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%% define your title in the usual way
\title{Cluster fan of {\bf z}-vectors and toric degenerations}

%% define your authors in the usual way
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\author{Gleb
Koshevoy\thanks{\href{mailto:koshevoy@cemi.rssi.ru}{koshevoy@cemi.rssi.ru}.
This research was  supported by RSF grant 16-11-10075.}\addressmark{1}}

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\address{\addressmark{1}CEMI RAS, MCCME,  Interdisciplinary Scientific
Center J.-V. Poncelet (CNRS UMI 2615)}

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\received{21.03.2018}

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\abstract{For a cluster algebra, we introduce and study a notion of {\bf z}-vectors.  
We prove that {\bf z}-vectors determine cluster variables. We associate to  a cluster seed  a simplicial cone being  span by {\bf z}-vectors of the cluster variables of this seed. These cones form a fan, a polyhedral complex, such that the dual graph of this complex is the Fomin--Zelevinsky exchange graph of the cluster algebra.
}

%% put your French abstract here, or comment this out if you don't have one
\resume{
\'Etant donn\'ee une alg\'ebre amass\'ee, nous introduisons et \'etudions une notion de  {\bf z}-vecteurs.
Nous montrons que les {\bf z}-vecteurs d\'etermin\'ent les variables d'amas. Nous associons \'a une graine d'amas un cone simplicial qui est engendr\'e par les {\bf z}-vecteurs des variables d'amas. Ces cones forment un drapeau, un complexe poly\'edral, tel que le graphe dual de ce complexe est le graphe d'\'echange de Fomin--Zelevinsky de l'alg\'ebre amass\'ee.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Cluster algebras, Laurent polynomials, Polyhedral complexes}

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\section{Introduction}

For a cluster algebra $\mathcal A$, we introduce a notion of $\mathbf z$-vectors. These vectors are associated to the cluster variables and  defined by choosing an initial seed and an ample total order on cluster variables of the seed.
 Namely, for an initial seed which possesses an ample total order on the set of  its variables, we define corresponding  term order on Laurent polynomials in variables of this set. For each cluster variable,  there is a Laurent polynomial expressing this cluster variable in  the variables of the initial seed. Such a polynomial exists due  to the Laurent phenomenon \cite{FZ1}. Then
the $\mathbf z$-vector of a cluster variable  is   the exponent of the leading monomial  of the corresponding  Laurent polynomial.
For an initial seed which does not possess an ample order, we expand the exchange matrix by adding frozen variables in order to get 
a seed with an ample order. Such an ample extension exists.  In such a case, we
define  $\mathbf z$-vectors as  projection along added frozen variables of 
$\mathbf z$-vectors of the extended cluster algebra.
A priori such defined $\mathbf z$-vectors depend on extensions, a conjecture  is that this is up a unimodular transformation.  

Our main result, Theorem \ref{mainth},  says that, if the Positivity conjecture holds true for a cluster algebra $\mathcal A$ (that is the case for example, for geometric cluster algebras \cite{GHKK, pos}), then $\mathbf z$-vectors uniquely determine the cluster variables and
the cluster monomials of $\mathcal A$, and the collection  of cones which are generated by $\mathbf z$-vectors of each seed of $\mathcal A$, form a fan, a polyhedral complex, which is dual to the Fomin--Zelevinsky exchange graph. Note that from this follows
a generalization of the linear independence in \cite{Keller}.


For a cluster
algebra of the coordinate ring of the basis affine space $\mathbb C[N_-\!\!\setminus\!\! GL_n]$, we show that  the cluster seeds corresponding to lexicographically minimal and maximal reduced decompositions of the longest element $w_0$ of the Weyl group $W$
%, the symmetric group $S_n$, 
are ample, and
the corresponding  fans, of cones are generated by $\mathbf z$-vectors, are unimodular equivalent to the fan defined in \cite{K14}, in particular, they 
are unimodular equivalent to the Gelfand--Tsetlin cone. 


\section{Ample orders and quivers}
A quiver $Q=(V,E)$ is a directed graph on the vertex set $V$ and edges set $E\subset V\times V$ without loops and $2$-cycles. As usual $(u,v)\in E$ means that $u=h(e)$ is the head of the edge $e$ and $v=t(e)$ is the tail.

An ice quiver is a quiver $Q$ such that some non-empty set of vertices $V_m\subseteq V$ is
distinguished as a set of mutable vertices, 
and the complement $V_f:=V\setminus V_m$ as frozen vertices.

We assign to each vertex $v\in V$, a formal variable $x_v$, such that $(x_v)_{v\in V}$ form a transcendental basis of the field $\mathbb C((x_v)_{v\in V})$.

A total order $>$  (a complete transitive antisymmetric binary relation)
on the set of variables $x_v$, $v\in V$ union an extra element $\mathbf 0$ ( corresponding to constants), is {\em ample} if, for every $v\in V_m$, it holds
\begin{equation}\label{ampledef}
\prod_{u: (v, u)\in E}x_u\gg \prod_{w: (w, v)\in E}x_w,
\end{equation}
where $\gg$ is the lexicographical extension $>$ to Laurent monomials. We denote also by $\gg$ the corresponding term order on  $\mathbb C[(x_v^{\pm 1})_{v\in V}]$. 
We set the product over the empty set to be $\mathbf 0$.  

\begin{example} a) For the line quiver with even number of vertices, the ample order is unique, for example, for the quiver $u\longrightarrow v$, $u$, $v\in V_m$, the order
$x_u<\mathbf 0<x_v$ is ample.

b) For the line quiver with odd number of variables an ample order does not exist. For example, for the quiver  $u\longrightarrow v\longrightarrow w$,  $u$, $v$, $w\in V_m$,  we have $\mathbf 0<v<\mathbf 0$,
that contradicts transitivity.

c) For any quiver, it suffices to add one more frozen vertex in order to get an ample order on the extended quiver. 
\end{example}
\section{Cluster algebras and $\mathbf z$-vectors}
Let us briefly recall the formalism of cluster algebras (for more details see
\cite{FZ4}) needed here.  For a positive integer $r$,   an ${r}$-regular
tree, denoted by $\mathbb T_r$, whose edges are labeled by  $1, \ldots , {r}$, so that the ${r}$ edges emanating
from each vertex receive different labels. 
We denote by $t_0$ the root of $\mathbb T_r$.  Then an edge of $\mathbb T_r$ is denoted by  $t { \rightarrow}|_k\, t'$,
indicating that vertices
$t, t'\in \mathbb T_r$ form an edge $(t,t')$ of $\mathbb T_r$ and  $k\in [{r}]$ is the color of this edge.

For a case of  geometric cluster algebras, a cluster seed is a pair: an ice quiver,  $Q=(V,E)$,
and a tuple of variables $\mathbf x = (x_{j}, \, j\in V)$, 
such that the collection $\{x_j, j\in V\}$ generates a field $\mathbf C[x_k,\,k\in V_f](x_j,\,j\in V_m)$.

Let us assign a cluster seed to a root $t_0$ of the tree $T_r$ with $r=|V_m|$, and  denote
by $(\mathbf x_{t_0}, Q_{t_0})$ this seed. 
A seed pattern is an assignment of a cluster seed $(\mathbf x_t=(x_{j;t})_{j\in V(Q_t)} , Q_t)$ to every vertex $t\in \mathbb T_m$, such
that the seeds assigned to the endpoints of any edge $t\to _k t'$
are obtained from each
other by the seed mutation $\mu_k$, $k\in V_m$.  The 
mutation $\mu_k$ transforms the quiver and variables. Namely the mutation sends $Q_t$ into a new quiver $Q_{t'} = \mu_k(Q_t)$ via a sequence of three steps. 
Firstly, for each oriented two-arrow path $u\to k\to w$, $u$, $w\in V(Q_t)$, add a new arrow
$u\to w$. Secondly, reverse the direction of all arrows incident to the vertex $v$. Finally,
repeatedly remove oriented 2-cycles until unable to do so.


The mutation $\mu_k$ assigns the variables to the vertices of $\mu_k(Q_t)$ by the following mutation rule: 
$\mu_k(x_{j;t})=x_{j;t}$ if $j\neq k$, and
\begin{equation}\label{mut1}
\mu_k(x_{k;t})=\frac{\prod_{(i, k)\in E(Q_t) }x_{i;t}+\prod_{(k, j)\in E(Q_t) }x_{j;t}}{x_{k;t}}.
\end{equation}
A $\mathbf d$-vector of a cluster variable (\cite{FZ4}) is the exponent of the denominators of the Laurent polynomial expressing this variables in cluster variables of the initial seed.  
 For a finite cluster algebra, $\mathbf d$-vectors uniquely determine cluster variables and cluster monomials, and, moreover, the collection of simplicial cones, such that a cone of the collection is generated by $\mathbf d$-vectors of cluster variables of a seed, forms a fan, while seeds run over the set of cluster seeds.
In this case, the union of the cones of this fan covers the whole space. 
For general cluster algebras $\mathbf d$-vectors may not determine cluster variables.

Fomin and Zelevinsky (\cite{FZ4}) introduced a notion of  $\mathbf g$-vectors, and 
Derksen et al (\cite{DWZ}) proved that $\mathbf g$-vectors uniquely determine cluster variables, and the cones, generated by $\mathbf g$-vectors of cluster variables of seeds, form a fan. Note that, for a 
cluster algebra which is not finite, the union of the cones of the fan does not coincide with the linear span of $\mathbf g$-vectors. 

We introduce a notion of $\mathbf z$-vectors, and show that for every cluster algebra for which the Positivity Conjecture holds true, $\mathbf z$-vectors 
share the same  properties as $\mathbf g$-vectors have.

For a geometric cluster algebra, we say that an initial seed is {\em ample} if its quiver is {\em ample}. For a general case, we
say that an initial seed with an exchange matrix $B_0$  of a cluster
algebra $\mathcal A$ is {\em ample}, if there exists a total order on the
set of the variables of the seed, such that (\ref{ampledef}) holds true
for any mutable variable $v$, where the left hand side of
(\ref{ampledef}) is replaced by the product $\prod_{u: b_{u,v}\ge 0}u$
and the right hand side is replaced by $\prod_{w: b_{u,w}\le 0}w$.


For a non ample quiver $Q=(V, E)$,  we call a quiver $\hat Q=(V', E')$ such that $V\subset V'$, $V_m=V'_m$, $\hat Q|_V=Q$, and $\hat Q$ is ample, an {\em ample extension} of $Q$. 
Obviously, any quiver possesses an ample extension. Similarly is defined an ample extension of an exchange matrix.


In what follows we consider  geometric cluster algebras. 
Let  $(\mathbf x_0, Q_0)$ be an initial  cluster seed,  without loss of generality we assume that $Q_0$ is an ample quiver.
Let $>$ be an ample order, and  denote by $\gg$ the corresponding term order  on $\mathcal F:=\mathbf C[x_k,\,k\in V_f](x_j,\,j\in V_m)$ defined with respect to the lexicographical extension $>$ to Laurent monomials. 

 
 
For a variable $x_{j;0}$, we set the $j$th basis vector $e_j=(0, \ldots, 0, 1, 0, \ldots, 0)\in \mathbb Z^V$ (with $1$ at the $j$th place) to be its $\mathbf z$-vector. 


Suppose that, for a cluster seed $(\mathbf x_t, Q_t)$, $t\in \mathbb T_{|V_m|}$,
the $\mathbf z$-vectors of its cluster variables are yet defined. Then, for a cluster  $\mu_j(\mathbf x_t, Q_t)$, $j\in V_m(Q_t)$, the $\mathbf z$-vectors of its variables
we define by the following rule:  For $k\neq j$, $\mathbf z(\mu_j(x_{k;t}))=\mathbf z(x_{k;t})$; and 
\begin{equation}\label{zmut1}\mathbf z(\mu_j(x_{j;t}))=
\begin{cases} 
-\mathbf z(x_{j;t})+\sum_{(k,j)\in E(Q_t)}\mathbf z(x_{k;t}) &
 if \,\,  \prod_{(k,j)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{k;t})}\gg \prod_{(j,l)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{l;t})} \cr
-\mathbf z(x_{j;t})+\sum_{(j,l)\in E(Q_t)}\mathbf z(x_{l;t}) & otherwise \cr
\end{cases}\end{equation}
Here are some useful properties of $\mathbf z$-vectors.
\begin{prop}\label{z-premier}\hfill\break
%\begin{enumerate}
%\item 
1. For  each  cluster seed $(\mathbf x_t, Q_t)$, the vectors
$\mathbf z (x_{j;t})$, $j\in V(Q_t)$, form a basis of $\mathbb Z^V$.\\
2. For any cluster seed $(\mathbf x_t, Q_t)$, and any vertex $j\in V_m(Q_t)$,
it holds 
\begin{equation}\label{zmainprop}
\frac{ \prod_{(k,j)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{k;t})}}{\prod_{(j,l)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{l;t})}}\mbox{ is a Laurent monomial in }  y_j, \, j\in V_m(Q_0),
\mbox{ where }
 y_j=\frac{ \prod_{(k,j)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{k;0})}}{\prod_{(j,l)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{l;0})}}.
 \end{equation}
3. For any cluster seed $(\mathbf x_t, Q_t)$, and any vertex $j\in V_m(Q_t)$,
we have
\begin{equation}\label{zrepr}
x_{j;t}=\mathbf x_0^{\mathbf z(x_{j;t})}(1+R_{j;t}( y_j, \, j\in V_m(Q_0))), 
\end{equation}
where $R_{j;t}( y_j, \, j\in V_m(Q_0))$ is a Laurent polynomial in variables
$( y_j, \, j\in V_m(Q_0))$ which  is dominated by $\mathbf 0$
with respect to term order $\gg$. 
\end{prop}

\begin{remark} For general cluster algebras, for validity of analogs of the above proposition and all  theorems  below, we have to  assume that  Positivity Conjecture holds true for such cluster algebras. This conjecture says that the Laurent polynomials, expressing 
cluster variables of one cluster seed in variables of another seed, have positive coefficients. This conjecture was recently proven for geometric cluster algebras.
\end{remark} 

\begin{theorem}[\cite{GHKK,pos}]\label{CP}
For a geometric cluster algebra, Positivity conjecture holds true. 
\end{theorem} 
%{\em Proof of Proposition \ref{z-premier}}. 
\begin{proof}[Proof of Theorem \ref{z-premier}]
The item 1 follows from that  the collection of vectors
$\mathbf z (x_{j;0})$, $j\in V(Q_0)$, is a basis of $\mathbb Z^V$, and the mutation rule (\ref{zmut1}) preserves unimodularity.

For proving the item 2, we proceed by induction on  distance between $t\in\mathbb T_m$ and the root.
Namely, let us denote by
$
P_{u;t}%(\hat y _j, j\in V_m(Q_0))
:=\frac{ \prod_{(k,u)\in E(Q_t) }\mathbf x_0^{\mathbf z(x_{k;t})}}{\prod_{(u,l)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{l;t})}}
$
the corresponding Laurent polynomial, $u\in  V_m(Q_t)$. 
Then, for the mutation $\mu_u$ of $(\mathbf x_t, Q_t)$, $u\in  V_m(Q_t)$, we have 
$
\mu_u(P_{u;t})=\frac 1{P_{u;t}}$,
and, for $v\neq u$, it might be  
\[
\mbox{either } \mu_u(P_{v;t})=P_{v;t}\mbox{ or }\mu_u(P_{v;t})=P_{v;t}
P_{u;t}^{|b_{u,v}|}\mbox{ or }\mu_u(P_{v;t})=P_{v;t}(P_{u;t})^{-|b_{u,v}|},
\]
where $b_{u,v}$ denotes the multiplicity of edges joining $u$ and $v$.
From this follows the statement of the item 2.

The item 3: by induction on the distance between $t$ and the root, we have 
\[
\mu_k(x_{k;t})=\frac{\prod_{(i,k)\in E(Q_t) }x_{i;t}+\prod_{(k, j)\in E(Q_t) }x_{j;t}}{x_{k;t}}=
\]
(this is due to the induction assumption)
\[
\frac{\prod_{(i,k)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{i;t})}(1+R_{i;t})+\prod_{(k,j)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{j;t})}(1+R_{j;t})}{\mathbf x_0^{\mathbf z(x_{k;t})}(1+R_{k;t})}=
\]
(here we assume $\prod_{(i,k)\in E(Q_t) }\mathbf x_0^{\mathbf z(x_{i;t})} \gg \prod_{(k,j)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{j;t})}$)
\[
\frac{\prod_{(i,k)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{i;t})}}{\mathbf x_0^{\mathbf z(x_{k;t})}}\left(
\frac{\prod_{(i,k)\in E(Q_t)}(1+R_{i;t})+
\frac{\prod_{(k,j)\in E(Q_t)}\mathbf x_0^{\mathbf z(x_{j;t})}}{\prod_{(i,k)\in E(Q_t) }\mathbf x_0^{\mathbf z(x_{i;t})}}\prod_{(k,j)\in E(Q_t)}(1+R_{j;t})}{(1+R_{k;t})}\right)=
\]
\[
\mathbf x_0^{\mathbf z(\mu_k(x_{k;t}))}
\frac{\prod_{(i,k)\in E(Q_t)}(1+R_{i;t})+
P_{k;t}^{-1}\prod_{(k,j)\in E(Q_t)}(1+R_{j;t})}{(1+R_{k;t})}
\]
Since $\mathbf 0\gg P_{k;t}^{-1}$, and $\mu_k(x_{k;t})$ is a Laurent polynomial in $x_{j;0}$, $j\in V(Q_0)$, we get that, due to Theorem \ref{CP}, 
\[
\frac{\prod_{(i, k)\in E(Q_t)}(1+R_{i;t})+
P_{k;t}^{-1}\prod_{(k, j)\in E(Q_t)}(1+R_{j;t})}{(1+R_{k;t})}
\]
is   a Laurent polynomial of the required form.\end{proof}

For a cluster seed $(\mathbf x_t, Q_t)$ with an ample initial seed, we denote by $\mathbf Z_t$ the cone generated by the vectors $\mathbf z(x_{j;t})$,
$j\in V(Q_t)$.

\begin{theorem}\label{mainth}\hfill
\begin{enumerate} 
\item For geometric cluster algebra $\mathcal A$ with an ample initial seed  $(\mathbf x_0, Q_0)$, the cones $\mathbf Z_t$, $t\in \mathbb T_{|V_m|}$, form a conical polyhedral complex, a fan $\mathcal Z(\mathcal A)$, and two cones coincide, $\mathbf Z_t=\mathbf Z_{t'}$, if and only if  the cluster seed $(\mathbf x_t, Q_t)$ is isomorphic to $(\mathbf x_{t'}, Q_{t'})$.
\item For a non ample cluster seed $(\mathbf x_0, Q_0)$ of $\mathcal A$, let $(\hat{\mathbf x}_0, \hat Q_0)$ be  an ample extension, and  $\hat{\mathcal A}$ be the corresponding  extension of $\mathcal A$ obtained by adding coefficients (frozen variables). Then 
the cones $\hat{\mathbf Z}_t$, $t\in \mathbb T_{|V_m|}$, form a fan
$\hat{\mathcal Z}(\mathcal A)$, and $\hat{\mathbf Z}_t=\hat{\mathbf
Z}_{t'}$ if and only if  the cluster seed $(\mathbf x_t, Q_t)$ is isomorphic to $(\mathbf x_{t'}, Q_{t'})$.
\end{enumerate}
\end{theorem}
For a case of non-ample initial quiver, the projection of the fan ${\mathcal Z}(\hat{\mathcal A})$ along the added frozen coordinates
turns out to be a fan, and the projections of any two different cones of
${\mathcal Z}(\hat{\mathcal A})$ are different cones.

Denote by ${\mathcal Z}_{\hat Q_0\to Q_0}({\mathcal A})$ such a projection of the fan
${\mathcal Z}(\hat{\mathcal A})$ and call the generators of one-dimensional rays of the former fan  $\mathbf z$-{\em vectors of $\mathcal A$ conditionally on} $\hat Q_0$.

Our conjecture is  that, for any two ample extensions of $Q_0$, $\hat Q_0$ and $Q'_0$, the fans ${\mathcal Z}_{\hat Q_0\to Q_0}({\mathcal A})$ and 
${\mathcal Z}_{ Q'_0\to Q_0}({\mathcal A})$ are unimodular equivalent.

Before going to prove Theorem \ref{mainth}, we pause for examples.
%\medskip

\begin{example}

a) Consider the cluster algebra with the initial seed 
 $x\longrightarrow y$. This quiver is ample with the ample order $y>\mathbf 0 > x$ and the corresponding $\mathbf z$-vectors are  depicted below. Note that the cluster cone is  the
normal fan to the pentagon.

b) Consider the cluster algebra with  the initial seed
 $x\rightrightarrows y$. This quiver is also ample,  $y> \mathbf 0 > x$ and corresponding $\mathbf z$-vectors are  depicted below on Picture b).
 \end{example}
\[
\begin{matrix}{
\unitlength=0.6mm \special{em:linewidth 0.4pt}
\linethickness{0.4pt}
\begin{picture}(105.00,105.00)
\put(80.00,60.00){\vector(1,0){30.00}}
\put(110.00,58.00){\makebox(0,0)[cc]{$x$}}
\put(80.00,60.00){\vector(-1,0){30.00}}
%
\put(80.00,60.00){\vector(0,1){30.00}}
\put(83.00,90.00){\makebox(0,0)[cc]{$y$}}
\put(80.00,60.00){\vector(0,-1){30.00}}
%
\put(80.00,60.00){\vector(-1,1){27.00}}

\put(70.00,5.00){\makebox(0,0)[cc]{$a$}}

\end{picture}
%\bigskip

}&{
\unitlength=0.6mm \special{em:linewidth 0.4pt}
\linethickness{0.4pt}
\begin{picture}(105.00,105.00)
%\put(40.00,60.00)
\put(80.00,60.00){\vector(1,0){10.00}}
\put(92.00,58.00){\makebox(0,0)[cc]{$x$}}
\put(80.00,60.00){\vector(-1,0){10.00}}
%
\put(80.00,60.00){\vector(0,1){10.00}}
\put(83.00,70.00){\makebox(0,0)[cc]{$y$}}
\put(80.00,60.00){\vector(0,-1){10.00}}
%
%\put(80.00,60.00){\vector(-1,1){45.00}}
\put(80.00,60.00){\vector(-1,2){10.00}}
\put(80.00,60.00){\vector(-2,3){20.00}}
\put(80.00,60.00){\vector(-3,4){30.00}}
\put(45.00,100.00){\makebox(0,0)[cc]{$\ldots$}}

%\put(80.00,60.00){\vector(-4,5){15.00}}
\put(80.00,60.00){\vector(-2,1){20.00}}
\put(80.00,60.00){\vector(-3,2){30.00}}
\put(80.00,60.00){\vector(-4,3){40.00}}

\put(40.00,96.00){\makebox(0,0)[cc]{$\vdots$}}
\put(70.00,5.00){\makebox(0,0)[cc]{$b$}}
\end{picture}
%\bigskip
}
\end{matrix}
\]


  

%\medskip



\begin{theorem}\label{M2} Let $\mathcal A$ be a geometric cluster algebra with an ample  $(\mathbf x_0, Q_0)$. Then 
\[
\mathbf z(x_{j;t})=\mathbf z(x_{j';t'})\mbox{ if and only if  } x_{j;t}=x_{j';t'}.
\]
\end{theorem}
%{\em Proof of Theorem \ref{M2}}. 
\begin{proof}[Proof of Theorem \ref{M2}]
Suppose  there are two seeds $\mathbf S_t=(\mathbf x_t, Q_t)$ and ${\mathbf S}_{t'}=(\mathbf x_{t'}, Q_{t'})$
and cluster variables $x_{j;t}$ and $x_{j';t'}$ in $\mathbf S_t$ and ${\mathbf S}_{t'}$, respectively, 
such that $\mathbf z(x_{j;t})=\mathbf z(x_{j';t'})$. Then 
$x_{j;t}$ and $x_{j';t'}$ have the same leading monomials with respect to $\gg$, and from (\ref{zrepr}),  we have 
$x_{j;t}=\mathbf x_0^{\mathbf z(x_{j;t})}(1+R_{j;t}({\mathbf y}))$, and 
$x_{j';t'}=\mathbf x_0^{\mathbf z(x_{j';t'})}(1+R_{j';t'}({\mathbf y_1'}) )$.

%Then by Laurentness property of cluster algebras \cite{FZ1},  
For each $j$,   $x_{j,;t}=P'_{j,t}(\mathbf x_{t'})$, where  $P'_{j,t}$ is a Laurent polynomial of the cluster variables $\mathbf x_{t'}$.
We claim that such a Laurent polynomial is of the form
\begin{equation}\label{sum1}
x_{j;t}=P'_{j,t}(\mathbf x_{t'})=x_{j';t'}+\overline{P}'(\mathbf x_{t'}\setminus x_{j';t'}),
\end{equation}
where $\overline{P}'(\mathbf x_{t'}\setminus x_{j';t'})$ is a Laurent polynomial which does not depend on the variable $x_{j';t'}$.

In fact, the leading monomial of $x_{j;t}$ with respect to  $\gg$  is equal to the leading monomial of  $x_{j';t'}$.  Hence, if  the leading monomial of $P_{j,t}(\mathbf x_{t'})$
% the Laurent expression of $x_{j;t}$ in the cluster variables $\mathbf x_{t'}$ 
would be different of  $x_{j';t'}$, we will get   linear dependence among the
 vectors $\mathbf z(x_{k,t'})$, $k\in V(Q_{t'})$.  This is not the case because of the item 1 of Proposition \ref{zmainprop}. 
 
By analogy, we have 
\begin{equation}\label{sum2}
x_{j';t'}=x_{j;t}+\overline{P}(\mathbf x_{t}\setminus x_{j;t}),
\end{equation}
where $\overline{P}(\mathbf x_{t}\setminus x_{j;t})$ is a Laurent polynomial which does not depend on the variable $x_{j;t}$.
Substituting  (\ref{sum1}) in (\ref{sum2}), we get
\begin{equation}\label{sum3}
\overline{P}'(\mathbf x_{t'}\setminus x_{j';t'}) +\overline{P}(\mathbf x_{t}\setminus x_{j;t}) =0.
\end{equation}
Because of positivity the coefficients of Laurent polynomials in (\ref{sum3}), we get 
%this is the case if and only if
$
\overline{P'}=\overline{P}=0$.
\end{proof}
%\hfill  $\Box$
% \medskip
\begin{proof}[Proof of Theorem \ref{mainth}]
%{\em Proof of Theorem \ref{mainth}}. 
Suppose there are two seeds, $\mathbf S_t$ and $\mathbf S_{t'}$,
such that the intersection of cones $C:=\mathbf Z_t$ and  $C':=\mathbf  Z_{t'}$  is not a common face. 
Let us firstly consider a case $int(C)\cap int(C')\neq\emptyset$, where $int(C')$ denotes interior of a cone.
Because the cones $C$ and $C'$ are unimodular simplicial cones (item 1 of Proposition \ref{zmainprop}), there is an integer vector $\mathbf z\in C\cap C'$. Hence there is a monomial $X$ of the cluster variables $\mathbf x_t$ such that it has the form, due  to (\ref{zrepr}), $
X=\mathbf x_0^{\mathbf z}(1+R( \mathbf y))$
with the leading monomial $\mathbf x_0^{\mathbf z}$.

Analogously, there is a monomial $X'$ of the cluster variables $\mathbf x_{t'}$ such that  $
X'=\mathbf x_0^{\mathbf z}(1+R '(\mathbf y ))$ 
with the same leading monomial $\mathbf x_0^{\mathbf z}$.


Follow the same reasoning as in the proof of Theorem \ref{M2}, we get $X=X'$.

%Because of this we have the following

{\em Claim}. Each cluster variable $x_{j;t}$  of the seed $\mathbf S_t$ is a Laurent monomial in the cluster variables $\mathbf x_{t'}$,
and vice versa, each cluster variable $x_{j';t'}$  of the seed $\mathbf S_{t'}$ is a Laurent monomial in cluster variables $\mathbf x_t$.

In fact,  $X=\prod_{j}x_{j;t}^{a_{j;t}}$ is a cluster monomial in variables of $\mathbf S_t$. 
Each $x_{j;t}$, $j\in V(Q_t)$  is a Laurent polynomial in the cluster variables $\mathbf x_{t'}$. Hence $X$ is a monomial of
the Laurent polynomials in variables of $\mathbf x_{t'}$.  Since $X=X'$ and $X'$ is a monomial in cluster variables of $\mathbf x_{t'}$, we conclude that each  $x_{j;t}$  of the seed $\mathbf S_t$ is a Laurent monomial in the cluster variables $\mathbf x_{t'}$.
That proves the claim.

%Analogously, each cluster variable in $\mathbf x_{t'}$ is a Laurent monomial in cluster variables of $\mathbf x_t$.

Since the cones $C$ and $C'$ have a common interior point, there is a facet
of one of these cones such that this facet has a common interior point with another cone.
Without loss of generality, suppose that such a facet $F$ is a facet of 
$C$. It is easy to see that the vertices of $C$ corresponding to $\mathbf z$-vectors of the frozen variables belong $F$. Let a vertex $\{u;t\}\in V(Q_t)$ be such that $\mathbf z(x_{u;t})$ spans an extreme  ray of $C$ which does not belong to $F$. Then denote by
$\mathbf S_{t''}$ the seed $\mu_{\{u;t\}}(\mathbf S_{t})$.
Then cones $C'':=\mathbf Z_{t''}$ and $C'$ have a common interior point (for example, such a point
exists in a small neighbor
of $F$).   Thus, we are in the situation of Claim, and, hence, the new cluster variable $\mu_u(x_{u;t})$ of 
the seed $\mathbf S_{t''}$
is a Laurent monomial in
cluster variable of $\mathbf S_{t'}$. But this can not be the case. In fact, all cluster variables in $\mathbf S_t$
are  Laurent monomials in cluster variables $\mathbf S_{t'}$, and hence, due to the mutation rule (\ref{mut1}), $\mu_u(x_{u;t})$  
has to be the sum of 
two Laurent monomials in cluster variables $\mathbf S_{t'}$. 
Thus, we get the statement under the assumption $int(C)\cap int(C')\neq \emptyset$.

Now, let $F_1$ and $F_2$ be minimal faces of $C$ and $C'$, respectively,
such that $C\cap C'\subset F_i$, $i=1,2$, and $F_1\neq F_2$.

 
Let the cluster variables $x_{f;t}$, $f\in L_1$,  be such that  the vectors $\mathbf z(x_{f;t})$, $f\in L_1$, span the cone $F_1$, and let the cluster variables $x_{f';t'}$, 
$f'\in L_2$,  be such that  the vectors $\mathbf z(x_{f';t'})$, $f'\in L_2$, span the cone $F_2$.
 

We claim that each cluster variable $x_{f;t}$, $f\in L_1$, is a Laurent monomial in the variables $x_{f';t'}$, 
$f'\in L_2$, and vice versa 
each cluster variable $x_{f';t'}$, $f'\in L_2$, is a Laurent monomial in the variables $x_{f;t}$, 
$f\in L_1$.  

From the contrary, suppose there is a cluster variable $x_{f;t}$ such that the corresponding Laurent monomials in  variables  of the cluster $\mathbf S_{t'}$, contains a variable $x_{u;t'}$,
which does not belong to the set  $x_{f';t'}$, $f'\in L_2$.
Then we consider a mutation $\mu_{\{u;t'\}}(\mathbf S_{t'})$.  Denote by $\mathbf  S_{t''}:=\mu_{\{u;t'\}}(\mathbf S_{t'})$ the new seed.

  Because $\mathbf z(x_{u;t'})\not\in F_2$, the cone $C''=\mathbf Z_{t''}$ contains the intersection $F_1\cap F_2$. 
Hence  each of the cluster variables $x_{f;t}$, $f\in L_1$, is a Laurent monomial in the variables $x_{j;t''}$ of the cluster $\mathbf S_{t''}$.

If the expression of  $x_{f;t}$  as a Laurent monomial in variables of $\mathbf S_{t''}$ contains $\mu_u(x_{u;t'})$,
then $\mu_u(x_{u;t'})$ can be expressed as a Laurent monomial in variables of the cluster $\mathbf S_{t'}$, that is not the case. 

Hence   the expression  of  $x_{f;t}$  as a Laurent monomial in variables of $\mathbf S_{t''}$ does not contain $\mu_u(x_{u;t'})$, and since $x_{u;t'}$ is not a variable of 
$\mathbf S_{t''}$, this expression of $x_{f;t}$ does not contain $x_{u;t'}$.
Thus we get the claim.


Because of this claim,  $F_1$ belong to the linear span of $F_2$, and $F_2$ belong to the linear span of $F_1$. This implies that $F_1\cap F_2$ is full-dimensional in each of the linear spans of $F_1$ and $F_2$.

Then by the same reasoning  as for the full-dimensional intersection $C_1\cap C_2$, we can find a cluster variable $x_{w;t}$ of $\mathbf S_t$, such that $\mathbf z(x_{w;t})$ is  a vertex  of $F_1$, such that, for the  cone $F_1'$ being span by 
$F_1\setminus \mathbf z(x_{w;t})$ and $\mathbf z(\mu_w(x_{w;t}))$, we have the full-dimensional (of dimension $F_2$)  intersection  $F_1'\cap int (F_2)$ (here $int(F_2)$  denotes the relative interior of $F_2$). 
Due to minimality of $F_1$,   $F_1'$ belongs to the linear span of $F_2$, otherwise $F_2$ would belong to the linear span of $F_1\setminus \mathbf z(x_{w;t})$.  
This implies that  the variable
$\mu_w(x_{w;t})$  can be expressed as a Laurent monomial in 
the variables $x_{f';t'}$, $f'\in L_2$, that is not the case due to (\ref{mut1}).
% \hfill $\Box$
\end{proof}

\section{Base affine space $N_-\!\!\setminus\!\! GL_n$ and the simplicial fan in the cone of semistandard Young tableaux}
For $GL_n(\mathbb C)$ and its unipotent subgroup $N_-$, consider an
action $N_-$ on $GL_n(\mathbb C)$ by multiplication on the left.
The set of regular functions on $GL_n(\mathbb C)$ which are invariant under such an action of $N_-$,
$\mathbb C[GL_n]^{N_-}=\mathbb C[x_{ij}]^{N_-}$,  is the coordinate ring of the {\em  base affine space}, the GIT  quotient $N_-\!\!\setminus\!\!\!\setminus\!\! GL_n$. 
The ring $\mathbb C[N_-\!\!\setminus\!\!  GL_n]$ plays an important role in the representation theory of $GL_n(\mathbb C)$: it naturally carries all its irreducible representations, each with multiplicity 1.

According to First Fundamental Theorem of invariant theory, $\mathbb C[N_-\!\!\setminus\!\! GL_n]$ is
generated by the {\em flag minors}. (A flag minor $\Delta_I$ of a matrix $x=(x_{ij})\in GL_n(\mathbb C)$ is a minor
occupying in  the first $|I|$
rows and in columns  of the set $I$.)
Moreover, there hold (see, for example \cite{Fulton}):
(i) The coordinate ring $\mathbb C[N_-\!\!\setminus\!\! GL_n]$ is a unique factorization domain.
(ii) For any $I\subset [n]$, the flag minor $\Delta_I$ is an irreducible polynomial in $\mathbb C[x_{ij}]$
(and hence in $\mathbb C [N_-\!\!\setminus\!\! GL_n]$).
(iii)  Flag minors $\Delta_I$ are non-equivalent among themselves. 

Let us recall a construction of a distinguished tableaux basis in 
 $\mathbb C[N_-\!\!\setminus\!\! GL_n]$.  
For a semistandard Young tableau $Y$, an element $\Delta_Y$
of this basis is  the
product of the flag minors with column sets 
corresponding to the fillings of columns of $Y$.
Because of the   D\'esam\'enien--Kung--Rota algorithm (\cite{DKR, MS}), the tableaux basis  is an additive basis in  $\mathbb C[N_-\!\!\setminus\!\! GL_n]$.


Recall (see, for example \cite[Theorem  14.11]{MS}), that the flag minors
form also a sagbi basis.\footnote{The term {\em sagbi} is an acronym for {\em subalgebra analogue of Grobner
bases for ideals}.} The existence of a finite sagbi basis is a special property for a
subalgebra $R$. Namely, such a sagbi basis defines a flat degeneration of $R$  the algebra of its leading term monomials. The latter is an affine toric variety, and hence a
finite sagbi basis provides a flat family connecting $Spec(R)$ and the affine toric variety.


For a cluster algebra on $\mathbb C [N_-\!\!\setminus\!\! GL_n]$ there are seeds labeled by reduced decompositions of $w_0$. Recall a construction of a cluster quiver  corresponding to a reduced decomposition of $w_0$ (\cite{BFZup}).

A reduced decomposition of $w\in W$ is a word $\mathbf i = (i_1\cdots i_l)$ in the alphabet $[n-1]=\{1,\ldots, n-1\}$, such that $w=s_{i_1}\cdots s_{i_l}$ gives a factorization of  smallest length. 
Let us denote by $|i_r|$ the letter of
$[n-1]$ at the position $i_r$ in $\mathbf i$. Denote by $\mathbf i_{\min}$ the lexicographical minimal reduced decomposition
$1,(2,1),(3,2,1),\ldots ,(n-1,n-2,\ldots ,2,1)$ of $w_0$, and by $\mathbf i_{\max}$ the lexicographical maximal  reduced decomposition
$n-1,(n-2,n-1),(n-3,n-2,n-1),\ldots ,(1, 2,\ldots, n-1)$.


Let us add $n$ additional entries $i_{-n+1}, \ldots , i_{-1}$ at the beginning of a reduced decomposition 
$\mathbf i$ of $w_0$, by setting $i_{-j} = -j$
for $j \in [ n]$. 
For $k\in -[ n-1]\cup [ l(w_0)]$, we denote by $k_
+ = k^+_{\mathbf i}$
the smallest index $r$ such
that $k < r$ and $|i_r| = |i_k|$; if $|i_k|\neq |i_r|$ for $k < r$, then we set $k_+ = l(w_0) + 1$. An index $k\in [ n]\cup [ l(w_0)]$ is $\mathbf i$-exchangeable if both $k$ and $k_+$ belong to $[l(w_0)]$.
Denote by $A=(a_{ij})$ the Cartan matrix for $SL_n$.

Now, the  vertices of the quiver
$ G ( \mathbf i)$ are labeled by elements of $-[ n]\cup [ l(w_0)]$;
vertices $i_k$ and $i_r$, with $k < r$, $k,\,r\in -[ n]\cup [ l(w_0)]$,
are connected by an
edge in the quiver if and only if either $k$ or $r$ (or both) are $\mathbf i$-exchangeable, and one of the
following  conditions is satisfied:
(1) $r = k_+$; (2) $r < k_+ < r_+$, $a_{|i_k|,|i_r|} < 0$, $k$, $r\in [l(w_0)]$;
(3) $r < r_+ < k_+$, $a_{|i_k|,|i_r|} < 0$, $k\in -[n-1]$.
The edges of type (1) are called {\em horizontal}, and  of types (2) and (3) are called
{\em inclined}. Horizontal edges are directed from smaller to bigger and inclined are directed vice versa.
%\medskip


The set of frozen vertices is constituted from nodes labeled by $i_{-j}$, $j\in [n]$, and
nodes  labeled by
$i_k$ with  $k_+ = l(w_0) + 1$, $k\in [n-1]$. So, there are $(n-1)(n-2)/2$ mutable nodes and $2n-1$ frozen ones 

Denote by $\mathcal P(\mathbf i)$  a cluster algebra on $\mathbb C [N_-\!\!\setminus\!\! GL_n]$ with the initial quiver $Q_0:=G(\mathbf i)$, and the following set of initial cluster variables:  to a vertex $i_r$, $r=1,\ldots , l(w_0)$,  is  associate the minor 
$\Delta_{s_{i_1}\ldots s_{i_r}(\omega_{i_r})}$, 
and  to the vertex $i_{-j} $, is associate the minors $\Delta_{\omega_j}$, $j\in [n-1]$, where $\omega_j$ denotes $j$th fundamental weight, $j\in [n-1]$, and we regard $\omega_j$ as a subset $[j]\subset [n]$. (For example, $\Delta_{s_1s_2\omega_2}$ is nothing but the flag minor $\Delta_{\{2,3\}}$.)

Denote by $\mathcal P^{w_0}(\mathbf i)$ a cluster algebra on $\mathbb C [N_-\!\!\setminus\!\! GL_n]$ with the initial quiver $Q_0:=G(\mathbf i)$, and the following set of initial cluster variables: to a vertex $i_r$, $r=1,\ldots , l(w_0)$,  is associate the minor 
$\Delta_{w_0s_{i_1}\ldots s_{i_r}(\omega_{i_r})}$, 
and  to the vertex $i_{-j} $, is associate the flag minors $\Delta_{w_0\omega_j}$, $j\in [n-1]$. 

The cluster algebras $\mathcal P(\mathbf i)$ and $\mathcal P^{w_0}(\mathbf i)$ are isomorphic. This isomorphism takes the form of reversing directions of edges in all quivers.

\subsection{The lexicographical minimal and maximal reduced decompositions of $w_0$}
Here we show that the quivers $ G(\mathbf i_{\min})$ and $ G(\mathbf i_{\max})$ are ample, and the corresponding fans of $\mathbf z$-vectors are unimodular equivalent to the fan constructed  in \cite{K14}. %in the cone of semistandard Young tableaux.

%Consider the seed  $\mathbf S_{t_0}$ with the quiver $\mathbf G(\mathbf i_{\min})$ as the initial seed in $\mathcal P^{w_0}(n)$. 
Consider the reverse row lexicographical order on matrix entries $x_{n n}>x_{n-1\, n}>\ldots x_{1\,n}>\ldots x_{n\,1}>\ldots x_{11}$, and  the corresponding term order $\succ^{rr}$ on $\mathbb C[x_{ij}]$.  

Then we have the following
\begin{lemma}\label{leadterm}
The  order on vertices of the quiver $  G(\mathbf i_{\min})$, defined by ordering  the minors associated to the vertices as above with respect to $\succ^{rr}$, is ample, and hence  $ G(\mathbf i_{\min})$ is ample.\end{lemma}
\begin{proof}
%{\em Proof}. 
The statement follows from that the vertices of $  G(\mathbf i_{\min})$ are labeled by interval flag minors.
In fact, since  product of the  elements of the main diagonal of a minor is the leading term with respect to $\succ^{rr}$, we get
 that  nodes of $  G(\mathbf i_{\min})$  
are ordered in descending from left to right and from top to bottom. Because of such descending order, we get the statement.
\end{proof}
%\hfill $\Box$
%\medskip

%Denote by $>>$ such an order on $\mathbf  G(\mathbf i_{\min})$. 

Then we have the following
\begin{prop}
For the  initial ample seed $  G(\mathbf i_{\min})$, % $>>$, 
the fan of $\mathbf z$-vectors of $\mathbf Z(\mathcal P^{w_0}(\mathbf i_{\min}))$ is unimodular equivalent to the fan constructed in \cite{K14}.
\end{prop} 
From this Proposition, we get that the cone generated by $\mathbf
z$-vectors is unimodular equivalent to the Gelfand--Tsetlin cone. 
Moreover, because of this we get that the cluster algebra $\mathcal
P^{w_0}(G(\mathbf i_{\min}))$ has a flat toric degeneration to the affine
toric variety defined by the GT-cone. This situation  corresponds to the
case of sagbi basis for antidiagonal term order in \cite[Theorem
14.11]{MS}.
%\medskip


The sagbi basis for diagonal term order in the loc. cit. corresponds to
the initial seed with $G(\mathbf i_{\max})$. 
Then we have the following analog to Lemma \ref{leadterm}.
\begin{lemma}\label{leadterm1}
The  order on vertices of the quiver $  G(\mathbf i_{\max})$ corresponding to ordering the minors, associated to the vertices as above with respect to $\succ^{rr}$, is ample, and hence  $  G(\mathbf i_{\max})$ is ample.\end{lemma}
\begin{proof}%[Proof of Theorem \ref{Tsample}]
%{\em Proof}. 
The statement follows from that the vertices of $\mathbf  G(\mathbf i_{\max})$ are labeled by complements to interval flag minors. In fact, since  product of the  elements of the main diagonal of a minor is the leading term with respect to $\succ^{rr}$, 
the nodes of $ G(\mathbf i_{\max})$ are ordered, follow $\succ^{rr}$,  in descending from right to left and from top to bottom. From that we get the statement.
%\hfill $\Box$\medskip
\end{proof}
 Then we have the following
\begin{prop}
For the  initial ample seed $  G(\mathbf i_{\max})$, 
the fan of $\mathbf z$-vectors, $\mathbf Z(\mathcal P (\mathbf i_{\max}))$, is unimodular equivalent to the fan constructed in \cite{K14}.
\end{prop}

%\medskip

 
From this Proposition, we get that the cone generated by $\mathbf
z$-vectors is unimodular equivalent to the Gelfand--Tsetlin cone.  Note
that the cone of $\mathbf g$-vectors studied in \cite{Magee} coincides with the cone of $\mathbf z$-vectors for the ample initials seed $  G(\mathbf i_{\max})$.


%\bigskip

In view of these results on sagbi bases, we can say that cluster algebras, possessing finitely generated cone of $\mathbf z$-vectors, have flat toric degenerations.




\acknowledgements{ I thank Arkady Berenstein,  Vladimir Danilov, Xin Fang, Sergey Fomin, Volker Genz, Peter Littelmann, and Bea Schumann for incentive discussions,  referees for useful remarks, and 
Mathematisches Institute Universit\"at zu K\"oln for hospitality. }

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\end{document}

The Lusztig basis is a basis in $\mathbb C[GL_n/U]$ labeled by the integer point of the cone $\mathbf D(n)$.
An element of Lusztig basis labeled by an integer point of $\mathbf D(n)$ has the form
$\Delta_T+$ lexicographically smaller terms, where $T$ is a semi-standard Young tableau corresponding to the point
in $\mathbf D(n)$.

The Lusztig basis and the tableau basis are different.
For example, for $GL_3$,  the Lusztig basis (discovered in  1985 by I.Gelfand and A.Zelevinsky
(\cite{GZ})) is the collection of monomials in $\Delta_{1}$, $\Delta_{2}$,
$\Delta_{3}$, $\Delta_{12}$, $\Delta_{23}$, $\Delta_{13}$, and $\Delta_{123}$ which do not contain the product
$\Delta_2\Delta_{13}$, while                                                                                              the tableaux basis is the collection of monomials in $\Delta_{1}$, $\Delta_{2}$, $\Delta_{3}$, $\Delta_{12}$,
$\Delta_{23}$, $\Delta_{13}$, and $\Delta_{123}$ which do not contain the monomial $\Delta_1\Delta_{23}$.


An element of the Lusztig basis is {\em  real} if
its square belongs to the Lusztig basis.  We call the semi-standard Young tableau {\em real}
if the corresponding element of the Lusztig basis is real.


Recently, in Kashiwara et al it was proven that  the cluster monomials correspond to the {\em real} elements of the dual canonical basis.
%\medskip
