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\newcommand{\mrm}{\mathrm}
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\newcommand{\R}{\mathbb{R}}
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\newcommand{\Brick}{\mathrm{Brick}}
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%%%

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\title[Breaking bricks into blocks]{A degeneration of the brick variety and a mixed subdivision of the associahedron into cubes}

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\author[]{\texorpdfstring{Raj
Gandhi\thanks{\href{mailto:rg593@cornell.edu}{rg593@cornell.edu} R.G. was partially supported by a Postgraduate Scholarship - Doctoral - from the Natural Sciences and Engineering Research Council of Canada.}\addressmark{1}
\and 
Gabe Udell\thanks{\href{mailto:gru5@cornell.edu}{gru5@cornell.edu}.  G.U. was supported by the National Science Foundation Graduate Research Fellowship}\addressmark{2}}{Raj
Gandhi
and 
Gabe Udell}}

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\address{\addressmark{1}Department of Mathematics, Cornell University, Ithaca, NY 14853, USA\\ \addressmark{2}Department of Mathematics, Cornell University, Ithaca, NY 14853, USA}

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\abstract{We study a degeneration of Escobar's brick variety in cases where it is toric. We show that the components of the central fibre of the degeneration are reduced toric products of Richardson varieties. We write the moment polytope of the variety, which has the same normal fan as the brick polytope of Pilaud and Santos, as a Minkowski sum of Bruhat interval polytopes and show that the polyhedral subdivision induced by the degeneration is a mixed subdivision. As corollaries, we obtain a characterization of toric Richardsons, and a subdivision of the associahedron into combinatorial cubes. %Along the way we prove a new characterization of toric Richardson varieties.
}

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\keywords{brick polytope, associahedron, polyhedral subdivision, Richardson variety}

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\begin{document}


\maketitle
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\section{Introduction}

    Toric geometry provides a bridge between algebraic geometry and discrete geometry. Every projectively embedded toric variety has a moment polytope, and every lattice polytope satisfying certain conditions comes from a toric variety. Characteristics of the moment polytope can be read off from its toric variety (e.g., the dimension of the moment polytope is the dimension of the effective torus action on the variety.)  

    \details{
For root independent words, the brick variety is toric, and for all words, the moment polytope of the brick variety is the brick polytope. For that word, the brick polytope is the Loday realization of the associahedron \cite{PS15}. c associahedron words are root independent \cite{PS15}. And can be done in other types. For that word based on a different Coxeter element $c$, it is the $c$-associahedron studied by Hugh Thomas, et al. }
    
    Our focus is on brick varieties and  brick polytopes. Let $Q$ be a word in the simple generators of $S_n$, and let $w\in S_n$. 
    The \textit{brick polytope} was defined by Pilaud and Santos in \cite{bricksort} as a polytopal realization of certain subword complexes $\Delta(Q,w)$; the \textit{subword complex} was originally defined by Knutson and Miller \cite{KM04}. The \textit{brick variety} $\mrm{Brick}^Q$, with respect to $Q$, was defined by Escobar in \cite{E16}. There, Escobar showed that, under certain conditions on $Q$, the brick variety $\mrm{Brick}^Q$ is a toric variety, and, moreover, that it is the toric variety of the brick polytope. We introduce a slight variation on the brick polytope called the \textit{bulky brick polytope} which is the moment polytope of the brick variety with respect to a different choice of projective embedding from \cite{E16}. Our first result is that the bulky brick polytope has a Minkowski sum decomposition into Bruhat interval polytopes (\Cref{thm:Minkowski}). Using this, we prove a new characterization of toric Richardson varieties (\Cref{thm:characterization}).

    In \cite{S91}, Sturmfels proved that Gr\"obner degenerations of toric varieties correspond to regular subdivisions of their moment polytopes. 
    Applying this principle, we describe a degeneration of $\Brick^Q$ in \Cref{section:degeneration} and study the corresponding subdivision of the brick polytope.
    %In \Cref{section:degeneration}, we define a degeneration of a toric brick variety $\mrm{Brick}^Q$.
    We show that the central fibre of this degeneration is a union of reduced products of toric Richardsons, and that the degeneration induces a  mixed subdivision of the corresponding brick polytope. In a special case, we apply results of \cite{LMP21} to produce a subdivision of the associahedron into pieces combinatorially equivalent to cubes. See \Cref{fig:enter-label} for illustrations of the subdivisions of the 2D and 3D associahedra.




    
    % We show in all cases that the induced subdivision is a mixed subdivision.
   % Using the fact that the components of the central fibre are smooth toric Richardsons, we apply recent results of Lee--Masuda--Park \cite{LMP21} to show that the components of the induced subdivision of the brick polytope are combinatorially equivalent to cubes. Furthermore, we show that the induced subdivision is a \textit{mixed} subdivision. For a special cas 
	\details{
	Slicing and dicing shapes and all is pretty cool, but what's even cooler than cubes? Ice cubes? Cubical popsicles? So cool! The purpose of this abstract is to slice and dice friends of the associahedra into cubes. Sorry friends! This is in line with loads of literature on this stuff. I mean, come on, look at [insert references here].
	
	Okay, before jumping the gun into the main results, let's just give some perspective here. The \textbf{brick polytope} was defined by Pilaud--Stump \cite{PS15} in their search for a polytopal realization of the subword complex of Knutson--Miller \cite{KM04}. Escobar gave geometric meaning to the brick polytope in \cite{E16}, where the brick manifold was defined, an algebraic symplectic variety with an algebraic symplectic torus action, whose moment polytope is the brick polytope. In certain cases, the brick manifold is isomorphic to Brion's resolution of the Richardson variety (see \cite[Section 3.4]{E16} and \cite{B05}). Certain subsets of the set of pseudoline arrangements in a sorting network end up corresponding to fixed points of the torus action on the brick manifold (\cite{E16}).
	
	It is well-known that every toric Schubert variety is smooth. To everyone's dismay, not every toric Richardson variety in smooth. \commentsG{The final version should definitely include the phrase ``to everyone's dismay'' somewhere} Lee--Masuda--Park have given characterizations of smooth toric Richardsons in terms of the dimensions of their moment polytopes (\cite{LMP21}). In particular, a toric Richardson variety is smooth if and only if its moment polytope is combinatorially equivalent to a cube. Results in \cite{LMP21} will be a main protagonist in many proofs in this paper (an antagonist to cube haters). So wait.. what are the results in question? 
    
	In Section XX, we degenerate Escobar's brick manifold into components, which end up being products of Richardson varieties. This degeneration is ``nice'' in the sense that it is a Grobner degeneration that commutes with torus actions. In particular, the components that appear after the degeneration, as well as the Richardons that appear as factors in each component, are toric. On the moment polytope side, this gives a slicing and dicing of the brick polytope into the moment polytopes of each component (see \cite{S91}). Our main result is that the Richardsons that appear in each component of the degeneration are smooth, and ultimately, that after Minkowski summing them, we get a dissection of the brick polytope into cubes. Cubes!!!
}


 \begin{figure}
    \centering \includegraphics[width=0.224\linewidth]{Assoc3.png}
\includegraphics[width=0.185\linewidth]{newnewassoc1.png}%{assoc4.png}
\includegraphics[width=0.185\linewidth]{newnewassoc2.png}%{assoc4more.png}
\includegraphics[width=0.243\linewidth]{bulky.png}%{assoc4more.png}
    \caption{Left: Subdivision of the 2D associahedron into combinatorial cubes. Center: Projection of subdivision of the 3D associahedron into combinatorial cubes. Right: Projections of the skeletons of the 3D associahedron and 3D bulky associahedron.}
    \label{fig:enter-label}
\end{figure}

    \section{The brick polytope}
    We recall the \textit{brick polytope} of \cite{bricksort} and \cite{PS15}, closely following the exposition in \cite{E16}, and we define the \textit{bulky brick polytope}, laying the foundations for this paper.
    
    Consider the symmetric group $S_n$ on $n$ letters. The simple transposition $s_i\in S_n$ swaps $i$ and $i+1$ and fixes all other $j$. Any element $w\in S_n$ can be expressed $w=s_{i_1}\dotsb s_{i_k}$, where the $s_{i_j}$ are simple transpositions in $S_n$. The \textbf{length} $\ell(w)$ of an element $w\in S_n$ is the minimal $k$ over all expressions $w=s_{i_1}\dotsb s_{i_k}$ of $w$ as a product of simple transpositions $s_{i_j}$ in $S_n$. Any expression $w=s_{i_1}\dotsb s_{i_k}$ with $k=\ell(w)$ is called \textbf{reduced}. A \textbf{word} $Q=(s_{q_1},\dotsc,s_{q_m})$ in $S_n$ is a sequence of simple transpositions $s_{q_i}\in S_n$, and a \textbf{subword} of $Q$ is a subsequence of $Q$. We can associate to any word $Q=(s_{q_1},\dotsc,s_{q_m})$ a permutation $s_{q_{1}}\dotsb s_{q_m}$. If $v\in S_n$, then a \textbf{word for $v$} is any word $Q=(s_{q_1},\dotsc,s_{q_m})$ in $S_n$ with $v=s_{q_1}\dotsb s_{q_m}$. 
    The \textbf{Bruhat order} $\leq$ on $S_n$ is the partial order on $S_n$ in which $u\leq v$ if and only if, for any word $Q$ of $u$, there is a subword of $Q$ that is a word for $u$. The \textbf{longest permutation} $w_0$ in $S_n$ sends $i$ to $n+1-i$ for all $i$. \details{The \textbf{one-line notation} for a permutation $\sigma\in S_n$ is $i_1 i_2\dotsb i_n$, where $\sigma(j)=i_j$.   
	\[u\leq v\iff v=tu\text{ for some $t\in S_n$ and } \ell(u)<\ell(v).\]}
    %We will often write permutations in one-line notation; that is, we will write a permutation $\pi\in S_n$ as $\pi(1)\pi(2)\dots \pi(n).$ For example, the permutation $v=231$ in $S_3$ is the one-line notation for the permutation sending $1$ to $2$, $2$ to $3$, and $3$ to 1. The longest permutation in $S_n$ is expressed in one-line notation as $w_0=n (n-1) (n-2)\dotsb 321$.
    \begin{defn}[{\cite{Kostant-Toda,TW15}}] The \textbf{Bruhat interval polytope} $P_{u,v}$ is the convex hull of the points $(w(1),w(2),\dotsc,w(n))$ over all $u\leq w\leq v$. 
    \end{defn}

    \begin{defn}[{\cite{flagPositroids}}]
    The \textbf{twisted Bruhat interval polytope} $\tilde{P}_{u,v}$ is the convex hull of the points $(n+1-w^{-1}(1),n+1-w^{-1}(2),\dotsc,n+1-w^{-1}(n))$ over all $u\leq w\leq v$.
    \end{defn}

    \begin{rmk}
        Observe that $\tilde{P}_{u,v}=P_{w_0v^{-1},w_0u^{-1}}$. A (twisted) Bruhat interval polytope $P_{u,v}$ ($\tilde{P}_{u,v}$) is \textbf{toric} if $\dim P_{u,v}=\ell(v)-\ell(u)$ ($\dim \tilde{P}_{u,v}=\ell(v)-\ell(u)$). When $P_{u,v}$ is toric, so are $P_{u^{-1},v^{-1}}$ and $\tilde{P}_{u,v}$, and vice versa \cite{LMP21}. 
    \end{rmk}    
    Fix a word $Q=(s_{q_1},\dotsc,s_{q_m})$ in $S_n$. We can view a subword $J$ of $Q$ as a sequence $(s_{j_1},\dotsc,s_{j_m})$ obtained from $Q$ by replacing some entries by $1$. Denote by $Q\setminus J$ the sequence of length $|Q|$ whose $k$-th entry equals $1$ if $j_k\neq 1$ and equals $s_{q_k}$ otherwise. Denote by $J_{(k)}$ the product of the leftmost $k$ entries in $J$, and $J_{(0)}=1$. %with $-$ behaving as the identity element if $k\geq 1$, and $J_{(0)}=1$. 

   % beepboop

 %   skoobop

    
    \begin{defn}[{\cite{KM04}}]
        The \textbf{subword complex} $\Delta(Q,w)$, with respect to a word $Q$ and $w\in S_n$, is the simplicial complex on vertex set $Q$ whose facets (faces) are the subwords $J$ of $Q$ such that the product $(Q\setminus J)_{(|Q|)}$ is a reduced expression (contains a reduced expression) for $w$. 
    \end{defn}
    \begin{defn}[{\cite{KM04}}]
            The \textbf{Demazure product} $\mrm{Dem}(Q)$ of a word $Q$ in $S_n$ is defined inductively: 
    \begin{itemize}
        \item $\mrm{Dem}(\text{empty word})=1$.
        \item $\mrm{Dem}((Q,s_i))=\begin{cases}
            \mrm{Dem}(Q)\cdot s_i,  & \text{if $\ell(\mrm{Dem}(Q)\cdot s_i)>\ell(\mrm{Dem}(Q))$,}\\
            \mrm{Dem}(Q), & \text{otherwise.}
            \end{cases}$
    \end{itemize}
    \end{defn}
    
\details{
The subword complex $\Delta(Q,w)$ is homeomorphic to a sphere if $\dem(Q)=w$ and to a ball otherwise \cite[Corollary 3.8]{KM04}. We only focus on non-empty spherical subword complexes $\Delta(Q,w_0)$ in this work, a choice partly justified by the fact that every spherical subword complex is isomorphic to $\Delta(Q',w_0)$ for some $Q'$ such that $\Dem(Q')=w_0$ \cite[Theorem 3.7]{CJP14}.
}

Define $[n]:=\{1,\dotsc,n\}$. Consider the standard basis vectors $p_i$ of $\mbb{R}^n$, with $1$ appearing in entry $i$ and $0$ appearing in all other entries. Define the vectors $\omega_i=p_1+\dotsb+p_i$ for $i\in[n]$ and $\alpha_i=p_i-p_{i+1}$ for $i\in[n-1]$. 
    Given a subword complex $\Delta(Q,w_0)$ and a face $J$ of $\Delta(Q,w_0)$, %, the \textbf{root function} $r(J,\cdot)$ and
     the \textbf{weight function} and \textbf{bulky weight function}, $w(J,\cdot)\colon\{\text{subwords of $Q$}\}\to \mbb{R}^n$ and $\mbf{w}(J,\cdot)\colon\{\text{subwords of $Q$}\}\to \mbb{R}^n,$ are defined by
    %\[r(J,\cdot)\colon\{\text{subwords of $Q$}\}\to \mbb{R}^n,\quad r(J,k)=(Q\setminus J)_{(k)}(\alpha_{q_k}),\]
   \[w(J,k):=(Q\setminus J)_{(k)}(\omega_{q_k}) \qquad \qquad \text{ and } \qquad \qquad \mbf{w}(J,k):=(Q\setminus J)_{(k)}\left(\sum_{i=1}^{n}\omega_{i}\right).\]
  \noindent The \textbf{brick vector} is $B(J):=\sum_{i=1}^{|Q|}w(J,k)$ and the \textbf{brick polytope} of $Q$ is  \par
         { \centering $ \displaystyle
            \B(Q):=\mrm{conv}\{B(J)\mid J\in\Delta(Q,w_0)\text{ and }(Q\setminus J)_{(|Q|)}=w_0\}. 
        $ \par}
        \noindent The \textbf{bulky brick vector} is $\mbf{B}(J)=\sum_{k=0}^{|Q|}\mbf{w}(J,k)$, and the \textbf{bulky brick polytope} of $Q$ is
        \begin{center}$
        \BB(Q):=\mrm{conv}\{\mbf B(J)\mid J\in\Delta(Q,w_0)\text{ and }(Q\setminus J)_{(|Q|)}=w_0\}.
       $ \end{center}
    \begin{rmk}
    The brick polytope was originally defined in \cite{bricksort}. The definitions of $w(J,k), B(J),$ and $\mathcal{B}(Q)$ come from \cite{PS15} with a modification made by \cite{E16}. We introduce the bulky versions $\mbf{w}(J,k),\mbf{B}(J),$ and $\BB(Q)$ to match our treatment of the brick variety.
    \end{rmk}

   %\end{defn}
    
    %\noindent We will use the following definitions in the upcoming sections.
    \begin{defn}[{\cite{PS15}}]
       Let $J$ be a face of $\Delta(Q,w_0)$. The \textbf{root function} is $r(J,\cdot)\colon$
       {\parfillskip=0pt \par} \noindent
       $\{\text{subwords of $Q$}\}\to \mbb{R}^n$, with $r(J,k):=(Q\setminus J)_{(k)}(\alpha_{q_k})$. We say $Q$ is \textbf{root independent} if for some facet (or all facets) $J$ of $\Delta(Q,w_0),$ the multiset $\{\!\{r(J,k):q_k\in J\}\!\}$ is linearly independent.
    \end{defn}



%The twisted Bruhat interval polytope has a 




%Equivalently, $P_{u,v}=\conv\{w\cdot (1,2, \dots, n):u\leq w \leq v\}$ and $\tilde{P}_{u,v}=\conv\{w^{\cdot (n,n-1,\dots, $



%The Bruhat interval polytope was introduced by Tsukerman and Williams \cite{TW15} in 2015. The \textbf{Bruhat interval polytope} $P_{v,w}$ is the convex hull of the points $(z(1),z(2),\dotsc,z(n))$ over all $v\leq z\leq w$. The Bruhat interval polytope is the moment map image of the Richardson variety $X_{v}^w$.




   
    
  
    \section{A variety of varieties}	
    We recall several geometric objects, including the \textit{brick variety} of \cite{E16}.
    
	Fix a basis $e=(e_1,\dotsc,e_n)$ for $\mbb{C}^n$. Let $G=\mrm{GL}_n(\mbb{C})$ be the set of $n\times n$ invertible matrices over $\mbb{C}$. Consider the set of upper (resp. lower) triangular matrices $B_+$ (resp. $B_-$) in $G$, and % set $T:= B_+\cap B_-$, 
    the set of diagonal matrices $T$ in $G$. For $i_1<\dotsb<i_k$, set $F_{\{i_1,\dotsc,i_k\}}:=\mrm{Span}( e_{i_1},\dotsc,e_{i_k})$. Let $\mrm{Gr}(k,n)$ be the \textbf{Grassmannian} of $k$-planes in $\mbb{C}^n$, and observe that there is a natural action of $G$ on $\mrm{Gr}(k,n)$. %Define $\sigma\cdot F_{\{1,\dotsc,n\}}:=F_{\sigma(1),\sigma$
   Coordinate subspaces of the form $F_{\{i_1,\dotsc,i_k\}}$ are exactly the $T$-fixed points $\Gr(k,n)^{T}$ of $\Gr(k,n)$.
   The \textbf{complete flag variety} $\mrm{Fl}_n$ is the variety of complete flags of subspaces in $\mbb{C}^n$:
	\[\mrm{Fl}_n=\{F_\bullet=(F_1\subseteq F_2\subseteq \dotsb \subseteq F_{n-1}\subseteq F_n=\mbb{C}^n)\mid \mrm{  dim}(F_i)=i\}.\]
    The standard flag in $\mbb{C}^n$ is $F_\bullet^{\text{st}}=\left(F_{\{1\}}\subseteq F_{\{1,2\}}\subseteq \dotsb\subseteq F_{\{1,\dotsc,n-1\}}\subseteq \mbb{C}^n\right)$, and the opposite flag in $\mbb{C}^n$ is $F_\bullet^{\text{op}}=\left(F_{\{n\}}\subseteq F_{\{n,n-1\}}\subseteq \dotsb\subseteq F_{\{n,n-1,n-2,\dotsc,2\}}\subseteq \mbb{C}^n\right)$.
    The group $G$ acts on $\mrm{Fl}_{n}$ by the action
	\[g\cdot F_\bullet:=((g\cdot F_1)\subseteq (g\cdot F_2)\subseteq \dotsb\subseteq (g\cdot F_{n-1})\subseteq\mbb{C}^n),\quad g\in G,\text{ }F_\bullet\in\mrm{Fl}_n.\]
 %The $T$-fixed points of $\Fl_n(\C)$ are of the form $wB_+/B_+$ for $w\in S_n$ where the $i$-dimensional subspace of .    
 For $w\in S_n$ we define the \textbf{Schubert variety} $X_w=\overline{B_{-}wB_+/B_+}$ and \textbf{opposite Schubert variety} $X^w=\overline{B_{+}wB_+/B_+}$ which have the properties that $X_w\subseteq X_v$ iff $w\geq v$, and $X^w \subseteq X^v$ iff $w\leq v.$ 
% There are two natural cell decompositions for $G/B_+$, both indexed by the symmetric group $S_n$:
%	\begin{itemize}
%		\item For each $w\in S_n$, we consider the Schubert cell $X_w^\circ=B_-wB_+/B_+$ in $G/B_+$. The Schubert cells $\{X_w^\circ\}_{w\in S_n}$ form a cell decomposition for $G/B_+$.
%		\item For each $w\in S_n$, we consider the opposite Schubert cell $X^w_\circ=B_+wB_+/B_+$ in $G/B_+$. The opposite Schubert cells $\{X^w_\circ\}_{w\in S_n}$ form a cell decomposition for $G/B_+$.
%	\end{itemize}	
%	For each $w\in S_n$, the \textbf{Schubert variety} $X_w$ is the closure $X_w=\overline{X_w^\circ}$, and the \textbf{opposite Schubert variety} $X^w$ is the closure $X^w=\overline{X^w_\circ}$. The Schubert variety $X_w$ and opposite Schubert variety $X^w$ are equal to the following sets:
%	\[X_w=\sqcup_{v\geq w} X_v^\circ\quad\text{and}\quad X^w=\sqcup_{v\leq w} X^v_\circ.\]
%	In particular,
%	\[X_w\subseteq X_v\iff w\geq v\quad\text{and}\quad X^w\subseteq X^v\iff w\leq v.\]
The \textbf{Richardson variety} $X_u^v:=X_u\cap X^v\neq\emptyset$ if and only if $u\leq v$.  %The $T$-fixed points $(\mrm{Gr}(k,n))^T$ of $\mrm{Gr}(k,n)$ are $\{\sigma\cdot F_{\{1,\dotsc,k\}}\mid \sigma\in S_n\}$. 
    For $\sigma\in S_n$, define $\sigma\cdot F_{\{1,\dotsc,n\}}:=F_{\{\sigma(1),\dotsc,\sigma(k)\}}$ and $\sigma\cdot F_\bullet^{\text{st}}:=((\sigma\cdot F_{\{1\}})\subseteq (\sigma\cdot F_{\{1,2\}})\subseteq \dotsb\subseteq (\sigma\cdot F_{\{1,\dotsc,n-1\}})\subseteq\mbb{C}^n)$. The $T$-fixed points $(\mrm{Fl}_n)^T$ of $\Fl_n$ are $\{\sigma\cdot F_\bullet^{\text{st}}\mid \sigma\in S_n\}$.

     \details{There is an action of $S_n$ on $\mbb{R}^n$, defined by $\sigma\cdot e_i:=e_{\sigma^{-1}(i)}$ for all $\sigma\in S_n$ and $i=1,\dotsc,n$, and extended by $\mbb{R}$-linearity to all of $\mbb{R}^n$. Observe that for $x=(x_1,\dotsc,x_n)\in\mbb{R}^n$, we have $\sigma\cdot x=(x_{\sigma(1)},\dotsc,x_{\sigma(n)})$. The $\mbb{R}$-linear action of $S_n$ on $\mbb{R}^n$ induces a natural action of $S_n$ on $\mrm{Gr}(k,n)$, where $\sigma\cdot \mrm{span}_{\mbb{R}}(v_1,\dotsc,v_k):=\mrm{span}_{\mbb{R}}(\sigma\cdot v_1,\dotsc,\sigma\cdot v_n)$ for all $v_i\in\mbb{R}$ and $\sigma\in S_n$, and there is an action of $S_n$ on $\mrm{Fl}_n(\mbb{C})$ defined by $\sigma\cdot ( F_1\subseteq F_2\subseteq \dotsb\subseteq  F_n):=((\sigma\cdot F_1)\subseteq (\sigma\cdot F_2) \dotsb \dotsb\subseteq (\sigma\cdot F_{n-1})\subseteq  (\sigma\cdot F_n))$ for all $\sigma\in S_n$. } 
    
    The group $G\times G$ acts on $\mrm{Fl}_n\times \mrm{Fl}_n$ by $(f,g)\cdot (F_\bullet,G_\bullet):=(f\cdot F_\bullet,g\cdot G_\bullet)$ for all $f,g\in G$, $F_\bullet,G_\bullet\in\mrm{Fl}_n$. Define $G_\Delta:=\{(g,g)\mid g\in G\}\subseteq G\times G$. For $\sigma\in S_n$, the set of pairs of flags in \textbf{relative position $\sigma$} is the $G_\Delta$-orbit of $(F_\bullet^{\text{st}},\sigma(F_\bullet^{\text{st}}))$ in $\mrm{Fl}_n\times \mrm{Fl}_n$, and we will denote this set by $D_\sigma^\circ$. %Observe that $D_\sigma^\circ$ can be characterized as the set of flags $(F_\bullet,G_\bullet)$ in $\mrm{Fl}_n(\mbb{C})\times \mrm{Fl}_n(\mbb{C})$ such that $\mrm{dim}(F_i\cap G_j)=|[1,i]\cap\sigma([1,j])|$ for all $i,j$. 
    The closure $D_{\sigma}$ of  $D_{\sigma}^\circ$ in $\mrm{Fl}_n\times \mrm{Fl}_n$ is $D_{\sigma}=\sqcup_{\sigma'\leq \sigma}D_{\sigma'}^\circ$ (see, e.g., \cite[Lemma 2.1]{Bilen}.)  
    
   
    
 \details{
 We say that $F_\bullet$ and $G_\bullet$ are in \textbf{relative position $\sigma$}, where $\sigma\in S_n$, if $\mrm{dim}(F_i\cap G_j)=|[1,i]\cap \sigma([1,j])|$. If $F_\bullet =(F_{\{1\}},\dotsc,F_{\{1,\dotsc,n\}})$ and $G_\bullet =\sigma(F_\bullet):=(F_{\{\sigma(1)\}},\dotsc,F_{\{\sigma(1),\dotsc,\sigma(n)\}})$, then $(F_\bullet, G_\bullet)$ are in relative position $\sigma$. Now, let $(F_\bullet,G_\bullet)$ be in the $G_\Delta$-orbit of $(F_\bullet,\sigma(F_\bullet))$. Then \[|[1,i]\cap\sigma([1,j])|=\mrm{dim}(F_i\cap G_j)=\mrm{dim}(g\cdot(F_i\cap G_j))=\mrm{dim}((g\cdot F_i)\cap(g\cdot F_j)).\]
 Therefore, the $G_\Delta$-orbit of $(F_\bullet,\sigma(F_\bullet))$ is contained in the set of flags in relative position $\sigma$. To see that the set of flags 
 Denote by $D_{\sigma}^\circ$ the subset of $\mrm{Fl}_n(\mbb{C})\times \mrm{Fl}_n(\mbb{C})$ consisting of pairs of complete flags $(F_\bullet,G_\bullet)$, with $F_\bullet$ and $G_\bullet$ in relative position $\sigma$. The group $G\times G$ acts on $\mrm{Fl}_n(\mbb{C})\times \mrm{Fl}_n(\mbb{C})$ by $(f,g)\cdot (F_\bullet,G_\bullet):=(f\cdot F_\bullet,g\cdot G_\bullet)$ for all $f,g\in G$, $F_\bullet, G_\bullet\in\mrm{Fl}_n(\mbb{C})$. Define $ G_\Delta:=\{(g,g)\mid g\in G\}$, the diagonal in $G\times G$. Observe that $D_{\sigma}^\circ$ is a $G_\Delta$-orbit in $\mrm{Fl}_n(\mbb{C})\times \mrm{Fl}_n(\mbb{C})$. 
	\begin{egg}
		Consider the standard flag $F_\bullet=(F_{\{1\}}\subseteq F_{\{1,2\}}\subseteq F_{\{1,2,3\}}\subseteq F_{\{1,2,3,4\}})$ in $\mbb{C}^4$. Given $\sigma\in S_n$, set $\sigma(F_\bullet):=(F_{\{\sigma(1)\}}\subseteq F_{\{\sigma(1,2)\}}\subseteq F_{\{\sigma(1,2,3)\}}\subseteq F_{\{\sigma(1,2,3,4)\}})$. Then $(F_\bullet,\sigma(F_\bullet))\in D_{\sigma}^\circ$.
\end{egg}
}

Let $Q=(s_{i_1},\dotsc,s_{i_k})$ be a word in $S_n$. The \textbf{brick variety} $\mrm{Brick}^Q$ of \cite{E16}  is the projective variety consisting of sequences of flags such that the first flag is $F_\bullet^{\text{st}}$ and the last flag is $F_\bullet^{\text{op}}$ and consecutive pairs of flags lie in $D_{s_{i_j}}$:
	\begin{align*}
		&\mrm{Brick}^Q=\{(F_\bullet^{0},F_\bullet^1,\dotsc,F_\bullet^k)\mid F_\bullet^0=F_\bullet^{\text{st}},\text{ }F_\bullet^k=F_\bullet^{\text{op}},\text{ and } (F_\bullet^j,F_\bullet^{j+1})\in D_{s_{i_{j+1}}}\text{,  $j=0,\dotsc,k-1$}\}\\
		& \quad =(D_{s_{i_1}}\times \mrm{Fl}_n^{k-1})\cap (\mrm{Fl}_n\times D_{s_{i_2}}\times \mrm{Fl}_n^{k-2})\cap \dotsb\cap (\mrm{Fl}_n^{k-1}\times D_{s_{i_k}})\cap (F_\bullet^{\text{st}}\times \mrm{Fl}_n^{k-1}\times F_\bullet^{\text{op}}).
	\end{align*}
    By \cite[Theorem 20]{E16}, the variety $\mrm{Brick}^Q$ is smooth and irreducible, of dimension $|Q|-\ell(w_0)$. %Moreover, there is a natural embedding of $\mrm{Brick}^Q$ into $(\mrm{Fl}_n)^{|Q|+1}$. %Finally, there is an action of $S_n$ on $\mrm{Brick}^Q$ defined by $\sigma(F_\bullet^0,\dotsc,F_\bullet^{|Q|}):=(\sigma\cdot F_\bullet^0,\dotsc,\sigma\cdot F_\bullet^{|Q|})$ for all $\sigma\in S_n$. 
    Consider the set $\mathcal{J}=\{J\in \Delta(Q,w_0): (Q\setminus J)_{(|Q|)}=w_0\}$. For $J\in\mathcal{J}$, define the $T$-fixed point $p_J$ of $\Brick^Q$ as
    \[p_J:=\left(F_\bullet^{\text{st}},((Q\setminus J)_{(1)})\cdot F_\bullet^{\text{st}},\dotsc,((Q\setminus J)_{(m-1)})\cdot F_\bullet^{\text{st}},F_\bullet^{\text{op}}\right).\] 
    By \cite[page 7]{E16}, the map $\mcal{J}\to(\Brick^Q)^T$ that sends $J$ to $p_J$ is bijective.
    
\begin{egg}[{\cite[Example 12]{E16}}]
        Consider the word $Q=(s_1,s_2,s_1,s_2,s_1)$ in $S_3$. The brick variety $\Brick^Q$ can be visualized with a \textbf{Magyar diagram}:
        \[\adjustbox{scale=0.7,center}{
        \begin{tikzcd}[row sep = 1.1em, column sep = 1.7em]
	   \mbb{C}^3 \\
	\langle e_1,e_2\rangle & V_2 & \langle e_2,e_3\rangle \\
	\langle e_1\rangle & V_1 & V_3 & \langle e_3\rangle \\
	(0)
	\arrow[no head, from=1-1, to=2-2]
	\arrow[no head, from=1-1, to=2-3]
	\arrow[no head, from=2-1, to=1-1]
	\arrow[no head, from=2-1, to=3-2]
	\arrow[no head, from=2-2, to=3-3]
	\arrow[no head, from=2-3, to=3-4]
	\arrow[no head, from=3-1, to=2-1]
	\arrow[no head, from=3-2, to=2-2]
	\arrow[no head, from=3-3, to=2-3]
	\arrow[no head, from=4-1, to=3-1]
	\arrow[no head, from=4-1, to=3-2]
	\arrow[no head, from=4-1, to=3-3]
	\arrow[no head, from=4-1, to=3-4]
\end{tikzcd}
}\]
The points $(F_\bullet^{\text{st}},F_\bullet^{1},\dotsc,F_\bullet^{4},F_\bullet^{\text{op}})$ of $\mrm{Brick}^Q$ between $F_\bullet^{\text{st}}$ and $F_\bullet^{\text{op}}$ are of the form $F_\bullet^{1}=(V_1\subseteq \langle e_1,e_2\rangle\subseteq \mbb{C}^3)$, $F_\bullet^{2}=(V_1\subseteq V_2\subseteq\mbb{C}^3)$, $F_\bullet^{3}=(V_3\subseteq V_2\subseteq \mbb{C}^3)$, and 
 $F_\bullet^{4}=(V_3\subseteq \langle e_2,e_3\rangle\subseteq\mbb{C}^3)$.
The point $p_J$ corresponding to $J=(-,-,s_1,s_2,-)$ has $V_1= \langle e_2\rangle$, $V_2=\langle e_2,e_3\rangle$, and $V_3=\langle e_2\rangle$. 
\end{egg}
    



%\commentsG{We may need to put the condition $\dem(Q)=w_0$ in a lot of places.}
%\begin{lem}
%     Let $\pi_i:\Fl_n^k\to \Fl_n$ be the projection of $\Fl_n^k$ onto its $i$th factor. For any word $Q,$ with $\dem(Q)=w_0$ $\pi_i(X)=$ If $\Brick^Q$ is a toric variety 
%\end{lem}



% Let $\pi_i\colon (\mrm{Fl}_n(\mbb{C}))^k\to \mrm{Fl}_n(\mbb{C})$ be the projection onto the $i$-th factor. Define   $\nu_i:\Fl_n(\C)\hookrightarrow \prod_{k=1}^n\mrm{Gr}(k,n)\to \Gr(i,n)$, where the first map is an embedding, and the second is the projection onto the $i$-th factor.

%\Gabe{I wonder if we can elide eneugh details to get away without this....}



 \section{Moment polytopes}

 \subsection{Background}
    We discuss the moment polytope of the brick variety, following the exposition in \cite{E16}. 
    \begin{defn}
        Let $T'=(\C^\times)^k$. An algebraic action of $T'$ on the projective space $\mbb{P}^m$ is always of the form $(t_1,\dots,t_k)\cdot (x_1:\dots:x_m)=(x_1\prod_{i=1}^k t_i^{w_{1,i}}:\dots:x_m\prod_{i=1}^k t_i^{w_{m,i}})$ with $w_{j,i}\in \mbb{Z}.$ In such a case we say the $T'$-\textbf{weight} of the fixed point $(0:\dots:0:x_j:0:\dots:0)$ is the vector $(w_{j,1},\dots,  w_{j,k})\in\mbb{Z}^k\subseteq \mbb{R}^k$. Suppose $X$ is a variety with an algebraic action of $T'$.
        \begin{itemize}
            \item If $T'$ acts on $X$ with finitely many $T'$-fixed points and there is a $T'$-equivariant embedding $f\colon X\hookrightarrow \mbb{P}^m$ for some $m$, then the \textbf{moment polytope} of $X$ with respect to $f$ is the convex hull of the $T'$-weights over the points $f(\gamma)$, where $\gamma$ is a $T'$-fixed point of $X$.
            \item The \textbf{effective torus} is $T'/\mrm{St}_{T'}(x)$, where $\mrm{St}_{T'}(x)$ is the $T'$-stabilizer of a \textit{general}\footnote{\textit{general} in the sense that there is an open subset $U\subseteq X$ such that $\mrm{St}_{T'}(x)$ is constant for all $x\in U$.}  $x\in X$.
            \item $X$ is a \textbf{toric variety} with respect to the action of $T'$ if $X$ contains a dense $T'$-orbit. Equivalently, by orbit-stabilizer, $X$ is toric if and only if $\dim(X)=\dim(T'/\mrm{St}_{T'}(x)).$ 
        \end{itemize}
    \end{defn}

    
    The $T$-weight of the image of the fixed point $\sigma \cdot F_{\{1,\dotsc,n\}}$ of $\mrm{Gr}(k,n)$ under the ``Pl\"ucker embedding'' $\psi_k\colon \mrm{Gr}(k,n)\hookrightarrow  \mbb{P}^{\binom{n}{k}-1}$ is $\sigma(\omega_k)$. The torus $T$ acts diagonally on the variety $\prod_{k=1}^{n}\mrm{Gr}(k,n)$, and there is a projective embedding $\psi$ of $\prod_{k=1}^{n}\mrm{Gr}(k,n)$ into a projective space, given by the product of Pl\"ucker embeddings $\psi_k\colon \mrm{Gr}(k,n)\hookrightarrow \mbb{P}^{\binom{n}{k}-1}$, followed by the ``Segre embedding''. The $T$-weight of the fixed point $(\sigma_1\cdot F_{\{1\}},\dotsc,\sigma_n\cdot F_{\{1,\dotsc,n\}})$ under $\psi$, where $\sigma_k\in S_n$ for $k=1,\dotsc,n$, is $\sum_{k=1}^n\sigma_k(\omega_k)$. There is a $T$-equivariant embedding $\phi_n\colon \mrm{Fl}_n\hookrightarrow \prod_{k=1}^{n}\mrm{Gr}(k,n)$, where $F_\bullet=(F_1\subseteq F_2\subseteq \dotsb \subseteq F_{n}\subseteq \mbb{C}^n)$ is sent to $(F_1, F_2, \dotsc, F_{n})$. The $T$-weight of the fixed point $\sigma\cdot F_\bullet^{\text{st}}$ under $\phi_n$ is $\sigma(\omega_1+\dotsb+\omega_n)$. 
    \begin{rmk}
        Henceforth, we assume that a $T$-invariant subvariety of $\mrm{Gr}(k,n)$ is embedded in $\mbb{P}^{\binom{n}{k}-1}$ via $\psi_k$, and a $T$-invariant subvariety of $\mrm{Fl}_n$ is embedded in $\prod_{k=1}^n\mrm{Gr}(k,n)$ via $\phi_n$. We will use $\mu_{Z}$ to denote the map sending a $T$-invariant subvariety $X$ (or $T$-fixed point) of $Z$ to its moment polytope (or $T$-weight), where $Z$ will always be a single Grassmannian, a product of Grassmannians, a single flag variety, or a product of flag varieties. We always have $\mu_Z(X)=\conv_{x\in X^T}\mu_Z(x)$, where $X^T$ denotes the set of $T$-fixed points of $X$.
        \end{rmk}
    \begin{rmk}
        Observe that $\sigma(\omega_1+\dotsb+\omega_{n})=\sigma(n,n-1,\dotsc,2,1)=(n+1-\sigma^{-1}(1),n+1-\sigma^{-1}(2),\dotsc,n+1-\sigma^{-1}(n))$. The $T$-fixed points in the Richardson variety $X_u^v$ are precisely the coordinate flags $w\cdot F_\bullet^{\text{st}}$, with $u\leq w\leq v$. Thus, the moment polytope of $X_u^v$ is the twisted Bruhat interval polytope $\tilde{P}_{u,v}$. The dimension of a $T$-invariant subvariety matches the dimension of its moment polytope precisely when the variety is toric, so $\tilde{P}_{u,v}$ is toric exactly when $X_u^v$ is. 
    \end{rmk}


  %  For any $T$-invariant subvariety of 

    

     
    Let $Q=(s_{q_1},\dots, s_{q_{|Q|}})$ be a word in $S_n$ with $\dem(Q)=w_0$. Then  $\Brick^Q$ is a $T$-invariant subvariety of $\Fl_n^{|Q|+1}$ and thus has a moment polytope $\mu_{\Fl_n^{|Q|+1}}(\Brick^Q)$ given by the convex hull of the resulting $T$-weights of its $T$-fixed points. %The moment po
    %There is a $T$-equivariant inclusion
	%\[\pi\colon \mrm{Brick}^Q\hookrightarrow \Fl_n(\mbb{C})^{|Q|}, \quad (F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (F_\bullet^0,\dotsc,F_\bullet^{|Q|}).\]
    
Alternatively, we may view $\Brick^Q$ more directly as a subvariety of a product of Grassmannians. We define $\pi_i:\Fl_n^{|Q|+1}\to \Fl_n$ as the projection onto the $i$-th $\Fl_n$ factor 
(with indexing of $i$ starting at 0) and $\nu_i:\Fl_n\xhookrightarrow{\phi_n} \prod_{j=1}^n \Gr(j,n) \to \Gr(i,n) $ as the composition of $\phi_n$ with the projection onto $\Gr(i,n).$ We define $\rho_i:\Brick^Q\to \Gr(q_i,n)$ by $\rho_i=\nu_{q_i}\circ \pi_i$ and obtain the $T$-equivariant embeddings  %$$\rho\colon \Brick^Q\hookrightarrow \prod_{i=1}^{|Q|} \Gr(q_i,n)$ given by $(F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\rho_1(F_\bullet^1),\dotsc,\rho_n(F_\bullet^{|Q|}))$ and $\pi\colon \Brick^Q \to \prod_{i=0}^{|Q|}\prod_{j=1}^n \Gr(j,n)$ given by $(F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\phi_n(\pi_0(F_\bullet^1)),\dotsc,\phi_n(\pi_{|Q|}(F_\bullet^{|Q|}))).$
    %\[(F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\rho_1(F_\bullet^1),\dotsc,\rho_n(F_\bullet^{|Q|})), \quad \quad (F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\phi_n(\pi_0(F_\bullet^1)),\dotsc,\phi_n(\pi_{|Q|}(F_\bullet^{|Q|})))\]
    %\vspace{-0.13in}
     \[\rho\colon \Brick^Q\hookrightarrow \prod_{i=1}^{|Q|} \Gr(q_i,n),\quad  \quad (F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\rho_1(F_\bullet^1),\dotsc,\rho_n(F_\bullet^{|Q|})),\]
     %    \vspace{-0.13in}
     \[\pi\colon \Brick^Q \to \prod_{i=0}^{|Q|}\prod_{j=1}^n \Gr(j,n),\quad  \quad (F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\phi_n(\pi_0(F_\bullet^1)),\dotsc,\phi_n(\pi_{|Q|}(F_\bullet^{|Q|}))).\]
      %   \vspace{-0.14in}
    %  \[\rho\colon \Brick^Q\hookrightarrow \prod_{i=1}^{|Q|} \Gr(q_i,n),\quad  \quad (F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\rho_1(F_\bullet^1),\dotsc,\rho_n(F_\bullet^{|Q|})).\]
   % Consider the projection $\pi_i\colon\mrm{Brick}^Q\hookrightarrow \mrm{Fl}_n(\mbb{C})^{|Q|}\to \mrm{Fl}_n(\mbb{C})$ onto the $i$-th flag and the composition $\nu_i:\Fl_n(\C)\hookrightarrow \prod_{k=1}^n\mrm{Gr}(k,n)\to \Gr(i,n)$. Define the map $\rho_i\colon\mrm{Brick}^Q\to \mrm{Gr}(q_i,n)$ by $\rho_i=\nu_{q_i}\circ \pi_i$. Then there is $T$-equivariant embedding
   % \[\rho\colon \Brick^Q\hookrightarrow \prod_{i=1}^{|Q|} \Gr(q_i,n),\quad (F_\bullet^0,\dotsc,F_\bullet^{|Q|})\mapsto (\rho_1(F_\bullet^1),\dotsc,\rho_n(F_\bullet^{|Q|})).\]
   
Let $J$ be a subword of $Q$, and recall the fixed point $p_J$ of $\Brick^Q$ corresponding to $J$.   
The $T$-weight of $p_J$ under the embedding $\rho$ is the brick vector $B(J)=\sum_{i=1}^{|Q|}(Q\setminus J)_{(i)}(\omega_{q_i})$, and the $T$-weight of the fixed point $p_J$ under the embedding $\pi$ is the bulky brick vector $\mbf{B}(J)=\sum_{i=0}^{|Q|}(Q\setminus J)_{(i)}(\omega_1+\dotsb+\omega_{n})$. Thus, the moment polytope of $\Brick^Q$ under $\rho$ is the brick polytope $\mcal{B}(Q)$, and the moment polytope of $\Brick^Q$ under the embedding obtained by applying $\phi_n$ to each $\Fl_n$ factor is the bulky brick polytope $\BB(Q)$. As the brick polytope and bulky brick polytope are moment polytopes of $\Brick^Q$ with respect to two different projective embeddings, it follows from the theory of toric varieties (see, e.g., \cite{F93}) that $\mcal{B}(Q)$ and $\BB(Q)$ have the same normal fan\footnote{For exposition on ``normal fans'', see, for example, \cite[Section 5.2]{PS15}.}.
   % \begin{thm}(\cite[Theorem 2.9 and Corollary 2.10]{E16})
    %    Suppose $Q$ is the concatenation $\mbf{c}+\mbf{w_0(c)}$. Then $\mrm{Brick}^Q$ is a toric variety, and $B(Q,w)$ is the Loday realization of the associahedron. Moreover, $\mrm{Brick}^Q$ is the toric variety of the brick polytope $B(Q)$. \end{thm}
    
    \begin{thm}[{\cite[Theorem 15]{E16}}]\label{torbrick}
    $\Brick^Q$ is a toric variety if and only if $Q$ is root independent and $\ell(w_0)$$<$$|Q|$$\leq$$ \ell(w_0)+\dim(T).$ Further, $\Brick^Q$ is the toric variety of $\B(Q)$ (and of $\BB(Q)$). 
\end{thm}

    \subsection{Minkowski decomposition}



    The main results of this subsection are \Cref{thm:Minkowski}, which gives a Minkowski decomposition of $\BB(Q)$, and \Cref{thm:characterization}, which characterizes toric Richardson varieties\footnote{In future work, we hope to explore how \Cref{thm:characterization} relates to the root theoretic characterization of toric Richardson varieties given in \cite[Theorem 4.7]{CS23}.}. 
 \details{ \begin{lem}\label{fix of proj}
     For any Richardson variety $X_u^v$, we have $\left(\nu_i\left(X_u^v\right)\right)^T=\nu_i\left(\left(X_u^v\right)^T\right)$, and for any word $Q$ in simple reflections, $\left(\pi_i\left(\Brick^Q\right)\right)^T=\pi_i\left(\left(\Brick^Q\right)^T\right)$. 
\end{lem}
\begin{proof}
    Apply Borel's fixed point theorem to the inverse image of a $T$-fixed point in $\pi_i(\Brick^Q)$ (or $\nu_i(X_u^v)$) under $\pi_i$ (or $\nu_i$). 
\end{proof}  
}

\begin{lem}\label{proj is Richardson}
    Let $\pi_i:\Fl_n^k\to \Fl_n$ be the projection of $\Fl_n^k$ onto its $i$-th factor (with indexing of $i$ starting at 0). Let $R,S$ be words in $S_n$. Set $i:=|R|$, $u:=w_0\dem(S)^{-1}$, and $v:=\dem(R)$.  
    \begin{enumerate}
        \item We have $\pi_{i}(\Brick^{R+S})=X_u^v$, where $+$ denotes concatenation of words.
        \item If $\Brick^{R+S}$ is toric, then $X_u^v$ is toric.
   \end{enumerate}

%        \item If $\mrm{X}_u^v$ is toric and $\mrm{dim}(X_u^v)=\mrm{dim}(\mrm{Brick}^{R+S})$, then $\mrm{Brick}^Q$ is toric and $\pi_i$ is a resolution of singularities.
\end{lem}
\begin{proof}
    (1) follows from slight modifications of the proof of \cite[Theorem 26]{E16}. For (2), assume $\mrm{Brick}^{R+S}$ is toric. Then it contains a dense $T$-orbit $\mcal{O}$. As $\pi_i$ is $T$-equivariant, surjective onto $X_u^v$, and continuous, it follows that $\pi_i(\mcal{O})$ is a dense $T$-orbit in $X_u^v$. 
    \end{proof}





\begin{thm}\label{thm:Minkowski}
For $\mathcal{J}=\{J\in \Delta(Q,w_0): (Q\setminus J)_{(|Q|)}=w_0\}$ and $\lambda_\ell=\min(\{k:k>\ell, q_k=q_\ell \} \cup \{|Q|\})-\ell$, we have
    \begin{align}
 \BB(Q)&=(|Q|+1)\omega_n+\sum_{j=1}^{n-1} \min(\{k:q_k=j\})\omega_j  +  \sum_{\ell=1}^{|Q|} \lambda_\ell\conv\limits_{J\in \mathcal{J}}w(J,\ell)\label{eq:pieces1}\\
%&=\sum_{k=0}^{|Q|}\sum_{i=1}^{|Q|} \mu_{?}(\nu_i(\pi_k(\Brick^Q)))\\
        %&=
        %\conv_{p_J\in (\nu_i(\pi_k(\Brick^Q)))^T} 
        &=\tilde{P}_{1,1}+\sum_{k=1}^{|Q|} \tilde{P}_{w_0\dem(s_{q_{k+1}},\dots, s_{q_{|Q|}})^{-1},\dem(s_{q_1},\dots, s_{q_k})}.\label{eq:pieces}
    \end{align}
\end{thm}
\begin{sketch}
    

    
    %Using Theorem 21 of \cite{E16} and some combinatorics of subword complexes, we can show that the convex hulls over facets can be replaced with convex hulls over faces. 
    
  
   Manipulating the formula for $\mbf{B}(J),$ one finds 
   % \begin{align*}
       \[ \mbf{B}(J)=\sum_{k=1}^{|Q|}\mbf{w}(J,k) =(|Q|+1)\omega_n+\left(\sum_{j=1}^{n-1} \min(\{k:q_k=j\})\omega_j \right) +  \sum_{\ell=1}^{|Q|} \lambda_\ell w(J,\ell).\]
%\label{bridgebricks}.\]
   % \end{align*}
When $L$ is the set of facets of $\Delta(Q,w_0)$, \cite{bricksort} showed that $\B(Q)=\conv_{J\in L}\sum_{k=1}^{|Q|}w(J,k) = \sum_{k=1}^{|Q|}\conv_{J\in L} w(J,k)$. Further arguments can show that Equation \eqref{eq:pieces1} follows from their result once we write $\BB(Q)$ as above. One can show that $\conv_{J\in \mathcal{J}} w(J,k)$ is the polytope $\mu_{\Gr(q_k,n)}\rho_k(\Brick^Q)$\footnote{The polytope $\mu_{\Gr(q_k,n)}\rho_k(\Brick^Q)$ happens to be a \textit{positroid polytope}, as defined in, e.g., \cite{flagPositroids}.}. Manipulating this expression (and using \Cref{proj is Richardson}), we eventually arrive at $\BB(Q)=\sum_{k=0}^{|Q|}\sum_{j=1}^n \mu_{\Gr(j,n)}(\nu_j(\pi_k(\Brick^Q))).$ Applying \cite[Proposition 2.7]{TW15}, we have $\mu_{\Fl_n(\C)}(X_u^v)=\sum_{i=1}^n \mu_{\Gr(i,n)}(\nu_i(X_u^v)),$
and obtain \[\BB(Q)=\sum_{k=0}^{|Q|}\mu_{\Fl_n}\pi_k(\Brick^Q)=\sum_{k=0}^{|Q|} \tilde{P}_{w_0\dem(s_{q_{k+1}},\dots, s_{q_{|Q|}})^{-1},\dem(s_{q_1},\dots, s_{q_k})}. \qedhere\]


%From \Cref{fix of proj} the discussion of $T$-weights above,
%it follows that $\conv_{J\in \mathcal{J}} w(J,k)$ is the moment polytope $\mu_{\Gr(q_k,n)}\rho_k(\Brick^Q)$ (which, being the moment polytope of the projection of a Richardson variety to a Grassmannian, is a positroid polytope). 
%Elementary arguments show that $\conv_{J}\sum_{k}w(J,k) = \sum_{k}\conv_{J} w(J,k)$ implies that we have $\conv_{J}\sum_{k}\lambda_k w(J,k)=\sum_{k}\conv_{J} \lambda_k w(J,k)$ so one gets that the bulky brick polytope can also be written as a (weighted) Minkowski sum of the positroid polytopes $\conv_{J} w(J,k)$ where $k$ ranges from $1$ to $|Q|.$ As a side product, this gives a more combinatorial proof that $\B(Q)$ and $\BB(Q)$ are normally equivalent. 
%Manipulating and regrouping the terms, and applying \Cref{fix of proj} and \Cref{proj is Richardson}, one obtains the formula $\BB(Q)=\sum_{k=0}^{|Q|}\sum_{j=1}^n \mu_{\Gr(j,n)}(\nu_j(\pi_k(\Brick^Q))).$ 
%Applying the result from \cite{TW15} and \cite{Kostant-Toda} that $\mu_{\Fl_n(\C)}(X_u^v)=\sum_{i=1}^n \mu_{\Gr(i,n)}(\nu_i(X_u^v)),$
%we can write \[\BB(Q)=\sum_{k=0}^{|Q|}\mu_{\Fl_n(\C)}\pi_k(\Brick^Q)=\sum_{k=0}^{|Q|} \tilde{P}_{w_0\dem(s_{q_{k+1}},\dots, s_{q_{|Q|}})^{-1},\dem(s_{q_1},\dots, s_{q_k})}. \qedhere\]
\end{sketch}






\begin{thm}\label{thm:characterization}
    Let $u\leq v\in S_n$. Let $R$ be any reduced word for $v$, let $S$ be any reduced word for $u^{-1}w_0$, and let $J$ be any facet of the subword complex $\Delta(S+R,w_0)$. The following are equivalent: 
    \begin{enumerate}
        \item  $\Brick^{R+S}$ is a toric variety.
        \item $X_u^v$ is a toric variety.
        \item $\ell(v)-\ell(u)\leq \dim(T)$ and the multiset $\{\!\{r(J,k):q_k\in J\}\!\}$ is linearly independent.
    \end{enumerate}
\end{thm}
\begin{proof}
    $(1)\iff(3)$ by \Cref{torbrick}, and $(1)\implies(2)$ by \Cref{proj is Richardson}. We will prove $(2)\implies(1)$. The hypotheses on $R$ and $S$ ensure $\Brick^{R+S}$ is a resolution of singularities of $X_u^v$ by \cite{E16}, so $\dim(\Brick^{R+S})=\dim(X_u^v)$. If $X_u^v$ is a toric variety, then  $\dim(\tilde{P}_{u,v})=\dim(X_u^v).$ Since $\BB(R+S)$ has $\tilde{P}_{u,v}$ as a Minkowski summand (by \Cref{thm:Minkowski}), we have $\dim(\BB(R+S))\geq \dim(X_u^v)=\dim(\Brick^{R+S})\geq \dim(\BB(R+S)).$ The dimension of the moment polytope of a variety is the same as the dimension of the effective torus. This means the effective torus acting on $\Brick^{R+S}$ has the same dimension as $\Brick^{R+S}$ and thus $\Brick^{R+S}$ is a toric variety whenever $X_u^v$ is. 
\end{proof}




%moment polytope of a Richardson variety is the minkowski sum of the positroid polytopes obtained by projecting to each 



%\section{Minkowski decompositions}\label{section:Minkowski}

\section{A degeneration of the brick variety}\label{section:degeneration}

\subsection{The degeneration}
We define a degeneration of the brick variety and describe its central fibre (\Cref{degen of brick}).

Say $G\times G$ has coordinates $((a_{i,j})_{i,j=1,\dotsc,n},(b_{i,j})_{i,j=1,\dotsc,n})$. Let $t\in\mbb{C}$. Let $G_t$ be the subvariety of $G\times G$ cut out by the equations
	\[
		a_{i,j}=t^{j-i}b_{i,j},\text{ }\quad i<j; \qquad \qquad 
		b_{j,i}=t^{j-i}a_{j,i},\text{ }\quad i<j; \qquad \qquad
		a_{i,i}=b_{i,i},\text{ }\quad i=1,\dotsc,n.
	\]
	Observe that $G_1=G_\Delta$ and $G_0=B_-\times_T B_+$ (i.e., $G_0$ is the set of elements in the product $B_-\times B_+$ whose diagonals are equal).
	Consider the embedding $\rho\colon \mbb{C}^\times \to T$, sending $t$ to the diagonal matrix whose diagonal is $(1, t^{-1},\dots , t^{-n+1}).$
    
	%\[\rho(t)=\begin{bmatrix}
	%	1 &  &  & \\ 
	%	& t^{-1} &  & \\ 
	%	&  &  \ddots & \\ 
	%	&  &   & t^{-n+1}
	%\end{bmatrix}.\]
	For $t\in\mbb{C}^\times$, we have $G_t=\{(g,\rho(t)g(\rho(t))^{-1})\mid g\in G\}\subseteq G\times G$.
	For $\sigma\in S_n$ and $t\in\mbb{C}^\times$, the variety $D_{\sigma(t)}:=(1,\rho(t))\cdot D_{\sigma}$ is invariant under the action of $G_t$ on $\mrm{Fl}_n\times \mrm{Fl}_n$:
	\[(g,\rho(t)g(\rho(t))^{-1})\cdot (F_\bullet,\rho(t)(G_\bullet))=((g\cdot F_\bullet),\rho(t)(g\cdot G_\bullet))\in D_{\sigma(t)},\quad g\in G, F_\bullet, G_\bullet\in \mrm{Fl}_n.\]
    Consider the following two varieties, $\mcal{G}:=\cup_{t\in\mbb{C}}(G_t\times \{t\})\subseteq (G^2)\times \mbb{A}^1$ and $\mcal{D}_\sigma^\circ:=\cup_{t\in\mbb{C}^\times}(D_{\sigma(t)}\times \{t\})\subseteq (\mrm{Fl}_n^2)\times \mbb{A}^1$.
    Let $\mcal{D}_\sigma$ be the closure of $\mcal{D}_\sigma^\circ$ inside $(\Fl_n^2)\times \mbb{A}^1$, and let $D_{\sigma(0)}$ be the fibre of $\mcal{D}_\sigma\to\mbb{A}^1$ over $0$.
    The action of $G_t$ on $D_{\sigma(t)}$ for each $t\in \mbb{C}^\times$ induces an action of $G_0$ on $D_{\sigma(0)}$ (cf. \cite[Proposition 4.1]{KMS06}) \footnote{The orbit degeneration technique of degenerating a group alongside a variety so that the degeneration of the variety is a union of orbit closures of the degeneration of the group comes from \cite{KMS06}, which employed the same family of groups $G_t\subseteq G\times G.$}. In particular, $D_{\sigma(0)}$ is a union of orbits of $G_0=B_{-}\times_T B_+$. To uncover which union of orbits, observe that $(D_{\sigma(1)})^T\subseteq (D_{\sigma(t)})^T$ for all $t\in \C$, and consider the $B_-\times_{T}B_+$ orbits of these $T$-fixed points. This yields the correct answer, given in \Cref{sigmadegen}; however, proving this requires geometric techniques.
    %Because the degeneration is by the action of $\C^\times \subseteq T$, $D_{\sigma})\subseteq G$ the $T$-fixed points of $D_{\sigma}^\circ$ include all pairs of permutation flags $(wB/B,w\sigma B/B),$  
    
         \begin{thm}\label{sigmadegen}
           The central fibre $D_{\sigma(0)}$ is reduced and equal to $\bigcup\limits_{w\in S_n\text{ s.t.  } \ell(w\sigma) = \ell(w)+\ell(\sigma)} X_w\times X^{w\sigma }$.
        \end{thm}


\begin{egg}
    In $S_3$, we have $D_{s_1(0)}=(X_{1}\times X^{s_1})\cup(X_{s_2}\times X^{s_2s_1})\cup(X_{s_1s_2}\times X^{s_1s_2s_1}) $. 
\end{egg}
Let $Q=(s_{q_1},\dotsc,s_{q_k})$ be a word in $S_n$.
\begin{defn}
         A sequence $(u_0,u_1,\dotsc,u_{|Q|})$ in $S_n$ is a \textbf{brickly sequence} with respect to $Q$ if $u_0=1$, $u_{|Q|}=w_0$, and $u_i< u_is_{q_{i+1}}$ and $u_{i+1}\leq u_i s_{q_{i+1}}$ for all $i=0,\dotsc,|Q|-1$. We will denote by $L(Q)$ the set of brickly sequences for $Q$.
    \end{defn}
    \begin{egg}
        In $S_3$, set $Q=(s_1,s_2,s_1,s_2,s_1)$. Then $u=(1,1,s_2,s_1,s_1s_2,s_1s_2s_1)$ is a brickly sequence with respect to $Q$.
    \end{egg}
The degeneration of $D_{s_{q_i}}$ induces a degeneration of $\mrm{Brick}^Q$ into a union of reduced products of Richardson varieties (\Cref{degen of brick}). For $t\in \mbb{C}$, we define $\mrm{Brick}^Q(t)$ as
%\begin{align*}\mrm{Brick}^Q(t):=\\&
\[
(D_{s_{q_1}}(t)\times \mrm{Fl}_n^{k-1})\cap (\mrm{Fl}_n\times D_{s_{q_2}}(t)\times \mrm{Fl}_n^{k-2})\cap \dotsb\cap (\mrm{Fl}_n^{k-1}\times D_{s_{i_k}}(t))\cap (F_\bullet^{\text{st}}\times \mrm{Fl}_n^{k-1}\times F_\bullet^{\text{op}}).
\]
%\end{align*}% By defining $\mcal{B}^Q:=\overline{\{\mrm{Brick}^Q(t)\times\{t\}\}_{t\in\mbb{C}}}\subseteq (\mrm{Fl}_n(\mbb{C}))^{|Q|+1}\times\mbb{A}^1_t$ we obtain the degeneration as $\Brick^Q(0);=\mcal{B}^Q\cap (\Fl_{n}(\C)^{|Q|+1}\times 0).$

%Thus, $\mrm{Brick}^Q(1)=\mrm{Brick}^Q$, and $\mrm{Brick}^Q(0)$ is the union of products of Richardson varieties. Define the subvariety $\mcal{B}^Q:=\{\mrm{Brick}^Q(t)\times\{t\}\}_{t\in\mbb{C}}\subseteq (\mrm{Fl}_n(\mbb{C}))^{|Q|+1}\times\mbb{A}^1_t$, giving the following flat family over $\mbb{A}^1_t$:
%	\[\mcal{B}^Q\to \mbb{A}^1_t.\]



    \begin{thm}\label{degen of brick}
    Suppose $\Brick^Q$ is a toric variety. For $u=(u_0,u_1,\dotsc,u_{|Q|})\in L(Q)$, set \[C_u:=X_{u_0}^{u_0}\times X_{u_1}^{u_0s_{q_1}}\times X_{u_2}^{u_{1}s_{q_2}}\times\dotsb\times X_{u_{|Q|}}^{u_{|Q|-1}s_{q_{|Q|}}}.\]
    Then $\mrm{Brick}^Q(0)=\bigcup_{u\in L(Q)}C_u$, and $C_u\cap C_{u'}$ is a common face of $C_u$ and $C_{u'}$ for $u,u'\in L(Q)$.
    \end{thm}
The first part of \Cref{degen of brick} follows from the definition of $\Brick^Q(t)$ and \Cref{sigmadegen}, while some of the latter parts require more geometric techniques to prove.
	
   % \section{Properties of the degeneration}\label{section:properties}
    
 \subsection{Bricks tricks for mixed subdivisions }

 


The main result of this subsection is \Cref{mixed}, which says that the degeneration of the brick variety of \Cref{degen of brick} induces a mixed subdivision of the brick polytope.

%\Gabe{This is in dire need of review. I cna do the review. I think it is uncareful about properties of bruhat order. I think maybe the proof should be using a property which it is not using.}
\begin{lem}\label{subrichardson}
Suppose $1=u_0,u_1,\dots, u_{|Q|}=w_0 \cdot s_1$ is a brickly sequence in $S_n$. %Then $u_i$ is the permutation of a subword of $(s_{q_1}s_{q_2}\dots s_{q_{i}})$, and $u_i^{-1}w_0$ is the permutation of a subword of $s_{q_{i+1}}s_{q_{i+2}}\dots s_{q_{|Q|}}$. Notably, for 
Then for $x=w_0\dem(s_{q_{i+1}},s_{q_{i+2}},\dots, s_{q_{|Q|}})^{-1}$ and $y=\dem(s_{q_1},s_{q_2},\dots s_{q_i})$, the Bruhat interval $[u_{i},u_{i-1}s_{q_i}]$ is contained in $[x,y]$; equivalently, $X_{u_{i}}^{u_{i-1}s_{q_{i}}} \subseteq X_{x}^{y}$.
 %X_{u_{i+1}}^{u_iq_i}$
\end{lem}
\begin{sketch}
The proof proceeds by induction using subword combinatorics.\qedhere
\details{
	For $i=0$ we can see $u_0 = \prod_{j=1}^0 q_j=1.$ Suppose for induction sake that $u_i$ can be written as a subword of  $q_1q_2\dots q_{i}.$ Then $u_{i}q_{i+1}$ is a subword of $q_1\dots q_{i+1}$. Since $u_{i+1}\leq u_iq_{i+1},$ it follows that $u_{i+1}$ can be written as a subword of $q_1\dots q_{i+1}.$
    	
	For $i=|Q|,$ we have that $u_{|Q|}^{-1}w_0=1=\prod_{j=|Q|+1}^{|Q|}s_{q_j}$. Suppose for (backwards) induction that $u_{i+1}^{-1}w_0$ can be written as a subword of $q_{i+2}\dots q_{|Q|}.$ 
	Well, $u_{i+1}<u_{i}q_{i+1}$ thus $u_{i+1}^{-1}<q_{i+1}u_i^{-1}$ and $u_{i+1}^{-1}w_0>q_{i+1}u_i^{-1}w_0$.  Hence, $q_{i+1}u_i^{-1}w_0$ can be written as a  subword of $q_{i+2}\dots q_{|Q|}$. Thus we can write $u_i^{-1}w_0=q_{i+1}q_{i+1}u_i^{-1}w_0$ as a subword of $q_{i+1}q_{i+2}\dots q_{|Q|}.$ 

    Since $u_{i-1}$ is a subword of $q_1q_2\dots q_{i-1}$, $u_{i-1}q_i\leq \dem(q_1q_2\dots q_{i-1}q_i).$ Since $u_{i}^{-1}w_0$ can be written as a subword of $q_{i+1}q_{i+2}\dots q_{|Q|},$ $u_{i+1}^{-1}w_0 \leq \dem(q_iq_{i+1}\dots q_{|Q|}).$ Hence, $u_{i+1}\geq w_0\dem(q_iq_{i+1}\dots q_{|Q|})^{-1}.$ Therefore, $[u_i,u_{i-1}s_{q_i}]\subseteq [w_0\dem(s_{q_{i+1}}s_{q_{i+1}}\dots s_{q_{|Q|}})^{-1}, \dem(s_{q_1}s_{q_2}\dots s_{q_})]$ for $1\leq i \leq |Q|.$ }
	
\end{sketch}

\begin{defn}
    A \textbf{full dimensional Minkowski cell} of a Minkowski sum of polytopes $P=\sum_{i=1}^n P_i$ is a polytope $C=\sum_{i=1}^n C_i$ such that $\dim(C)=\dim(P)$ and $C_i$ is a face of $P_i$ for all $i$. A mixed subdivision of $P$ is a set of  full dimensional Minkowski cells $\{C_j=\sum_{i=1}^n C_{j,i}:j\in J\}$ for some index set $J$ such that $P=\bigcup_{j\in J} C_j,$ and $C_j\cap C_k$ is a common face of $C_j$ and $C_k$ for any $j,k$.
\end{defn}


\begin{thm}\label{mixed}
    If $\Brick^Q$ is toric, then, in the notation of \Cref{subrichardson}, the moment polytopes of the components of $\Brick^Q(0)\hookrightarrow \Fl_n(\C)^{|Q|}$ give a mixed subdivision of $\BB(Q)=\sum_{i=0}^{|Q|} \tilde{P}_{x,y}.$ 
\end{thm}
\begin{sketch}
    Let $u_\bullet,v_\bullet \in L(Q)$ be arbitrary brickly sequences. Using the fact that $C_u=X_1^1\times X_{u_1}^{u_0s_{q_1}}\times \dots \times X_{u_{|Q|}}^{u_{|Q|-1}q_{|Q|}} $ is a toric variety, we can show $\dim(\mu_{\Fl_n^{|Q|+1}}(C_u))$$=\dim\tilde{P}_{1,1}+\sum_{i=1}^{|Q|} \dim \tilde{P}_{u_{i},u_{i-1}s_{q_{i}}}$. The dimension of a Minkowski sum is only equal to the sum of the dimensions of its summands when the summands are affinely independent. Hence, the summands of $\mu_{\Fl_n^{|Q|+1}}(C_u)$ are affinely independent; this implies the linear map from the product $\tilde{P}_{1,1}\times \tilde{P}_{u_1,u_0s_{q_1}}\times \dots\times \tilde{P}_{u_{|Q|},u_{Q-1}s_{|Q|-1}}$ to $\mu_{\Fl_n^{|Q|+1}}(C_u)$ is an isomorphism. 
    
    %$\tilde{P}_{u_{i+1},u_is_{q_i}}$ 
    By \Cref{subrichardson}, we have that  $[u_{i},u_{i-1}s_{q_{i}}]$ is a subinterval of $[x,y]$. By \cite{LMP21}, subintervals of a toric Bruhat interval correspond to faces of the (twisted) Bruhat interval polytope. Hence, each $\tilde{P}_{u_{i},u_{i-1}s_{q_i}}$ is a face of $\mu_{\Fl_n^{|Q|+1}}(\pi_i(\Brick^Q))$, and $\mu_{\Fl_n^{|Q|+1}}(C_u)$ is a full-dimensional Minkowski cell. These cells union to give all of $\BB(Q)$ by \Cref{degen of brick}.  %Because $\mu(C_u)$ is linearly isomorphic to the product of its summands, the Minkowski sum of any collection  
    
    It was proved in \cite[Theorem 1.1]{CS23} that toric Bruhat intervals are lattices. The intersection of two intervals contained in a lattice is always an interval (or empty), so $[u_{i},u_{i-1}s_{q_i}]\cap [v_{i},v_{i-1}s_{q_{i}}] = [a_i,b_i]$ for some $a_i,b_i\in S_n.$ Clearly $X_{a_i}^{b_i}\subseteq X_{u_{i+1}}^{u_is_{q_i}} \cap X_{v_{i+1}}^{v_is_{q_i}}$ as it is a sub-Richardson variety. Any Richardson variety contained in $X_{u_{i+1}}^{u_is_{q_i}} \cap X_{v_{i+1}}^{v_is_{q_i}}$ will have fixed points indexed by permutations contained in $[a_i,b_i]$ and hence it will be contained in $X_{a_i}^{b_i}.$ Since intersections of Richardson varieties are reduced unions of Richardson varieties \cite[Theorem 2.74]{S24}, we conclude that $X_{a_i}^{b_i}$ is the desired intersection. So $\mu_{\Fl_n}(X_{a_i}^{b_i})=\tilde{P}_{a_i,b_i}$ will then be a common face of $\tilde{P}_{u_{i},u_{i-1} s_{q_i}}$ and $\tilde{P}_{v_{i},v_{i-1} s_{q_i}}.$ A product of faces, each drawn from a different polytope, is a face of the product of those larger polytopes. Ergo, using the fact that the Minkowski sums $\sum \tilde{P}_{u_{i},u_{i-1} s_{q_i}}$ and $\sum \tilde{P}_{v_{i},v_{i-1} s_{q_i}}$ are both linearly isomorphic to the products of the same polytopes, $\mu_{\Fl_n^{|Q|+1}}(C_u\cap C_v)=\sum \tilde{P}_{a_i,b_i}$ will be a face of both $\mu_{\Fl_n^{|Q|+1}}(C_u)$ and $\mu_{\Fl_n^{|Q|+1}}(C_v).$ 
\end{sketch}

 \subsection{Subdividing the associahedron into cubes}

The main result of this subsection is \Cref{thm:cubes}, which describes a subdivision of the associahedron into pieces that are combinatorially equivalent to cubes.




\details{
\begin{thm}[{\cite[Theorem 1.1 and  Corollary 6.10]{PS15}}]
    For any root independent word $Q,$ the polar of the brick polytope $\mathcal{B}(Q)$ realizes the subword complex $\Delta(Q,w).$ The word $Q=\mbf{c}+\mbf{w_0(c)}$, where $+$ denotes concatenation, is root independent for any choice of reduced word $\mbf{c}$ for a Coxeter element $c.$ For such $Q=\mbf{c}+\mbf{w_0(c)},$ the brick polytope $\mathcal{B}(Q)$ coincides up to translation with the $c-$associahedron of \cite{generalizedAssociahedra}. For the choice $\mbf{c}=(s_{1},s_{2},\dots, s_{n-1}),$   $\mathcal{B}(Q)$ is Loday's realization of the associahedron.
\end{thm}
}

\begin{thm}[{\cite{LMP21}}]\label{smoothtoric}
    The twisted Bruhat interval polytope $\tilde{P}_{u,v}$ is combinatorially equivalent to a cube if and only if $X_u^v$ is smooth and toric. If $v=cu$ or $v=uc$ for $c=s_1s_2\dotsb s_j$ or $c=(s_1s_2\dotsb s_j)^{-1}$, with $j\leq n-1$, and $\ell(v)-\ell(u)=j$, then $X_u^v$ is smooth and toric. 
\end{thm}
\begin{rmk}
    By \cite{LMP21}, for $X_a^b\subseteq X_{c}^d$ with $X_{c}^d$ toric, $\tilde{P}_{a,b}$ will be a face of $\tilde{P}_{c,d}.$ So when $X_{c}^d$ is also smooth, $\tilde{P}_{a,b}$ will be a face of a cube and therefore be a cube. This shows that any sub-Richardson of a smooth toric Richardson is smooth and toric.
\end{rmk}


Let $\textbf{c}$ be a \textbf{Coxeter element} in $S_n$, i.e., a word in $S_n$ containing all simple reflections exactly once. Denote by $\textbf{c}^\infty$ the concatenation of $\textbf{c}$ to itself ad infinitum. Define $\mbf{w_0(c)}$ to be the lex-first subword of $\textbf{c}^\infty$ that is a reduced word for $w_0$. %For $Q$ the concatenation $\mbf{c}+\mbf{w_0(c)}$, $\Brick^Q$ is the toric variety of Loday's realization of the associahedron \cite[Corollary 16]{E16}.

   
\begin{thm}\label{thm:cubes}
   Fix $\mbf c=(s_1,s_2,\dots, s_{n-1})$ and $\mbf c'=(s_{n-1}, s_{n-2},  \dots, s_1)$. Let $Q$ be the concatenation $\mbf{w_0(c)}+\mbf{c'}$. Then $\Brick^Q$ is toric and $\Brick^Q\simeq \Brick^{\mbf{c}+\mbf{w_0(c)}}$. Moreover, we have
   \begin{itemize}
       \item $\pi_i(\Brick^Q)$ is a smooth toric Richardson variety for each $i$, and each component $C_u$ of $\Brick^Q(0)$ is a product of smooth toric Richardsons.
       \item $\B(Q)$ is a translation of Loday's realization of the associahedron, $\BB(Q)$ has the same normal fan as Loday's realization, the summands of the Minkowski decomposition of $\BB(Q)$ (\Cref{eq:pieces}) are combinatorial cubes, and the moment polytopes of the components of $\Brick^Q(0)$ give a mixed subdivision of $\BB(Q)$ into pieces which are combinatorially equivalent to cubes.
   \end{itemize} 
\end{thm}
\begin{sketch}
    For $Q=\mbf{c}+\mbf{w_0(c)}$, $\Brick^Q$ is the toric variety of Loday's realization of the associahedron \cite[Corollary 16]{E16}. The first step here is to prove that $Q=\mbf{w_0(c)}+\mbf{c'}$ can be obtained from $\mbf{c}+\mbf{w_0(c)}$ by commuting moves (e.g. $s_1s_3=s_3s_1$). Commuting moves shuffle the order of the Grassmannian factors of the product $\prod_{i=1}^{|Q|} \Gr_{q_i,n} \ni \rho(\Brick^Q)$ but do not change the incidence structure recorded by the Magyar diagram (unlike arbitrary permutations of the letters of $Q$) and thus leave $\rho(\Brick^Q)$ unchanged aside from this permutation of its factors.
    % preserve the incidence structure 
    %moves have no impact on $\rho(\Brick^Q)\in \prod_{i=1}^{|Q|} \Gr_{q_i,n}$ except to shuffle the order of the Grassmannian factors of the product (otherwise permuting the letters of $Q$ could change the incidence structure recorded by the Magyar diagram and braid moves change which Grassmannian factors appear). 
    Hence, $\Brick^Q\cong \Brick^{\mbf{c}+\mbf{w_0(c)}},$ and  $\B(\mbf{w_0(c)}+\mbf{c'})=\B(\mbf{c}+\mbf{w_0(c)})$ is the same as Loday's realization of the associahedron up to translation \cite{E16}. 

    To show that $\pi_i(\Brick^Q)$ is smooth and toric, there are three cases. For $i<n,$ we can show $\pi_i(\Brick^Q)\subseteq X_{1}^{s_1s_2\dots s_{n-1}}.$  When $i\geq |Q|-n,$ we show $\pi_i(\Brick^Q) \subseteq X_{w_0(s_{1}s_2\dots s_{n-1})}^{w_0}.$ Both  $X_1^{s_1s_2\dots s_{n-1}}$ and $X_{w_0(s_{1}s_2\dots s_{n-1})}^{w_0}$ are smooth and toric by \Cref{smoothtoric}, so $\pi_i(\Brick^Q)$ is smooth and toric. For $ n\leq i\leq |Q|-n-1,$ we can show $\pi_i(\Brick^Q)=X_{u}^v$ with 
   $u=w_0(s_{q_{i+1}}\dots s_{q_{|Q|}})^{-1}$ and $v=s_{q_1}s_{q_2}\dots s_{q_{i}}.$ Thus  $vu^{-1}=s_{q_1}s_{q_2}\dots s_{q_{i}}s_{q_{i+1}}\dots s_{q_{|Q|}}w_0$ and from that we can obtain $vu^{-1}=s_1\dots s_{n-1}$ and thus $v=s_1\dots s_{n-1}u.$ Thus applying \Cref{smoothtoric}, we conclude that for all $i$, $\pi_i(\Brick^Q)$ is smooth and toric.

    Since $\pi_i(\Brick^Q)$ is smooth and toric, and $\pi_i(C_u)$ is a sub-Richardson of $\pi_i(\Brick^Q)$ for each $i$, we get that $\pi_i(C_u)$ is a smooth toric Richardson. The remaining combinatorial statements within the theorem can be proven from what we have presented already.
\end{sketch}









\details{
    One can prove by elementary arguments that $Q=\mbf{w_0(c)}+\mbf{c'}$ can be obtained from $\mbf{c}+\mbf{w_0(c)}$ by commuting moves (e.g. $s_1s_3=s_3s_1$) and that the subword $(s_{q_{n}},s_{q_{n+1}},\dots s_{q_{|Q|}})$ is a reduced word for $w_0.$   
    %One can prove by induction that $\mbf{w_0(c_{n-1})c_n'}$ can be obtained from $\mbf{w_0(c_n)}$ by commuting moves (e.g. $s_1s_3=s_3s_1$) and use this to show that $\mbf{w_0(c_n)c_n'}=\mbf{c_nw_0(c_{n-1})}$ can be obtained from $\mbf{c_nw_0(c_n)}$ by commuting moves. 
    Commuting moves do not change the interpretation represented by the Magyar diagram of $\Brick^Q$ as a subvariety of $\prod_i \Gr(q_i,n)$, so $\B(\mbf{w_0(c)}+\mbf{c'})=\B(\mbf{c}+\mbf{w_0(c)})$ which is the same as Lodays realization of the associahedron up to translation \cite{E16}, and $\BB(Q)$ will have the same normal fan. 
    Since $\mbf{w_0(c)}$ is a reduced word for $w_0$, $(s_{q_1},s_{q_2},\dots,s_{q_i})$ will be a reduced word whenever $i\leq |Q|-n-1$. 
    Since $(s_{q_n},s_{q_{n+1}},\dots, s_{q_{|Q|}})$ is a reduced word for $w_0$,  $(s_{q_{i+1}},\dots, s_{q_{|Q|}})$ will be a reduced word whenever $i\geq n.$ Consider $n\leq i\leq |Q|-n-1$. If we let $u=w_0\dem(s_{q_{i+1}},\dots, s_{q_{|Q|}})^{-1}$ and $v=\dem(s_{q_1},s_{q_2},\dots, s_{q_{i}}),$ then $vu^{-1}=s_{q_1}s_{q_2}\dots s_{q_{i}}s_{q_{i+1}}\dots s_{q_{|Q|}}w_0=(\prod (\mbf{w_0(c)}+\mbf{c'}))w_0=s_{q_1}s_{q_2}\dots s_{q_{n-1}} w_0w_0=s_1s_2\dots s_{n-1}.$ Thus $v=s_1s_2\dots s_{n-1} u$ so by \Cref{smoothtoric}, $\pi_i(\Brick^Q)=X_u^v$ is smooth and toric. 
    
    $X^{s_{q_1}s_{q_2}\dots s_{q_{n-1}}}=X^{s_{1}s_{2}\dots s_{n-1}}$ is smooth and toric \cite{idk} and for $i\leq n-1,$ $\pi_i(\Brick^Q)=X_{w_0\dem(s_{q_{i+1}},\dots, s_{q_{|Q|}})^{-1}}\cap X^{\dem(s_{q_1}s_{q_2}\dots s_{q_i})}\subseteq X^{s_{q_1}s_{q_2}\dots s_{q_{n-1}}}$ so $\pi_i(\Brick^Q)$ will be smooth and toric for each $i\leq n-1.$ \commentsG{I'm being kinda reckless about citing results without taking care to translate them properly} 

    In general $X^u=w_0\cdot X_{w_0u}.$ With this in mind we find that $X_{w_0(s_{|Q|-n+1}\dots s_{|Q|})^{-1}}= X_{w_0s_{1}\dots s_{n-1}}=w_0\cdot X^{s_1\dots s_{n-1}}$ is smooth and toric because $X^{s_1\dots s_{n-1}}$ is. Thus for $i\geq |Q|-n,$ $\pi_i(\Brick^Q)=X_{w_0\dem(s_{q_{i+1}},s_{q_{i+2}},\dots, s_{q_{|Q|}})^{-1}}^{\dem(s_{q_1},s_{q_2},\dots,s_{q_i})}\subseteq X_{w_0(s_{|Q|-n+1}\dots s_{|Q|})^{-1}}$ so $\pi_i(\Brick^Q)$ is smooth and toric for $i\geq |Q|-n.$ Thus we have shown $\pi_i(\Brick^Q)$ is always smooth and toric.

    Since $\pi_i(\Brick^Q)$ is smooth and toric, and $\pi_i(C_u)$ is a sub-Richardson of $\pi_i(\Brick^Q)$ for each $i$, we get that $\pi_i(C_u)$ is always a smooth toric Richardson. The remaining combinatorial statements within the theorem can be proven from what we have presented already.}




%why \mu(A \cap B)=\mu(A)\cap \mu(B)
 %no. thats not it. 
 %
%\lipsum[3]

%\begin{equation}\label{eqn:eq1}
%\int_0^\infty \exp(-x) \, \mathrm{d}x = 1.
%\end{equation}

%\begin{lem}\label{lem:lem1}
%\lipsum[11]
%\end{lem}

%\lipsum[10]

%\begin{figure}
%\centering
%\includegraphics[width=0.5\textwidth]{plot}
%\caption{A plot of a function.}
%\label{fig:plot}
%\end{figure}

%\subsection{A subsection}

%\lipsum[5]

%\begin{align*}
%1 + 1 + 1 &= 2 + 1 \\ &= 3.
%\end{align*}

%\subsubsection{A subsubsection}

%\lipsum[17]

%Here is a citation~\cite{greenwade93} with URL. Here is another
%citation~\cite{MR4}, which is the earliest-numbered Math Review with a
%functioning DOI: MR0000004. Finally, a reference to
%equation~\eqref{eqn:eq1}. And a reference to \Cref{fig:plot}.
























\acknowledgements{The authors thank their advisor Allen Knutson for proposing the problem of indexing the components of $\Brick^Q(0)$, and for his ongoing insights and support. We thank Mario Sanchez for suggesting we use the word $\mbf{w_0(c)c'}$ instead of $\mbf{cw_0(c)}$ as well as other crucial insights. We thank Spencer Backman for suggesting we look for a Minkowski decomposition for the bulky brick polytope. We also thank Ben Adenbaum, Christian Gaetz, Karola M\'esz\'aros, and Vasu Tewari for helpful conversations. }%\lipsum[20]}

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\printbibliography



\end{document}







   % This theorem tells us why people like to focus on things of the form $\Delta(Q,w_0).$


%This next one tells us we can perform commuting moves on words and the subword complex doesn't change.

%\begin{thm}(\cite[Proposition 3.8]{CJP14})
%	If two words $Q$ and $Q'$ coincide up to commutations then $\Delta(Q,w_0)\cong \Delta(Q',w_0).$
%\end{thm}








%For $Q=\mbf{cw_0(c)}$, $\mrm{Brick}^Q$ coincides up to translation with the $c$-associahedron. \commentsG{Citation needed perhaps but might be hard cuz apparently the related literature confused past me.}
    



%The following modification of the brick polytope is the main object of this paper:


%We define a bulky weight function 
%\begin{defn}
%      The \textbf{bulky weight function} is 
 %      \[\mbf w(J,\cdot)\colon\{\text{subwords of $Q$}\}\to \mbb{R}^n,\quad \mbf{w}(J,k)=(Q\setminus J)_{(k)}(\sum_{i=1}^{n-1}\omega_{i}).\]