1 \oplus P_2$ on $E^1 \sqcup E^2$ with rank function
\[
  \rk_{P_1 \oplus P_2}(S) = \rk_{P_1}(S \cap E^1) + \rk_{P_2}(S \cap E^2).
\]
There is a lattice isomorphism
\[
  \L_{P_1} \times \L_{P_2} \to \L_{P_1 \oplus P_2}, \quad (F, G) \mapsto F \sqcup G.
  \]


\begin{lem}\label{lem:sum}
  If $P_1$ and $P_2$ are polymatroids, then
    $\widetilde{P_1 \oplus P_2} = \widetilde P_1 \oplus \widetilde P_2.$
\end{lem}
\begin{proof}
  From Corollary \ref{cor:polymat} and the definition of $P_1 \oplus P_2$, it follows that $\widetilde P_1 \oplus \widetilde P_2$ lifts $P_1 \oplus P_2$.
  If $\Gamma_1$ and $\Gamma_2$ are the groups acting on $\widetilde P_1$ and $\widetilde P_2$, then $\Gamma_1 \times \Gamma_2$ acts on $\widetilde P_1 \oplus \widetilde P_2$, so it is multisymmetric.
  It is minimal by minimality of $\widetilde P_1$ and $\widetilde P_2$, so the lemma holds by the uniqueness statement of \cref{lift}.
\end{proof}

\begin{lem}\label{lem:restriction}
  If $F$ is a flat of a polymatroid $P$, then
  $\widetilde{P|_F} = \widetilde P|_{\pi^{-1}(F)}.$
\end{lem}
\begin{proof}
  The rank functions of both sides are obtained by restriction, so $\widetilde P|_{\pi^{-1}(F)}$ lifts $\widetilde{P|_F}$.
  If $\Gamma$ acts on $\widetilde P$, then a subset of $\Gamma$'s factors acts on $\widetilde{P}|_{\pi^{-1}(F)}$, so it is multisymmetric.
  It is minimal because $\widetilde P$ is, so \cref{lift} implies the lemma.
\end{proof}


\section{Building sets and Bergman fans}\label{sec:buildingsets}
Here, we recall the combinatorics of geometric building sets. We then generalize the definition of the Bergman fan of a polymatroid given in the introduction by associating a fan to each polymatroid equipped with a geometric building set.
\subsection{Geometric building sets}
Let $P$ be a polymatroid.
If $\mathscr{G} \subseteq \L_P$ and $F$ is a flat, let
\[ \mathscr{G}_{\leq F} \coloneqq \{ G \in \mathscr{G} : G \leq F\} \]
and write $\max \mathscr{G}$ for the set of maximal elements of $\mathscr{G}$.
\begin{dfn}
  A \word{geometric building set}\footnote{Our terminology is that of \cite[Definition 4.4]{FK04}. In \cite{PP21}, geometric building sets are called ``combinatorial building sets'', but \cite{FK04} uses this term for collections that satisfy only the isomorphism condition. 
  The ``combinatorial building sets'' of \cite{FK04} are the same as the ``building sets'' of \cite{FY04}. Postnikov considers only building sets for the Boolean lattice in \cite[Definition 7.1]{P09}; all such building sets are geometric.} of a polymatroid $P$ is a collection $\G$ of nonempty flats such that for all $F \in \L_P \setminus \{\emptyset\}$, the map
  \begin{equation*}
    \prod_{G \in \max \G_{\leq F}} \L_{P|_G} \to \L_{P|_F} \label{factorization}
    \end{equation*}
  is an isomorphism, and
  \begin{equation*}
 \sum_{G \in \max \G_{\leq F}} \rk(G) = \rk(F). \label{combobuildingset}
\end{equation*}
  If $E \in \G$, then a \word{nested set} of $P$ with respect to $\G$ is a subset $\N \subseteq  \G$ such that for all $\{F_1, \ldots, F_k\} \subseteq \N$ pairwise incomparable with $k \ge 2$, we have that 
  \[ \cl_P(F_1 \cup \cdots \cup F_k)  \not \in \G.\]
\end{dfn}
With respect to a fixed geometric building set, a subset of a nested set is nested, so nested sets form a simplicial complex.
All building sets are henceforth assumed to contain $E$\footnote{In the realizable case, this assumption guarantees the associated wonderful compactification is smooth and can be described as an iterated blow-up \cite[\S4.1]{dCP95}.  Combinatorially, we lose nothing by this assumption \cite[Remark 4.1]{PP21}.}.
\begin{example}
  The maximal geometric building set of $P$ is the collection of all nonempty flats.
  With respect to this building set, the nested sets are flags of nonempty flats.
\end{example}

\begin{lem}\label{liftedbs}
  Let $P$ be a polymatroid, $\widetilde P$ its minimal multisymmetric lift, and $\pi$ as in \cref{lift}.
  If $\mathscr G$ is a geometric building set for $P$, then 
\[ \widetilde{\mathscr G} = \{ \pi^{-1}(G) : G \in \mathscr G\} \cup \{\text{atoms of $\mathscr L_{\widetilde P}$}\} \]
is a geometric building set for $\widetilde P$.
\end{lem}
\begin{proof}
Let $F$ be a flat of $M = \tilde P$.
By \cref{lem:sum} and \cref{lem:restriction}, the map
\[ \prod_{G \in \max \widetilde \G_{\leq F^{\hspace{0.05ex}\mathrm{geo}}}} \L_{M|_G} \to \L_{M|_F} \]
  factors into a chain of isomorphisms
  \[
    \prod_{G \in \max \widetilde \G_{\leq F^{\hspace{0.05ex}\mathrm{geo}}}} \L_{M|_G}
    \cong \prod_{H \in \max \G_{\leq \pi(F^{\hspace{0.05ex}\mathrm{geo}})}} \L_{\widetilde{P|_H}}
    \cong \L_{\widetilde{P|_{\pi(F^{\hspace{0.05ex}\mathrm{geo}})}}} \cong \L_{M|_{F^{\hspace{0.05ex}\mathrm{geo}}}},
\]
and by \cref{Gfixed},
\[
  \sum_{G \in \max \widetilde \G_{\leq F^{\hspace{0.05ex}\mathrm{geo}}}} \rk_{M}(G)
  = \sum_{G \in \max \widetilde \G_{\leq F^{\hspace{0.05ex}\mathrm{geo}}}} \rk_{P}(\pi(G))
  = \sum_{H \in \max \G_{\leq \pi(F^{\hspace{0.05ex}\mathrm{geo}})}} \rk_{P}(H)
  = \rk_P(\pi(F^{\hspace{0.05ex}\mathrm{geo}})) = \rk_M(F^{\hspace{0.05ex}\mathrm{geo}}).
\]
Consequently, by \cref{lem:closure}\cref{lem:closurerank},
\[
  \prod_{G \in \max \widetilde\G_{\leq F}} \L_{M|_G}
   = \prod_{G \in \max \widetilde\G_{\leq F^{\hspace{0.05ex}\mathrm{geo}}}}  \L_{M|_G} \times \prod_{i \in F \setminus F^{\hspace{0.05ex}\mathrm{geo}}}  \L_{M|_{\{i\}}}
  \cong  \L_{M|_{F^{\hspace{0.05ex}\mathrm{geo}}}} \times \prod_{i \in F \setminus F^{\hspace{0.05ex}\mathrm{geo}}} \L_{M|_{\{i\}}} \cong \L_{M|_F},
\]
and
\[
  \sum_{G \in \max \widetilde{\mathscr{G}}_{\leq F}} \rk_{M}(G)
  = \rk_M(F^{\hspace{0.05ex}\mathrm{geo}}) + \sum_{i \in F \setminus F^{\hspace{0.05ex}\mathrm{geo}}} \rk_{M}(i)
  = \rk_M(F^{\hspace{0.05ex}\mathrm{geo}})+|F \setminus F^{\hspace{0.05ex}\mathrm{geo}}|
  = \rk_M(F),
\]
as desired.
\end{proof}
\begin{lem}\label{lem:nested}
  If $\G$ is a geometric building set for $P$, then $\N \subseteq \G$ is $\G$-nested if and only if $\widetilde \N = \{ \pi^{-1}(F) : F \in \N\}$ is $\widetilde \G$-nested.
\end{lem}
\begin{proof}
  Set $M = \widetilde P$.
  Let $\{F_1, \ldots, F_k\}\subseteq \mathscr N$ be a set of pairwise incomparable flats.
  By \cref{cor:sublattice}, the geometric flats of $M$ form a sublattice of $\L_M$, isomorphic to $\L_P$ by \cref{lift}. Hence,
  \[
    \cl_M(\pi^{-1}(F_1) \cup \cdots \pi^{-1}(F_k)) \in \widetilde \G  \iff  \cl_P(F_1 \cup \cdots \cup F_k) \in \G.  \qedhere
  \]
\end{proof}

\begin{rmk}[Geometry of building sets]\label{rmk:geomOfBS}
  Let $\{V_i\}_i$ be a subspace arrangement in $V$, defining a polymatroid $P$.
  Let $\G$ be a building set for $P$.
  By \cite[\S1.6]{dCP95}, the \textit{wonderful compactification} of $\{V_i\}_i$ with respect to $\G$ can be constructed by blowing up $\P(V)$ along all subspaces $\cap_{i \in F} \P(V_i)$ with $F \in \G$, first blowing up those of dimension 0, then those of dimension 1, and so on.
Let $\{V_{ij}\}_{ij}$ be a hyperplane arrangement realizing $\widetilde P$, as in \cref{rmk:why-geometric}.
  Blowing up a codimension 1 subvariety is an isomorphism, so \cref{lem:nested} implies that the wonderful compactification of $\{V_i\}_i$ with respect to a building set $\G$ is isomorphic to the wonderful compactification of $\{V_{ij}\}_{ij}$ with respect to $\widetilde \G$.
\end{rmk}

\subsection{Bergman fans}
Let $P$ be a polymatroid on $E$, with minimal lift $\widetilde{P}$ on $\widetilde E$.
Let $\R^{\widetilde E}$ be the vector space spanned by $\e_i$ for $i \in \widetilde E$, and write $\e_S \coloneqq \sum_{i \in S} \e_i$ for $S \subseteq \widetilde E$. 
If $\mathscr S \subseteq 2^{\widetilde{E}}$ is a collection of subsets, write
\[
  \sigma_{\mathscr S} \coloneqq \cone(\e_S : S \in \mathscr S) \subseteq \R^{\widetilde E} / \R (1,1,\ldots,1).
\]
\begin{dfn}\label{dfn:bergmanfan}
  Let $P$ be a polymatroid on $E$, and $\G \subseteq \L_P$ a geometric building set of $P$.
  The \word{Bergman fan} associated to $(P, \G)$ is
  $\Sigma_{P, \G} \coloneqq \{\sigma_{\N}\}_\N$,
  where $\N$ ranges over all $\widetilde\G$-nested sets of $\widetilde P$ such that $\widetilde E \not \in \N$.
\end{dfn}
If $P$ is a matroid and $\G$ is the maximal geometric building set of $P$, then $\widetilde P = P$ and $\widetilde \G = \G$. In this case, $\Sigma_{P,\G}$ coincides with the Bergman fan of \cite[Definition 3.2]{AHK18}.

\begin{lem}
  If $P$ is a polymatroid and $\G$ is the maximal geometric building set, then $\Sigma_{P, \G}$ coincides with the Bergman fan $\Sigma_P$ defined in the introduction (\cref{dfn:maximalBergman}). 
\end{lem}

\begin{proof}
  Fix a proper flat $F$ of $P$ and a subset $S \subseteq \widetilde{E}$. It suffices to show that the closure of $\pi^{-1}(F) \cup T$ is not geometric for all nonempty $T \subseteq S \setminus \pi^{-1}(F)$ if and only if the following inequality holds for all nonempty $T \subseteq S \setminus \pi^{-1}(F)$,
  \[
  \rk_P(F \cup \pi(T)) >\rk_P(F) + |T|.
  \]
  First, assume the displayed inequality holds for all such $T$. The inequality's left-hand side is the rank of the smallest geometric flat containing $\pi^{-1}(F) \cup T$. Because the right-hand side is at least $\rk_P(\pi^{-1}(F) \cup T)$, the closure of $\pi^{-1}(F) \cup T$ is not geometric. 
  
  Conversely, assume there is a nonempty $T\subseteq S \setminus \pi^{-1}(F)$ for which the inequality fails. We claim there is a nonempty $T'\subseteq S \setminus \pi^{-1}(F)$ such that the closure of $\pi^{-1}(F)\cup T'$ is geometric. If $\rk_P(F \cup \pi(T)) = \rk_P(F) + |T|$, then take $T'=T$. Otherwise, if $\rk_P(F \cup \pi(T)) < \rk_P(F) + |T|$, then there is a circuit $C \subseteq \pi^{-1}(F) \cup T$ such that $C \setminus \pi^{-1}(F)$ is nonempty. Writing $C \setminus \pi^{-1}(F) = \{c_1,\ldots,c_k\}$, we have $c_i \in \cl_{\widetilde{P}}(\pi^{-1}(F) \cup C \setminus c_i)$ for all $i$ (see \cite[Lemma 1.4.2]{O11}). Therefore by symmetry, $\pi^{-1}(\pi(c_i)) \subseteq \cl_{\widetilde{P}}(\pi^{-1}(F) \cup C)$ for all $i$, so the closure of $\pi^{-1}(F) \cup C$ is geometric. Now we can take $T'=C\setminus \pi^{-1}(F)$.
\end{proof}



\medskip
\begin{lem}\label{polytope}
  If $P$ is a polymatroid and $\G$ is a geometric building set of $P$, then $\Sigma_{P, \G}$ is a subfan of the normal fan of a convex polytope.
\end{lem}
\begin{proof}
  Let $E$ be the ground set of $P$ and $B$ be the Boolean polymatroid with $\rk_B(i) = \rk_P(i)$ for all $i \in E$.
  Suppose $\G$ is a building set for $P$. The definition of geometric building set implies for any flat $F$ of $P$, $\max \G_{\leq F}$ is a partition of $F$. Hence, $\widetilde \G$ is a building set of both $\widetilde B$ and $\widetilde P$, and there is an inclusion of fans
  $\Sigma_{P, \G} = \Sigma_{\widetilde P, \widetilde \G} \subseteq \Sigma_{\widetilde B, \widetilde \G}$ (In fact, if all single-element subsets of $E$ are flats of $P$, then $\G$ is a geometric building set for $B$, and we obtain $\Sigma_{P,\G} \subset \Sigma_{\widetilde B, \widetilde \G} = \Sigma_{B, \G}$).
  The lattice of flats of $\widetilde B$ is isomorphic to the lattice of subsets of $\widetilde E$,  so $\Sigma_{\widetilde B, \widetilde \G}$ is the normal fan of a convex polytope by \cite[Theorem 7.4]{P09}.
\end{proof}



By \cite{Bri96}, for any polymatroid $P$ and a geometric building set $\G$, $A(\Sigma_{P,\G})$ has the following presentation.
\begin{prop}\label{fanchow}
  The \word{Chow ring} of $\Sigma_{P,\G}$ satisfies 
  \[
    A(\Sigma_{P,\G}) = \Z[z_G : G \in \widetilde \G \setminus \{\widetilde E\}] / I_{P, \G}
  \]
  where $I_{P, \G}$ is the ideal generated by 
  \begin{align*}
      z_{G_1} \cdots z_{G_k} \quad &\text{for any not $\widetilde \G$-nested collection $\{G_1, \ldots, G_k\}$},\\
      \sum_{i \in G} z_G - \sum_{j \in F} z_F \quad &\text{for any } i,j \in \widetilde E.
  \end{align*}
In particular, the Chow ring of $\Sigma_P$ satisfies
\[
  A(\Sigma_P) = \Z[z_F : F \text{ nonempty proper flat of $P$}] \otimes \Z[z_i : i \in \widetilde E] / I_P,
\]
where $I_P$ is an ideal generated by the following polynomials where $z_\emptyset$ is replaced by 1 wherever it appears:
\begin{align*}
  z_{F_1} z_{F_2},  \quad &\text{$F_1$ and $F_2$ are incomparable proper flats of $P$,} \\
  z_F \prod_{i \in T} z_i,  \quad &\text{$F$ is a proper flat and  $T\subseteq \widetilde E\setminus \pi^{-1}(F)$ is nonempty satisfying $\rk(F \cup \pi(T)) \leq \rk(F) + |T|$,} \\
  \sum_{i \in F} z_F - \sum_{j \in G} z_G,  \quad &\text{$i$ and $j$ are elements of $\widetilde E$.}
\end{align*}
 
 
 
\end{prop}

We identify $A^1(\Sigma_{P,\G})$ with the space of piecewise linear functions on the support of $\Sigma_{P, \G}$, modulo global linear functions. Explicitly, a piecewise linear function $\ell$ is a representative of 
\[
  \sum_{\mathbf{u}_F} \ell(\mathbf{u}_F) z_F \in A^1(\Sigma_{P, \G}),
\]
where the sum is over all primitive ray generators $\mathbf{u}_F$ of rays of $\Sigma_{P, \G}$.

\begin{rmk}\label{fypresentation}
  A slightly different presentation of $A(\Sigma_{P, \G})$ is used by \cite{FY04}. 
  For $F \in \widetilde \G$, set
    \[ y_F = \begin{cases} -\sum_{i \in G} z_G, &F = \widetilde E \\ z_F, &\text{otherwise.} \end{cases} \]
    In terms of the $y_F$'s, $A(\Sigma_{P, \G})$ is defined by the ideal $I_{FY}$ generated by
  \begin{align*}
    y_{G_1} \cdots &y_{G_k}, \quad \text{$\{G_1, \ldots, G_k\}$ not $\widetilde \G$-nested and }\\
    \sum_{i \in G}& y_G, \quad i \in \widetilde E.
  \end{align*}
\end{rmk}
\begin{rmk}[Tropicalization and the Bergman fan]\label{tropicalization}
  Suppose $P$ is the matroid realized by an essential hyperplane arrangement $V_1,\ldots,V_n$ in $V$, defined by linear functionals $\ell_1,\ldots,\ell_n$.
  The inclusion
  \[ \P(V \setminus \cup_i V_i) \hookrightarrow \P((\mathbb{F}^*)^n), \quad v \mapsto [\ell_1(v): \ldots: \ell_n(v)] \]
  shows that $\P(V \setminus \cup_i V_i)$ is a very affine variety. Its tropicalization is the support of $\Sigma_P$, by \cite[\S9.3]{S02} and \cite[\S3]{AK06} (see also \cite[Theorem 4.1]{FS05}).
  The corresponding statement for realizable polymatroids does not make sense: the complement of a subspace arrangement may not be very affine, so tropicalization  \cite[Definition 3.2.1]{MS15} is not defined. Nevertheless, if $\{V_i\}_i$ is a subspace arrangement (so $P$ is a polymatroid), then generic hyperplanes $\{V_{ij}\}_{ij}$ realizing $\widetilde P$ as in \cref{rmk:why-geometric} define a subtorus of $\P(\prod_i V/V_i)$. Tropicalizing this torus's intersection with $\P(V \setminus \cup_i V_i)$
  gives the support of $\Sigma_{\widetilde P}$, which coincides with that of $\Sigma_P$.
 
\end{rmk}

\section{Chow rings of polymatroids}\label{sec:chow}
If $\cS$ is a collection of sets, write $\cup \cS$ for the union of the elements of $\cS$.
In \cite[Section 4]{PP21}, Pagaria and Pezzoli define the Chow ring of a polymatroid as follows\footnote{Unlike \cite{PP21}, we use $\Z$-coefficients.}.

\begin{dfn}\label{dfn:ppchow}
 Let $P$ be a polymatroid and $\G \subseteq \L_P$ a geometric building set. 
 The \word{Chow ring} of $(P, \G)$ is
 \[
   \DP(P, \G) \coloneqq \Z[x_F : F \in \G] / I,
 \]
 where $I$ is the ideal generated by
\[
x_{G_1} \cdots x_{G_k} \Big(\sum_{H \geq G} x_H \Big)^b
\]
for $G \in \G$, $\mathscr S = \{G_1, \ldots, G_k\} \subseteq \G$, and
$b \geq \rk_P(G) - \rk_P( \cup \mathscr S_{<G} ).$
\end{dfn}
When $P$ is realizable by an arrangement of subspaces, $\DP(P, \G)$ is the cohomology ring of De Concini \& Procesi's wonderful compactification of the arrangement complement.
In this section, we make use of the theory of Gr\"obner bases. For background on this subject, see \cite[Chapter 15]{E95}.

\begin{thm}\label{thm:isomorphism}
  Let $P$ be a polymatroid on $E$, $\widetilde E$ the ground set of $\widetilde P$, and $\pi: \widetilde E \to E$ the projection.
  There is an isomorphism $\operatorname{DP}(P, \mathscr G) \cong A(\Sigma_{P, \G})$ sending $x_F$ to $y_{\pi^{-1}(F)}$.
\end{thm}
\begin{proof}
    Let $I_{DP} \subseteq \Z[x_F : F \in \G]$ be the defining ideal of 
  $\DP(P, \G)$, and let $I_{FY} \subseteq \Z[y_F : F \in \widetilde \G]$ be the defining ideal of $A(\Sigma_{P,\G})$ as in \cite{FY04} (See
  \cref{fypresentation}). We define the following map on polynomial rings.
  \[
   \varphi: \Z[x_F : F \in \G] \to \Z[y_F : F \in \widetilde \G], \quad x_F \mapsto y_{\pi^{-1}(F)}
  \]
    First we show that $\varphi(I_{DP}) \subseteq I_{FY}$. Write 
  $f \in I_{DP}$ for one of the defining relations of $I_{DP}$:
\[ 
  f = \Big(\prod_{F \in \mathscr S} x_F \Big)\Big(\sum_{\mathscr G \ni 
  H \geq G} x_H \Big)^b.
  \]
  By \cite[Theorems 1 and 3]{FY04}, $I_{FY}$ contains the following two types of polynomials:
  \begin{align*}
\prod_{F \in \mathscr S}& y_F, \qquad \mathscr S \text{ not $\widetilde{\mathscr{G}}$-nested,} \\
    \prod_{F \in \N} y_F \Big( \sum_{H \geq G}& y_G \Big)^d, \quad \text{$\N$ a nested antichain, $\cup \N < G$, and $d = \rk(G) - \rk(\cup N)$.} 
\end{align*}
If $\mathscr S$ is not $\mathscr G$-nested, then $\tilde{\mathscr{S}} \coloneqq \{ \pi^{-1}(F) : F \in \mathscr S\}$ is not $\tilde{\mathscr G}$-nested by \cref{lem:nested}. Hence, $\varphi(f)$ is divisible by a relation of the first type. 
Otherwise, $\mathscr S$ is $\mathscr{G}$-nested, so $\tilde {\mathscr {S}}$ is $\widetilde{\mathscr{G}}$-nested.
In this case, $\varphi(f)$ is divisible by a relation of the second type because
\begin{align*}
 b &\geq
\rk_P(G) - \rk_P(\cup \mathscr S_{<G}) 
  = \rk_{\tilde P}(\pi^{-1}(G)) - \rk_{\tilde P}(\cup \mathscr{\tilde S}_{< \pi^{-1}(G)}) 
  = \rk_{\tilde P}(\pi^{-1}(G)) - \rk_{\tilde P}(\cup \max \mathscr{\tilde S}_{<\pi^{-1}(G)}).
\end{align*}
This proves that $\varphi(I_{DP}) \subseteq I_{FY}$, so $\varphi$ descends to
$\bar \varphi: \DP(P,\G) \to A(\Sigma_{P,\G})$.

If $F \in \widetilde \G$ is a flat of rank greater than 1, then $y_{F}$ is in the image of $\bar \varphi$.
By the linear relation $\sum_{i \in G} y_G = 0$, it follows that $y_i$ is also in the image of $\bar\varphi$. Therefore, $\bar\varphi$ is surjective. 
It remains to show that $\bar \varphi$ is injective.
By \cite[Theorem 2]{FY04}, the generators of $I_{FY}$ in the previous paragraph
are a Gr\"obner basis with respect to any lexicographic monomial order $<$ in which $F_1 \subseteq F_2$ implies $y_{F_1} > y_{F_2}$.
Any such order is an elimination order with respect to $\{y_i: i \in \widetilde E\}$.
By \cite[Proposition 15.29]{E95}\footnote{Eisenbud's proof of this statement works over $\Z$ because all leading coefficients in our Gr\"obner basis are 1.\label{zgbnote}}, 
the generators of $I_{FY}$ in the previous paragraph
that do not involve any $y_i$, $i \in \widetilde E$, are a Gr\"obner basis for $\im(\varphi) \cap I_{FY}$.
Any such polynomial is the image of a generator of $I_{DP}$, so $\varphi^{-1}(I_{FY}) = I_{DP}$.
This implies $\bar \varphi$ is an isomorphism.
\end{proof}
The proof of \cref{thm:isomorphism} also shows the following, originally obtained from \cite[Corollary 2.8]{PP21}.
\begin{cor}\label{cor:gb}
  The defining relations of $\DP(P, \G)$ in 
  \cref{dfn:ppchow} form a Gr\"obner basis with respect to any lexicographic $<$ such that $x_{F_1} < x_{F_2}$ whenever $F_1 \supsetneq F_2$.
\end{cor}
This recovers the monomial basis of \cite{PP21}.
\begin{cor}\label{cor:basis}
  The following monomials are a $\Z$-basis of $\DP(P,\G)$:
  \[
    x_{G_1}^{a_1} \cdots x_{G_k}^{a_k}
  \]
  where $\N = \{G_1, \ldots, G_k\}$ is a nested set of $\G$, and 
  \[ 1 \leq a_i < \rk(F_i) - \rk(\cup \N_{<F_i})\quad \text{for all $1 \leq i \leq k$.} \]
   
\end{cor}
\begin{proof}
Immediate from \cref{thm:isomorphism} and \cite[Corollary 1]{FY04} (or \cref{cor:gb} and \cite[Theorem 15.3]{E95}).
\end{proof}
\begin{rmk}
  In \cite[Corollary 2.8]{PP21}, degree-lexicographic order (also called ``homogeneous lexicographic'' or ``graded lexicographic'' order) is used.
  Since the defining relations of $\DP(P,\G)$ in 
  \cref{dfn:ppchow} are all homogeneous, their initial terms with respect to the lex or degree-lex orders are the the same. 
  Consequently, they are a Gr\"obner basis with respect to one order if and only if they are with respect to the other.
\end{rmk}
\begin{rmk}\label{rmk:geoProof}
      Suppose $P$ is a polymatroid with building set $\G$, realized by an arrangement $\{V_i\}_i$ in $V$. Let $\{V_{ij}\}_{ij}$ realize $\widetilde P$ as in \cref{rmk:why-geometric}. In this case, an alternate proof of \cref{thm:isomorphism} is possible.
  By \cite[\S4]{FY04}, $A(\Sigma_{P, \G})$ is isomorphic to the Chow ring of the wonderful compactification of $\{V_{ij}\}_{ij}$ with respect to $\widetilde \G$.
  By \cref{rmk:geomOfBS}, the wonderful compactifications of $\{V_{ij}\}_{ij}$ with respect to $\widetilde \G$ and $\{V_i\}_i$ with respect to $\G$ are isomorphic. Hence, their Chow rings are isomorphic. The Chow ring of the latter space is isomorphic to $\DP(P, \G)$ by a comparison of the presentations in \cref{dfn:ppchow} and \cite[Theorem 5.2]{dCP95}. 
\end{rmk}
In the remainder of this section, we recover \cite[Theorem 4.7]{PP21} and generalize \cite[Theorem 4.21]{PP21} using \cref{thm:isomorphism} and the tropical Hodge theory of \cite{ADH20}.
If $R$ is a $\Z$-algebra, define $R_\Q \coloneqq R \otimes_\Z \Q$ and $R_\R$ likewise.
\begin{cor}\label{cor:hodge}
  \newcommand{\mathbb{K}}{\mathbb{K}}
  Let $\mathbb{K} = \Q, \R$.
  Let $P$ be a polymatroid of rank $r$ and $\G$ a geometric building set.
  Let $\ell$ be any $\mathbb{K}$-valued strictly convex piecewise linear function on $\Sigma_{P,\G}$, viewed as an element of $A^1(\Sigma_{P,\G})_\mathbb{K} \cong \DP^1(P,\G)_\mathbb{K}$.
  \begin{enumerate}\itemsep 5pt
  \item \label{poincare} (Poincar\'e duality) There is an isomorphism 
    \[ \deg: \DP(P,\G)^{r-1} \to \Z, \]
    and for all $0 \leq k < r/2$, the pairing
      \[ \DP^k(P,\G) \times \DP^{r-k-1}(P, \G) \to \Z, \quad (a,b) \mapsto \deg(ab)  \]
    is non-degenerate.
\item \label{lefschetz} (Hard Lefschetz) For every $0 \leq k < r/2$, the multiplication map
  \[ \DP^k(P,\G)_\mathbb{K} \to \DP^{r-k-1}(P,\G)_\mathbb{K}, \quad a \mapsto \ell^{r - 2k-1} a \]
  is an isomorphism.
\item \label{hodge-riemann} (Hodge-Riemann) For every $0 \leq k < r/2$, the bilinear form
  \[
    \DP^k(P,\G)_\mathbb{K} \times \DP^k(P,\G)_\mathbb{K} \to \mathbb{K}, \quad (a,b) \mapsto (-1)^k \deg(\ell^{r-2k-1} ab)
  \]
  is positive definite on the kernel of multiplication by $\ell^{r-2k}$.
  \end{enumerate}
\end{cor}
\begin{proof}
  Let $M = \widetilde P$.
  By \cite[Theorem 4]{FY04}, $\Sigma_M$ is a refinement of $\Sigma_{P,\G}$. In particular, the two fans have the same support.
  By \cite[Proposition 2.4]{AHK18} and \cref{polytope}, both fans are subfans of the normal fans of convex polytopes, so both fans support strictly convex piecewise linear functions. By \cite[Proposition 2]{FY04}, $\Sigma_{P, \mathscr{G}}$ is a smooth fan. 

  The desired statements now follow by applying \cite[Theorem 1.6]{ADH20}, \cite[Proposition 5.2]{AHK18}, and \cite[Theorems 6.19 and 8.8]{AHK18}.
  Loosely, \cite[Theorem 1.6]{ADH20} says properties \cref{poincare}, \cref{lefschetz}, and \cref{hodge-riemann} hold for $A(\Sigma_{P,\G})_\R \cong \DP(P,\G)_\R$ if and only if they hold for $A(\Sigma_M)_\R$, and \cite{AHK18} verifies them for $A(\Sigma_M)_\R$.

  Properties \cref{lefschetz} and \cref{hodge-riemann} for $\DP(P, \G)_\Q$ follow immediately from those for $\DP(P, \G)_\R$.
  For \cref{poincare}, note that \cite{ADH20}'s Poincar\'e duality arguments go through over $\Z$. (Explicitly, one must check statements 6.6--6.9, and Propositions 6.16, 6.17 of \cite{ADH20}.)
\end{proof}
\begin{rmk}\label{ConeRemark}
  In \cite{PP21}, \cref{cor:hodge} is proved for $\ell$ in the \word{$\sigma$-cone} \cite[Definition 4.15]{PP21}, the positive span of 
  \[
    - \sum_{G \in \G_{\geq F}} x_G, \quad F \in \G
  \]
  in $\DP(P, \G)$. The $\sigma$-cone is generally a proper subset of the cone of strictly convex piecewise linear functions on $\Sigma_{P, \G}$.
  For example, if $M$ is a loopless matroid on $E$ and $\G$ is its maximal building set, then for any $i \in E$,
  \[
    \beta \coloneqq \sum_{i \not \in F} z_F = - \sum_{|G| > 1} \big(|G| - 1\big) y_G
  \]
  is in the closure of the cone of strictly convex piecewise linear functions on $\Sigma_M$ \cite[Proposition 4.3, Lemma 9.7]{AHK18}.
  However, $\beta$ may not be in the closure of the $\sigma$-cone, e.g. when $M$ is Boolean of rank at least 3.
  For comparison of the $\sigma$-cone to the ample cone of the wonderful compactification, see \cite[Remark~4.22]{PP21}.
\end{rmk}
