# Geometry of weighted Homogeneous spaces

Mohammad Reza Rahmati<EMAIL_ADDRESS>
###### Abstract.

We define and describe the basic properties of weighted homogeneous spaces
(WHS). A weighted homogeneous variety can be defined via an element in the
highest Weyl chamber giving a weight system. Besides, it is presented a
criterion when two WHS with possibly different weight systems are isomorphic.
We also explain invariant Kähler differentials on WHS and study the coordinate
ring of a WPS by cluster algebras associated with weighted quivers.

###### 2010 Mathematics Subject Classification:

Primary

##  1\. Introduction

A homogeneous space is the quotient of a reductive Lie group $G$ by a
parabolic subgroup $P$ up to conjugation denoted by, $X=G/P$. It can also be
described as a projective subvariety of the projectivization of some
irreducible representation of $G$. Abe et. al [1, 2] has studied a weighted
analog of homogeneous spaces, [see also [19, 64]]. Their way of definition
considers embedding a weighted homogeneous space (WHS) into a weighted
projective space. However, configuration implies that one needs to add a high
enough number to all the original weights of the torus action, to make all of
them positive. This is because we naturally expect that a WHS should be
compact. One may compare a weighted Homogeneous space to the weighted
projective space, already well known. A weighted projective space is an
orbifold, i.e., it can be covered by orbifold charts, where each is isomorphic
to the quotient of an affine space by a finite group.

In this text, we present a notion of weighted homogeneous spaces
$\frac{G}{P}\left[\psi_{H}\right]$, where $G$ is a complex semisimple Lie
group, $P$ is a parabolic subgroup of $G$ and $H$ is an element of highest
Weyl chamber, $\psi_{H}$ is the weight system [[7]-see also Section 3 below]
defined by $H$. A weight system gives rise to an action of the maximal torus
on the Bruhat cells $\cong\mathbb{C}^{|R(P)|}$. Let $T$ be a maximal torus of
$G$ and $\mathfrak{h}$ be the corresponding Cartan subalgebra of
$\mathfrak{g}=Lie(G)$ and $R=R_{-}\cup R_{+}$ the associated root system,
$\Pi$ the simple roots for $\mathfrak{g}$. Suppose

(1.1) $C_{+}=\\{H\in\mathfrak{h}_{\mathbb{R}}|\langle\alpha,H\rangle\geq
0,\forall\alpha\in\Pi\\}$

is the highest Weyl chamber. Assume $H\in C_{+}$ is so that
$\langle-,H\rangle$ is a dominant integral weight. The weight system is given
by the integral dominant weight function $\psi_{H}:\Pi\to\mathbb{Z}_{\geq 0},\
H\in C_{+}$. The map $\psi_{H}$ together with the twisted action of the
maximal torus $T$ of $G$ on the different Bruhat cells, which are affine
spaces [see section 3], defines the weighted homogeneous space $\frac{G}{B}$
(denoted $\frac{G}{B}[\psi_{H}]$). Specifically, we define the weighted
homogeneous space as

(1.2)
$\frac{G}{P}\big{[}\psi_{H}\big{]}=\left[Cone\left(\dfrac{G}{P}\right)\diagdown
0\right]\big{/}_{H}T.$

The action of the maximal torus $T$ is just defined on different open Bruhat
cells. Here there is needed an explanation for the meaning of the action of
the torus $T$. Here the torus $T$ has never the total dimension of the cone
where it is acting. The precise meaning of making quotient in (1.2) is through
the total action of $T$ on the Bruhat cells [19, 64] [see the corresponding
section in the text]. Weighted homogeneous spaces (WHS) are in many senses
related to ordinary homogeneous spaces $G/P$. We have a locally finite
quotient map
$\pi_{H}:\frac{G}{P}\longrightarrow\frac{G}{P}\left[\psi_{H}\right]$. The map
$\pi_{H}$ is continuous in quotient topology. However, it has critical values
at the singularities of $\frac{G}{P}\left[\psi_{H}\right]$. The singularities
are finite quotient singularities and provide the structure of an orbifold on
WHS. Compared to standard homogeneous manifolds, a WHS behaves very much like
the weighted projective spaces (WPS) compared to projective spaces. One
illustrates a commutative diagram.

(1.3) ${G/P}$${\left[\mathcal{C}one\left(\dfrac{G}{P}\right)\smallsetminus
0\right]}$${\dfrac{G}{P}\left[\psi_{H}\right]}$$\scriptstyle{/\times_{\alpha}\mathbb{Z}_{\langle\alpha,H\rangle}}$$\scriptstyle{/\mathbb{C}^{*}}$$\scriptstyle{/[\psi_{H}]}$

which can be compared with its analog for weighted projective spaces (WPS)
[see remark 3.3]. The quotient map $\pi_{H}$ can be explained on the Bruhat
cell. This has also been done in the elder setup of [19, 64]. The Bruhat
decomposition and the Bruhat cells can also be naturally explained in the
weighted case. Then the Bruhat decomposition of $G/P$ descends to the one for
$\frac{G}{P}\left[\psi_{H}\right]$ by the map $\pi_{H}$. In other words, we
briefly have

(1.4) $\frac{G}{P}\big{[}\psi_{H}\big{]}=\coprod_{w\in
W/W_{L}}X_{w}[\psi_{H}],\qquad\pi_{H}:X_{w}\to X_{w}[\psi_{H}]$

where $X_{w}$’s are the Bruhat cells [see Sections 2 and 3 for notations].

There are certain circumstances where two WPS are equal. In fact, one can
associate a fan of rank $n$ to a WPS [a WPS is toric]. Then two WPS with two
weight systems are isomorphic if and only if the two fans can be transformed
to one another by a matrix in $GL_{n}\mathbb{Z}$ [69]. We generalize this
argument to weighted homogeneous manifolds. In fact to a WHS
$\frac{G}{P}\left[\psi_{H}\right]$ we may associate a combinatorial data
$\Sigma[H]$. The association of a $\Sigma[H]$ to a WHS is a formal way to
record the weights in a convex polytope or coordinate axes of a lattice. This
is a simple way we record this data in a way to be able to express the theorem
when two WHS with different lattices might be isomorphic. In order to study
the isomorphism question above we may consider relative maps
$f:\frac{G}{P}[\psi_{H_{1}}]\to\frac{G}{P}[\psi_{H_{2}}]$. The map exists only
if there is a homomorphism $\beta_{f}:\Sigma[H_{1}]\longmapsto\Sigma[H_{2}]$.

We classify the $K$-invariant Kähler forms on
$\frac{G}{P}\left[\psi_{H}\right]$. The result is a generalization of that in
[7]. The Kähler differentials on WHS $\frac{G}{P}\left[\psi_{H}\right]$ can be
defined by pluri-subharmonic potentials

(1.5) $\displaystyle\phi:$ $\displaystyle G\to\mathbb{R}\cup-\infty$
$\displaystyle\phi(g)$
$\displaystyle=\sum_{J(S)}c_{\alpha}\log\|\rho_{\alpha}(g).v_{\alpha}\|.$

The representation $\rho_{\alpha}$ is the irreducible representation of $G$ of
highest weight $w_{\alpha}=$ the fundamental weights of $G$, and $v_{\alpha}$
is the highest weight vector. The Kähler forms are pulled back to the Kähler
forms on $G/P$, via the locally quotient map $\pi_{H}$. We shall explain in
Section 5 that how the equation (1.5) may be compared to the non-weighted case
[see Theorem 4.2]. As partial contributions, we generalize some of the results
in [7].

We discuss the coordinate rings of homogeneous varieties as a cluster algebra
and its mutations via the cluster complex. It is known that the coordinate
ring of $G/P$ is a cluster algebra, [40], [30], [35]. These cluster algebras
are of finite type. That is, the adjacency matrix is defined via a Cartan
matrix. An example is the Grassmann cluster algebras, i.e., the coordinate
ring of the Grassmannians $Gr(k,n)$, where cluster complex can be obtained as
a dual to a $(k,n)$-diagram on the disc. There exists embedding of a WHS into
a product of weighted projective spaces on highest weight representations,
where we obtain a multi-graded algebra structure on the coordinate ring of
$\mathbb{C}\left[\frac{G}{P}\left[\psi_{H}\right]\right]$. The coordinate ring
of a Homogeneous space is a finite extension of the weighted one. The degree
of the extension is equal to the product of the weights at all the roots
involved in the quotient $G/P$.

Organization and References: Sections 2 and 3 are classical. The references
for section 1 are [51, 74, 76, 75, 38, 15, 13, 25, 44, 47, 9, 43, 56] as well
as various other classical texts in Lie algebras and their representations.
Section 3 explains the basics of cluster algebras. The standard references for
section 3 are [8, 23, 30, 31, 32, 33, 48, 49, 50, 53, 54, 78, 10, 34, 35, 36].
In section 4 we present some new definitions and contributions on weighted
homogeneous spaces. The relevant references are [29, 45, 11, 20, 21, 27, 28,
42, 46, 12, 19, 62, 63, 65, 66, 67]. In section 5, we discuss invariant Kahler
forms on WHS. The relevant references are [16, 18, 55, 5, 22, 7, 45, 77].
Finally, in section 6, we discuss weighted quivers and the coordinate ring of
weighted homogeneous spaces.

##  2\. Homogeneous Spaces

The material of this section is well known, the references are [38, 13, 76,
74, 9]. Assume $\mathfrak{g}=Lie(G)$ is a semisimple Lie algebra,
$\mathfrak{h}=\langle H_{\alpha_{1}},...,H_{\alpha_{n}}\rangle$ be a Cartan
and $R=R_{+}\bigcup R_{-}$ the associated root system with
$\Pi=\\{\alpha_{1},...,\alpha_{n}\\}$ the simple roots. Let

(2.1) $\mathfrak{g}=\mathfrak{h}\oplus\ \bigoplus_{\alpha\in
R}\mathfrak{g}_{\alpha}$

be the root decomposition. Let

(2.2) $\mathfrak{b}=\mathfrak{h}\oplus\bigoplus_{\alpha\in
R_{+}}\mathfrak{g}_{\alpha}$

the maximal solvable (Borel) subalgebra with corresponding Lie group $B$. Then
$B$ is a closed subgroup, and $G/B$ is a compact topological space. In fact,
$G/B$ can be embedded in a Grassmannian, where $B$ is the stabilizer of
$\wedge_{\alpha\in R_{+}}\alpha$ for the action

(2.3)
$G\times\mathbb{P}(\Lambda^{|R_{+}|}\mathfrak{g})\to\mathbb{P}(\Lambda^{|R_{+}|}\mathfrak{g}).$

A closed connected subgroup of $G$ is parabolic if $G/P$ can be realized as
the orbit of the action of $G$ on $\mathbb{P}(V)$ for some representation $V$
of $G$. Thus $G/P$ is a projective variety. Any Borel is Parabolic, and a
parabolic subgroup is a subgroup containing some Borel. In other words, Borels
are minimal parabolic subgroups. For the corresponding Lie algebras we have
$\mathfrak{b}\subset\mathfrak{p}\subset\mathfrak{g}$. Therefore

(2.4) $\mathfrak{p}=\mathfrak{h}\oplus\bigoplus_{\alpha\in
T}\mathfrak{g}_{\alpha},\qquad T\supset R_{+}$

and for $\mathfrak{p}$ to be a subalgebra, $T$ must be closed under addition.
It follows that the parabolic subgroups of the simple group $G$ are in 1-1
correspondence with subsets of nodes of the Dynkin diagram, i.e., simple
roots.

Assume $V_{\lambda}$ is the irreducible representation of $G$ with the highest
weight $\lambda$. Consider the action of $G$ on $\mathbb{P}(V_{\lambda})$, and
let $p\in\mathbb{P}(V_{\lambda})$ be the point corresponded as the subspace of
weight $\lambda$. One can show that $G.p$ is the unique closed orbit of the
action of $G$ and $P_{\lambda}$ is the parabolic corresponded to the subset of
simple roots perpendicular to $\lambda$, i.e., the faces of Weyl chambers.
Therefore there is a 1-1 correspondence $\\{\text{parabolic subgroups of}\
G\\}\ \stackrel{{\scriptstyle 1-1}}{{\longleftrightarrow}}\ \\{\text{faces of
the Weyl chambers}\\}$ where the face corresponding to $P_{\lambda}$ is that
containing $\lambda$. The weights in interior of Weyl chamber correspond to
$B$. The Weyl group can be written as $W=N_{G}(T)/T$ where $N(T)$ is the
normalizer of $T$ in $G$. For $w\in W$ one can choose a representative
$\dot{w}$. One can show that the double coset $B\dot{w}B$ is independent of
the lift.

######  Theorem 2.1.

(Bruhat decomposition) We have a disjoint union:

(2.5) $G=\coprod_{w\in W}B\dot{w}B.$

Let $U$ (resp. $U^{-}$) be the unipotent radical of $B$ (resp. opposite Borel
$B^{-}$). Define

(2.6) $U_{w}=U\cap\dot{w}U^{-}\dot{w}^{-1},\qquad
U_{w}^{\prime}=U\cap\dot{w}U\dot{w}^{-1}$

Again independent of the lift. The Lie algebra of $U_{w}$ is the sum of all
root spaces $\mathfrak{g}_{\alpha}$ such that $\alpha$ is positive but
$w^{-1}\alpha$ is negative. For $U_{w}^{\prime}$ it is the sum of those where
$\alpha$ and $w^{-1}\alpha$ are both positive. Thus $U_{w}U_{w}^{\prime}T=B$.
Replacing this identity, one gets

(2.7) $B\dot{w}B=U\dot{w}B.$

It is not hard to check that the presentation of an element in $B\dot{w}B$ in
the form of the above product $u.\dot{w}.b$ is unique.

######  Proposition 2.2.

The dimension of $U_{w}$ is the cardinality $R_{+}\cap w(R_{-})$. In fact,
$U_{w}=\mathbb{C}^{l(w)}$ is affine.

######  Theorem 2.3.

(Bruhat decomposition of $G/B$) We have a disjoint union of cosets

(2.8) $G/B=\coprod_{w}B\dot{w}B/B,\qquad X_{w}=B\dot{w}B/B$

where $X_{w}\cong\mathbb{C}^{l(w)}$ are Bruhat cells, and $l(w)=\sharp\\{\
\alpha\in R_{+}\ |\ w\alpha<0\ \\}$.

Bruhat decomposition provides a 1-1 correspondence between

(2.9) $B\setminus G/B\ \ \longleftrightarrow\ \ W=N_{G}(H)/H$

which is the origin to define different kinds of orders on the Bruhat cells.
One can also express the Bruhat decomposition for $P\setminus G/B$. One has

(2.10) $P=\coprod_{w\in W_{L}}B\dot{w}B$

where $L$ is the Levi component in Levi decomposition $P=LN$ ($N$ is
unipotent), and $W_{L}=N_{L}(T)/T$.

######  Theorem 2.4.

(Bruhat decomposition of $G/P$) We have the decomposition

(2.11) $\ P\dot{w}B=\coprod_{x\in W_{L}}BxwB.$

It follows that;

(2.12) $G/P=\coprod_{w\in W/W_{L}}P\dot{w}B.$

The correspondence in (2.12) is replaced with

(2.13) $P\setminus G/B\ \longleftrightarrow\ W(P)=W/W_{L}=N_{L}(T)/T.$

There is a cross-action of the Weyl group on $P\setminus G/B$ that is
generated by the following action of simple reflections on the Bruhat cells

(2.14) $s_{\alpha}\times PwB\longmapsto
Pws_{\alpha}^{-1}B,\qquad\alpha\in\Pi.$

We shall consider the Bruhat cells as the manifold charts for $G/P$ and also
for $\frac{G}{P}\left[\psi_{H}\right]$. First, it would be essential to
understand how the coordinate changes are formulated by the action of the Weyl
group on the Bruhat cells. The cross action on the Bruhat cells is the basic
building block of these coordinate changes.

######  Example 2.5.

  * (1)

Consider $G=SL_{m}\mathbb{C}$, with standard Borel $B$ of upper triangular
matrices. We have $W=S_{m}$ the symmetric group on $m$ elements, $U_{w}$ are
matrices with $1$ on the diagonal and $0$ in the $(i,j)$ coordinates where
$i>j,\ w^{-1}(i)<w^{-1}(j)$ and its dimension equals

(2.15) $l(w)=\sharp\\{\ (i,j)\ |\ i>j,\ w(i)<w(j)\ \\}.$

The Bruhat cell $X_{w}$ of $SL_{m}\mathbb{C}/B$ consists of the flags such
that the dimension of the intersections with standard flag are governed by
$w\in S_{m}$.

  * (2)

The Bruhat decomposition of $Sp_{2n}\mathbb{C}$ is obtained from that of
$SL_{2n}\mathbb{C}$ provided a change of the order on basis
$e_{1},...,e_{n},e_{2n},...,e_{n+1}$. Then $B$ is matrices with upper left
block upper triangular and lower right block is lower triangular, with $0$
other places. Its Weyl group is

(2.16) $W=\\{\ w\in S_{2n}\ |\ w(n+i)=w(i)+n\ \\}.$

The Bruhat decomposition of $SO_{n}\mathbb{C}$ has a similar explanation.

Let $\theta:G\to G$ be a Cartan involution of $G$, and $K=G^{\theta}$ and
$B=TU$ a $\theta$-stable Borel of $G$. Suppose

(2.17) $\tilde{V}=\left\\{x\in G\ |\ \frac{\theta(x)}{x}\in N(T)\ \right\\}.$

There is an action of $K\times T$ on $\tilde{V}$ by

(2.18) $(K\times T)\times\tilde{V}\to\tilde{V},\qquad(k,t)\times x\mapsto
kxt^{-1}.$

Let $V$ be the orbit space of the action. Denote a representative of $v$ as
$\dot{v}$.

######  Theorem 2.6.

(T. A. Springer [74]) There is a bijection

(2.19) $V\stackrel{{\scriptstyle 1-1}}{{\longleftrightarrow}}K\setminus G/B.$

The cross action of $W$ on $K\setminus G/B$ is given by

(2.20) $s_{\alpha}\times K\dot{v}B\mapsto K\dot{v}s_{\alpha}^{-1}B$

A similar formula can be stated for the Parabolic $P$, [see [79] for details].

######  Theorem 2.7.

[61] There is a bijection

(2.21) $K\setminus
G/P\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\left(L^{\theta}\setminus
L/P\cap L\right)\times_{W_{L}}W$

where the fiber product is with respect to the cross action of $W_{L}$ on
$L^{\theta}\setminus L/P\cap L$.

##  3\. Weighted Homogeneous Spaces

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{h}$ a
Cartan, and $R=R_{+}\cup R_{-}$ the associated root system and $\Pi$ the
simple roots. Set

(3.1) $\mathfrak{h}_{\mathbb{R}}=\\{\ H\in\mathfrak{h}\ |\
\alpha(H)\in\mathbb{R},\ \forall\alpha\in R\ \\}$

and $C_{+}$ the highest Weyl chamber i.e.

(3.2) $C_{+}=\\{H\in\mathfrak{h}|\alpha(H)\geq 0,\forall\alpha\in R_{+}\\}.$

For any $H\in\mathfrak{h}$ define
$\psi_{H}:\Pi\to\mathbb{R},\qquad\alpha\mapsto\alpha(H)$. The weight system is
given by the integral dominant weight function

(3.3) $\psi_{H}:\Pi\to\mathbb{Z}_{\geq 0},\qquad H\in C_{+}.$

Then $\psi_{H}$ assigns a positive weight to each simple root i.e. the nodes
of the Dynkin diagram. Let $\ \\{X_{\alpha},H_{\alpha},X_{-\alpha}\\}$ be an
$\mathfrak{sl}_{2}$-triple in $\mathfrak{g}$. The root groups are defined via
the maps

(3.4) $\phi_{\alpha}:SL_{2}\mathbb{C}\to L_{\alpha},\qquad
Lie(L_{\alpha})=\langle X_{\alpha},H_{\alpha},X_{-\alpha}\rangle.$

On $G$ we shall consider

$\displaystyle z_{\alpha}(z)=\phi_{\alpha}\begin{pmatrix}1&z\\\
0&1\end{pmatrix},\ z_{-\alpha}(z)=\phi_{\alpha}\begin{pmatrix}1&0\\\
z&1\end{pmatrix},$ (3.5)
$\displaystyle\check{\alpha}(z)=\phi_{\alpha}\begin{pmatrix}z&0\\\
0&z^{-1}\end{pmatrix}.$

Let us describe how this defines the weighted $\frac{G}{B}$ (denoted
$\frac{G}{B}[\psi_{H}]$). Set $U$ to be the unipotent radical of the Borel
$B$. There is a $T$-equivariant isomorphism $U\cong\mathbb{C}^{|R_{+}|}$,
where $T$ acts on $\mathbb{C}^{|R_{+}|}$ by

(3.6) $t.\ x_{\alpha}=\alpha(t)\ x_{\alpha},\qquad\alpha\in R_{+}.$

Thus a $1$-parameter subgroups gives rise to an action on $U$ defined by,

(3.7) $t._{H}\ x_{\alpha}=t^{\ \langle\alpha,H\rangle}\
x_{\alpha},\qquad\alpha\in R.$

The Weyl group $W=N_{G}(T)/T$ acts on $T$ by conjugation and the action of $W$
on the weight lattice $\mathcal{P}$ is given by

(3.8) $t^{w(\lambda)}=(w^{-1}tw)^{\lambda},\qquad w\in W,\
\lambda\in\mathcal{P},\ t\in T$

where $\lambda$ is regarded as a character of the torus $T$ by

(3.9) $\lambda(\exp(t))=\exp(t)^{\lambda}=e^{\lambda(t)}.$

The Weyl group also acts on the cells $U$, so that it permutes the Bruhat
cells (cross action) where the $T$-action gets twisted as

(3.10) $t._{H}\ x_{\alpha}^{w}=\
\big{(}w^{-1}t^{\langle\alpha,H\rangle}w\big{)}\
x_{\alpha}^{w},\qquad\alpha\in R_{+},\ x_{\alpha}^{w}\in w.U$

where $x_{\alpha}^{w}$ states the coordinate $x_{\alpha}$ in $wU$. Remember
that $H$ was chosen so that $\langle-,H\rangle$ was acting as an integral
dominant weight, thus we get all the above weights positive. Let us define

(3.11)
$\frac{G}{B}\big{[}\psi_{H}\big{]}=\left[Cone\left(\dfrac{G}{B}\right)\diagdown
0\right]\big{/}_{H}T,$

more generally the above analogy recommends the following.

######  Definition 3.1.

Assume $\psi_{H}\in Hom(\Pi,\mathbb{Z}_{+})$ be a positive dominant integral
weight corresponding to $H\in C_{+}$. Define

(3.12)
$\frac{G}{P}\big{[}\psi_{H}\big{]}=\left[Cone\left(\dfrac{G}{P}\right)\diagdown
0\right]\big{/}_{H}T$

where the action of $T$ is given by (3.7) and (3.10).

Assume the parabolic subgroup $P$ corresponds to the set of simple roots
$\Pi\setminus I=J$. Let $R(P)$ be the roots with components in $J$ and denote
by$U(P)$ be the Bruhat cell corresponding to the identity. Then
$U(P)\cong\mathbb{C}^{|R(P)|}$. The $T$-action on the coordinates $x_{\alpha}$
are given by the same identity as (3.7). The actions of the Weyl group behaves
as change of coordinates.

######  Theorem 3.2.

We have

(3.13) $\frac{G}{P}\left[\psi_{H}\right]=\left[\times_{\alpha\in
R(P)_{+}}\mathbb{Z}_{\langle\alpha,H\rangle}\right]\setminus G/P.$

###### Proof.

We first explain how the following diagram is well-defined

(3.14) ${G/P}$${\left[Cone\left(\dfrac{G}{P}\right)\diagdown
0\right]}$${\dfrac{G}{P}\big{[}\psi_{H}\big{]}.}$$\scriptstyle{/\times_{\alpha\in
R(P)_{+}}\mathbb{Z}_{\langle\alpha,H\rangle}}$$\scriptstyle{/\mathbb{C}^{*}}$$\scriptstyle{/[\psi_{H}]}$

In order to check this, we note that over each Bruhat cell, the right edge of
the triangle is a simple quotient map by a finite group induced by the maps
$x_{\alpha}\mapsto x_{\alpha}^{\langle\alpha,H\rangle}$. Because the charts
are compatible with the left edge cone construction, we obtain the triangle
commutativity. Next, we stress that the claim in (3.13) is to express the WHS
as the global quotient of the projective variety $G/P$ by the finite group
$\left[\times_{\alpha\in R(P)_{+}}\mathbb{Z}_{\langle\alpha,H\rangle}\right]$.
In order to show the isomorphism of the two projective varieties on both sides
of (3.13), one can prove that their corresponding homogeneous coordinate rings
are isomorphic [the same method has been used in [27] for weighted projective
spaces]. Now, we note that the argument used in [27] (please refer to Sec.
1.2.2 of that reference) extends over here. Here, the action of the group
scheme $\mu=[\times_{\alpha\in R(P)_{+}}\mu_{\langle\alpha,H\rangle}]$ on
$G/P$ is induced by the action of $\mu$ on
$\mathbb{C}\left[\frac{G}{P}\right]$ and is defined by the $x_{\alpha}\mapsto
x_{\alpha}\times\overline{Y_{\alpha}}$ where
$\overline{Y_{\alpha}}=Y_{\alpha}\ \mod(Y^{\langle\alpha,H\rangle}-1)$. This
shows the induced map

(3.15)
$\phi:\mathbb{C}\left[\frac{G}{P}\big{[}\psi_{H}\big{]}\right]\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathbb{C}\left[\frac{G}{P}\right]^{\times_{\alpha}\mu_{\alpha}},\
(\mu_{\alpha}=\langle\alpha,H\rangle)$

is an isomorphism, where the superfix means elements fixed by the action.
Therefore

(3.16) $\displaystyle\frac{G}{P}\big{[}\psi_{H}\big{]}$
$\displaystyle=proj\left(\mathbb{C}\left[\frac{G}{P}\big{[}\psi_{H}\big{]}\right]\right)$
$\displaystyle=\left[\times_{\alpha\in
R(P)_{+}}\mathbb{Z}_{\langle\alpha,H\rangle}\right]\setminus G/P.$

∎

######  Corollary 3.2.1.

There is a well-defined surjective continuous map

(3.17) $\pi_{H}:\frac{G}{P}\to\frac{G}{P}\big{[}\psi_{H}\big{]}$

that is a continuous locally finite quotient map with a finite number of
cyclic singularities.

###### Proof.

The map $\pi_{H}$ is nothing but the right edge of the triangle in (1.3) or
(3.14) and was explained along with the definition of WHS locally. Its global
definition was given in the proof of Theorem 3.2. ∎

######  Remark 3.3.

The above map has critical values and hence singularities. The singularities
are quotient singularities. They are also called orbifold singularities. One
can compare the commutative diagram (3.14) with its well-known analogue

(3.18) ${\mathbb{P}^{n}}$${\mathbb{C}^{n+1}\diagdown
0}$${\mathbb{P}[\lambda_{0},...,\lambda_{n}].}$$\scriptstyle{/\mathbb{Z}_{\lambda_{0}}\times...\times\mathbb{Z}_{\lambda_{n}}}$$\scriptstyle{/\mathbb{C}^{*}}$$\scriptstyle{/\mathbb{C}^{*}}$

We have to ensure that the action is well understood on all coordinate charts
in the case of WHS.

In weighted projective spaces (WPS) theory, there is a well-known method to
determine when two WPS are isomorphic. We can associate to
$\mathbb{P}[\lambda_{0},...,\lambda_{n}]$ a fan $\Sigma$ in
$\mathbb{R}^{n}=:N$ on the standard basis by defining

(3.19) $v_{i}=\frac{1}{\lambda_{i}}e_{i},\qquad e_{i}\ \text{ standard basis
in}\ \mathbb{R}^{n}.$

One may add another vector

(3.20) $v_{0}=-\frac{1}{\lambda_{0}}(e_{1}+e_{2}+...+e_{n})$

to build up a fan (uniquely for the fan condition). We call it the fan
associated with the WPS, [see [69] for more details]. The aforementioned fan
makes the WPS a toric variety. Many of the geometric properties of the WPS can
be characterised from the combinatorics of their toric fans. It is possible
that two weighted projective space with different weight systems be
algebraically isomorphic. First one may reduce the weights as follows. If we
set,

(3.21) $\displaystyle
d_{j}=lcm(\lambda_{0},\dots,\widehat{\lambda_{j}},\dots,\lambda_{n})$
$\displaystyle a_{j}=gcd(d_{0},\dots,\widehat{d_{j}},\dots,d_{n})$
$\displaystyle(\lambda_{0}^{\prime},\dots,\lambda_{n}^{\prime})=(\lambda_{0}/a_{0},\dots,\lambda_{n}/a_{n}),$

then one has
$\mathbb{P}[\lambda_{0},\dots,\lambda_{n}]\cong\mathbb{P}[\lambda_{0}^{\prime},\dots,\lambda_{n}^{\prime}]$
[[69], proposition 1.26], and the latter WPS is called the reduction of the
first (also called reduced). A major question is to characterize the WPS with
different weight systems that are isomorphic. Two reduced WPS are isomorphic
if and only if their fans are equivalent. The latter is stated in the
following theorem.

######  Theorem 3.4 ([69] proposition 2.7).

Two reduced WPS $\mathbb{P}[\lambda_{0},\dots,\lambda_{n}]$ and
$\mathbb{P}[\lambda_{0}^{\prime},\dots,\lambda_{n}^{\prime}]$ with the
associated fans $[v_{1},\dots,v_{n},v_{0}]$ and
$[v_{1}^{\prime},\dots,v_{n}^{\prime},v_{0}^{\prime}]$ are isomorphic iff
there exists a matrix in $GL_{n}\mathbb{Z}$ which exchanges their associated
fans of the two in $\mathbb{R}^{n}$, i.e if there exists $\phi\in
GL_{n}\mathbb{Z}$ such that $\phi v_{i}=v_{i}^{\prime}$ [as explained above].
It will be the same if we just consider the transfer over the vectors
$v_{1},...,v_{n}$.

AQUI ME QUEDE, QUEDAMOS EN QUE MOHAMMAD VA A REVISAR NUEVAMENTE TODO EL PAPER.

One has an equivalent formulation via the polytopes associated to these fans
[cf. [69]]. If one put the vectors $v_{i}$ as the columns of a matrix
$[v_{0},...,v_{n}]$, the aforementioned condition means that both of the
weight matrices obtained this way have the same Hermit canonical forms, [see
[69] for details]. Theorem (3.4) characterizes; when two weighted projective
spaces (with possibly different weight systems) are isomorphic. We shall
determine when two weighted homogeneous spaces of the same type
$\frac{G}{P}[\psi_{H_{1}}]$ and $\frac{G}{P}[\psi_{H_{2}}]$ are algebraically
isomorphic.

######  Definition 3.5.

Assume $H\in C_{+}$ and $\psi_{H}$ the associated weight on $\Pi$. We
associate a fan (this is not a toric fan) on the coroot lattice by defining

(3.22) $v_{i}:=\frac{1}{\langle\alpha,H\rangle}{\alpha}^{\vee},\qquad
i=1,...,n$

and add a vector $v_{0}$ (in a unique way) to the above vectors to make it a
fan in $N=\mathbb{R}^{n}$. Denote this fan by $\Sigma_{H}(\Pi)$. In fact this
fan provides all the data of a bigger fan on $\mathbb{R}^{|R(P)_{+}|}$,
consisting of all the weights on the axis $x_{\alpha}$, by the vectors

(3.23) $v_{\alpha}=\frac{1}{\langle\alpha,H_{j}\rangle}e_{\alpha},$

where $e_{\alpha}$ are the standard basis of $\mathbb{R}^{|R(P)_{+}|}$ and add
the additional vector

(3.24) $v_{0}=\frac{1}{\lambda_{0}}\sum_{\alpha\in R_{+}}e_{\alpha}$

to the $v_{\alpha}$ (unique) to satisfy the fan condition. We denote this fan
by $\Sigma_{H}(R(P))$.

The above fan is a combinatorial framework we record the weights of a WHS, and
by no means may characterize the structure of WHS.

######  Theorem 3.6.

The two weighted homogeneous spaces $\frac{G}{P}[\psi_{H_{1}}]$ and
$\frac{G}{P}[\psi_{H_{2}}]$ are isomorphic if and only if there exists a
matrix in $GL_{n}\mathbb{Z}$ exchanging their fans.

###### Proof.

Consider the extended fan to a fan in $\mathbb{R}^{|R(P)_{+}|}$ on the
standard basis labeled by the positive roots $\alpha\in R(P)$ as

(3.25) $v_{\alpha}^{j}=\frac{1}{\langle\alpha,H_{j}\rangle}e_{\alpha},\qquad
j=1,2$

where $e_{\alpha}$ are the standard basis of $\mathbb{R}^{|R(P)_{+}|}$. One
can also add the additional vector

(3.26) $v_{0}^{j}=\frac{1}{\lambda_{0}}\sum_{\alpha\in R(P)_{+}}e_{\alpha}$

to the $v_{\alpha}$ (in a unique way) to satisfy the fan condition. From our
definition of weighted homogeneous space it follows that; if
$\Sigma_{H_{1}}(R(P))\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\Sigma_{H_{2}}(R(P))$
the two weighted homogeneous spaces are isomorphic
$\frac{G}{P}[\psi_{H_{1}}]\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\frac{G}{P}[\psi_{H_{2}}]$.
For the converse implication assume we have the isomorphism of two WHS
associated to $\psi_{H_{1}}$ and $\psi_{H_{2}}$. Then the same way as
explained before Theorem 3.4 first we can make the weights on the simple roots
reduced. So we may assume the isomorphism already is given over the reduced
weight systems. The isomorphism induces an isomorphism of
$\mathcal{O}_{\frac{G}{P}[\psi_{H_{1}}]}\cong\mathcal{O}_{\frac{G}{P}[\psi_{H_{2}}]}$.
Any Cartier divisor representing these line bundles can be given as
$\sum_{j}a_{j}D_{j}$ where $(a_{1},\dots,a_{R(P)_{+}})$ is an integer point in
the hyperplane $V=\sum_{j}\lambda_{j}X_{j}-\delta=0$ [resp.
$V^{\prime}=\sum_{j}\lambda_{j}X_{j}-\delta^{\prime}=0$, where $\delta$ and
$\delta^{\prime}$ denotes the least common multiple of the weights]. because
these representatives are getting exchanged under the given isomorphism, it
follows that any integer point of the $V$ and $V^{\prime}$ can be transferred
to each other. This proves the existence of the desire isomorphism $\phi\in
GL_{R(P)_{+}}(\mathbb{Z})$ on lattices. Therefore

(3.27)
$\frac{G}{P}[\psi_{H_{1}}]\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\frac{G}{P}[\psi_{H_{2}}]\Leftrightarrow\Sigma_{H_{1}}(R(P))\stackrel{{\scriptstyle\phi}}{{\longrightarrow}}\Sigma_{H_{2}}(R(P))$

where $\phi\in GL_{|R(P)_{+}|}\mathbb{Z}$. Because of the formal enlargement
of the space of lattices and the linearity

(3.28)
$\Sigma_{H_{1}}(R(P))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma_{H_{2}}(R(P))\Leftrightarrow\Sigma_{H_{1}}(\Pi)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\Sigma_{H_{2}}(\Pi)$

exchanging $v_{\alpha}^{(i)},\ i=1,2,\ \alpha\in\Pi$. ∎

######  Proposition 3.7.

There is a map

(3.29) $\frac{G}{P}[\psi_{H_{1}}]\longrightarrow\frac{G}{P}[\psi_{H_{2}}]$

precisely when we have a homomorphism of fans

(3.30) $\displaystyle\Sigma_{H_{1}}(\Pi)$
$\displaystyle\longrightarrow\Sigma_{H_{2}}(\Pi)$ $\displaystyle
v_{\alpha}^{(1)}$ $\displaystyle\longmapsto v_{\alpha}^{(2)}$

###### Proof.

As in the proof of the previous theorem a map
$\Sigma_{H_{1}}(\Pi)\longrightarrow\Sigma_{H_{2}}(\Pi)$ induces also a map on
the larger fans on the weights on all roots (in fact coroots by our
construction). Writing the coordinates out of weights then each coordinate
$x_{\alpha}$ is replaced by a variable $z_{\alpha}^{\langle\alpha,H\rangle}$.
Then in order to have a well defined map

(3.31)
$\mathbb{C}\left[\frac{G}{P}[\psi_{H_{2}}]\right]\longrightarrow\mathbb{C}\left[\frac{G}{P}[\psi_{H_{1}}]\right]$

the components of the map must be functions of
$z_{\alpha}^{\langle\alpha,H\rangle}$. Therefore the weights in the right hand
side of this map are divisible by those in the left. This is precisely when we
have $\Sigma_{H_{1}}(\Pi)\longrightarrow\Sigma_{H_{2}}(\Pi)$. More precisely a
map in (3.31) is a polynomial map given locally as

(3.32) $\displaystyle f:\frac{G}{P}[\psi_{H_{1}}]$
$\displaystyle\longrightarrow\ \frac{G}{P}[\psi_{H_{2}}],$
$\displaystyle(x_{\alpha})$ $\displaystyle\longmapsto(P_{\alpha}(\dots
x_{\beta}^{m_{\alpha\beta}}\dots)),$

In a way that the exponents be matched with the weights. ∎

##  4\. Invariant Kahler forms on $\frac{G}{P}[\psi_{H}]$

Assume $K$ is a maximal compact subgroup of the complex reductive Lie group
$G$. For an irreducible representation $\varrho$ of $G$ of highest weight
$\lambda$ and highest weight vector $v_{\lambda}$ the function

(4.1) $\displaystyle\phi$ $\displaystyle:G\to\mathbb{R}\cup-\infty$
$\displaystyle\phi$ $\displaystyle(g)=\log|\varrho(g)v_{\lambda}|$

is $K$-invariant, [7]. The highest weight representations $\rho_{\alpha}$
associated to the fundamental weights are defined by

(4.2) $\varrho_{\alpha}(\ \check{\beta}(z))=z^{\delta_{\alpha\beta}}$

where we are using the notation in (31). The associated potential is defined
by $\phi_{\alpha}(\check{\beta}(z))=\delta_{\alpha\beta}\log|z|$. The Kähler
forms defined by the potentials $\phi_{\alpha}$ for a basis for
$H^{2}(G/P,\mathbb{R})$. At a point $[x_{0}]\in G/P$ one defines the Dynkin
line $\mathbb{P}_{\alpha}=L_{\alpha}.[x_{0}]\ \cong\mathbb{P}(\mathbb{C})$, as
homology classes where the differential 1-forms can be integrated. In [7] is
shown that $\int_{\mathbb{P}_{\alpha}}w_{\beta}=\delta_{\alpha\beta}$. In
order to compare this notion with the weighted one, we should consider the
above forms as elements in $H^{2}(G/P,\mathbb{Z})$. In fact the aforementioned
classes can be calculated as Chern classes of $K$-invariant line bundles over
$G/P$. A closed $K$-invariant $(1,1)$-form $\omega$ on $G/P$ is given by a
form satisfying the identity
$pr^{*}\omega=\sqrt{-1}\partial\bar{\partial}\phi$, where
$\phi:G\to\mathbb{R}\cup-\infty$ is given as above, [[7]].

######  Remark 4.1.

[26] In the classical case where $G/P$ is given as a flag variety of
descending subspaces $V_{j},\ 1\leq j\leq r$ of the vector space $V$, one can
calculate the aforementioned Kähler forms as the curvature of certain
determinant bundles. The subspaces $V_{j}$ define tautological bundles
$\mathcal{V}_{j}$ on $G/P$. Define

(4.3) $L_{j}=\text{det}(\mathcal{V}_{j-1}/\mathcal{V}_{j})$

The curvature of line bundles $L_{j}$ can be calculated [cf. [26]] as

(4.4) $\Theta(L_{j})_{0}=-\sum_{i<j}dz_{ij}\wedge
d\bar{z}_{ij}+\sum_{i>j}dz_{ij}\wedge d\bar{z}_{ij},$

where $(z_{ij})_{i<j}$ are the coordinates on $U$. Set

(4.5) $L^{a}=L_{1}^{a_{1}}\otimes...\otimes L_{r}^{a_{r}}$

The curvature of the line bundle $L^{a}$ can be calculated from

(4.6) $\Theta(L^{a})_{e}=\sum_{i<j}(a_{i}-a_{j})dz_{ij}\wedge d\bar{z}_{ij}$

Thus $L^{a}$ is Kähler if and only if $a_{i}>a_{j},\ i<j$.

######  Theorem 4.2.

If $\omega$ is a closed $K$-invariant $(1,1)$-form on $\frac{G}{P}[\psi_{H}]$,
then its pull back over $G$ via the projection map can be written as

(4.7) $pr^{*}\omega=\sqrt{-1}\partial\bar{\partial}\phi$

where $pr:G\to G/P\to\frac{G}{P}[\psi_{H}]$ is the composition map and

(4.8) $\displaystyle\phi$ $\displaystyle:G\to\mathbb{R}\cup-\infty$
$\displaystyle\phi$
$\displaystyle(g)=\sum_{J(P)}c_{\alpha}\log\|\rho_{\alpha}(g).v_{\alpha}\|$

The form $\omega$ is Kähler iff $c_{\alpha}>0$.

###### Proof.

Assume $K$ is maximal compact of $G$. Fix the two root decompositions

(4.9) $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in
R}\mathbb{C}E_{\alpha},\qquad\mathfrak{k}=\mathfrak{h}\oplus\bigoplus_{\beta\in
R_{c}}\mathbb{C}E_{\beta}$

and let $\Pi$ and $\Pi_{c}$ the corresponding set of simple roots,
respectively. Set $R^{\prime}=R\setminus R_{c},\
\Pi^{\prime}=\Pi\setminus\Pi_{c}$. Then the complex tangent bundle of $X$ is
given by

(4.10) $T_{0}X=\sum_{\alpha\in\Pi\setminus\Pi_{c}}\mathbb{C}E_{\alpha}$

Let $P=LN$ be the Levi decomposition of $P$. By Theorem 2.6 we have

(4.11) $K\setminus
G/P\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\left(L^{\theta}\setminus
L/P\cap L\right)\times_{W_{L}}W$

where the fiber product is with respect to the cross action of $W_{L}$ on
$L^{\theta}\setminus L/P\cap L$. Because the roots that are fixed by $\theta$
are the compact roots i.e the roots in $R^{c}$, the roots involving in the
right hand side are those in $R^{\prime}$. Thus according to [5],
$K$-invariant form has the following standard form

(4.12) $w=\frac{i}{2\pi}\sum_{\alpha\in R^{\prime}}c_{\alpha}w^{\alpha}\wedge
w^{-\alpha}$

where $w^{\alpha}$ are 1-forms dual to $E_{\alpha}$. In fact there is a 1-1
correspondence between elements of $\mathfrak{h}_{\mathbb{R}}$ and the
$K$-invariant forms.

(4.13) $\mathfrak{t}^{*}\ni\xi=\sum c_{r}\tilde{\alpha}_{r}\ \
\stackrel{{\scriptstyle 1-1}}{{\Leftrightarrow}}\ \
\frac{i}{2\pi}\sum_{\alpha\in R_{+}^{\prime}}\langle\alpha|\xi\rangle
w^{\alpha}\wedge w^{-\alpha}$

[see also [5] page 27]. The potentials in (66) when expanded has the form
(71). ∎

An invariant line bundle $L$ on $\frac{G}{P}\left[\psi_{H}\right]$ can also be
pulled back to an invariant line bundle on $G/P$ through the finite map
$\pi_{H}$. In fact line bundles on WHS are those line bundles that are
invariant under the action of the finite group $\times_{\alpha}\mu_{\alpha}$.
Therefore we can also parametrize the $K$-invariant line bundles over
$\frac{G}{P}\left[\psi_{H}\right]$ by the characters of $P$.

######  Corollary 4.2.1.

Assume $L_{\chi}$ is a line bundle on $\frac{G}{P}\left[\psi_{H}\right]$
corresponding to the character $\chi:P\to\mathbb{C}^{*}$. Then

(4.14)
$c_{1}(L_{\chi})=-\sum_{\alpha\in\Pi(J)}\langle\chi,\check{\alpha}\rangle\langle\alpha,H\rangle[w_{\alpha}]$

In particular, the first Chern class of the canonical line bundle
$K_{\psi_{H}}$ is equal to

(4.15) $c_{1}(K_{\psi_{H}})=-\sum_{\alpha\in
J}\langle\rho,\check{\alpha}\rangle\langle\alpha,H\rangle[w_{\alpha}]$

where $\rho$ is the sum of all positive roots in $J$.

###### Proof.

We shall consider the differential forms on the smooth locus of
$\frac{G}{P}\left[\psi_{H}\right]$. Under the projection
$\pi_{H}:\frac{G}{P}\longrightarrow\frac{G}{P}\left[\psi_{H}\right]$, the pull
back of an invariant Kähler differential is defined by the potentials (66). As
the form of $\phi$ shows the potential defines the invariant form on
$\frac{G}{P}\left[\psi_{H}\right]$ if the exponents of the variable of
logarithms in (66) is divisible by the corresponding weights. Naturally we can
also suppose that the pull back of the Dynkin lines through $\pi_{H}$ are the
Dynkin lines of $G/P$. The proof is a repetition of the method in [7], noting
that the weight of the action on the root coordinate $\check{\alpha}$ is
multiplied by $\langle\alpha,H\rangle$ i.e the weight in the 1-parameter
subgroup $\check{\alpha}(z)$. ∎

######  Corollary 4.2.2.

Assume $P\subset Q$. The $K$-invariant Kähler-Einstein metric of $G/P$
restricted to any fiber of the fibration
$\frac{G}{P}\left[\psi_{H}\right]\longrightarrow\frac{G}{P}\left[\psi_{H}\right]$
is Kähler-Einstein

###### Proof.

The proof is the same as that in [7] in the unweighted case, and just all the
formulas have to be replaced by the one in (66). ∎

######  Example 4.3.

We choose an example of [24]. Lets consider $X=Sp(4)/B$ where $B$ is standard
Borel. $B$ consist of $4\times 4$ matrices with upper left block upper
triangular and lower right block lower triangular. Then
$\mathfrak{n}^{-}=Lie(N^{-})$ consists of matrices of the form

$\displaystyle\mathfrak{n}^{-}$
$\displaystyle=\left\\{\begin{pmatrix}0&0&0&0\\\ x_{1}&0&0&0\\\
x_{2}&x_{3}&0&-x_{1}\\\ x_{3}&x_{4}&0&0\end{pmatrix}\right\\}\ \
\stackrel{{\scriptstyle\exp}}{{\longrightarrow}}$ $\displaystyle N^{-}$
$\displaystyle=\left\\{\begin{pmatrix}0&0&0&0\\\ x_{1}&0&0&0\\\
P_{1}(x)&P_{3}(x)&0&-x_{1}\\\ P_{2}(x)&x_{4}&0&0\end{pmatrix}\right\\}$

where $P_{1},P_{2},P_{3}$ can be calculated specifically by exponentiation of
the matrix in the left. We have two simple roots
$\\{\alpha_{1}=\epsilon_{1}-\epsilon_{2},\ 2\epsilon_{2}\\}$ and $4$ positive
roots
$\\{\alpha_{1},\alpha_{2},\alpha_{1}+\alpha_{2},2\alpha_{1}+\alpha_{2}\\}$.
The vectors $v_{\alpha}$ in the Theorem are given by $e_{1},\ e_{1}\wedge
e_{2}$ in standard basis of $\mathbb{C}^{4}$, with corresponding highest
weight representations to be $\mathbb{C}^{4},\ \bigwedge^{2}\mathbb{C}^{4}$.
Therefore a potential for the Kähler form can be written

(4.16)
$\phi=\frac{c_{1}}{2\pi}\log\|g.e_{1}\|+\frac{c_{2}}{2\pi}\log\|g.(e_{1}\wedge
e_{2})\|,\ c_{1},c_{2}>0$

expansion gives (cf. [24])

$\displaystyle\phi(z)$
$\displaystyle=\frac{c_{1}}{2\pi}\log(1+|z_{1}|^{2}+|P_{1}(z)|^{2}+|P_{2}(z)|^{2})$
$\displaystyle+\frac{c_{2}}{2\pi}\log{\left(\sum_{i<j}\det\begin{pmatrix}1&0\\\
z_{1}&1\\\ P_{1}(z)&P_{3}(z)\\\ P_{2}(z)&z_{4}\end{pmatrix}^{2}\right)}$

##  5\. Cluster algebras

This section is classical in the theory of cluster algebras. The major
references are [19, 30, 31, 32, 33, 35, 36]. Consider a triangulation $T$ of
an $(n+3)$-gon and mark the $n$ diagonal by 1,2,…,n and the $n+3$ sides by
$n+1,...,2n+3$. The combinatorics of $T$ can be described by a $(2n+3)\times
n$ adjacency matrix $\hat{B}=(b_{ij})$,

(5.1) $b_{ij}=\begin{cases}1,\ \text{if $i,j$ label two sides of a triangle
clockwise}\\\ -1,\ \text{similar but counter clockwise}\\\ 0,\
\text{otherwise}\end{cases}$

Notice that $i$ is assigned to a side but $j$ to a diagonal. The principal
part of the adjacencies is encoded in the $n\times n$ matrix $B=(b_{i,j})$
that encodes adjacencies of the diagonals. We describe matrix mutation
operations between these triangulation by diagonal flips (in direction $k$) by
$\mu_{k}:\hat{B}\longmapsto\hat{B}^{\prime}$,

(5.2) $b_{ij}^{\prime}=\begin{cases}-b_{ij},\ \qquad\qquad\ k\in{i,j}\\\
b_{ij}+|b_{ik}|b_{kj},\quad k\notin{i,j},\ b_{ik}b_{kj}>0\\\
b_{ij}\qquad\qquad\qquad\text{otherwise}\end{cases}$

and write $\hat{B}^{\prime}=(b_{ij}^{\prime})=\mu_{k}(\hat{B}),\ \text{the
same for}\ B=(b_{ij})$. The operation $\mu_{k}$ on adjacency matrix $B$
corresponds to the flipping the diagonal number $k$ that is removing it and
adding other opposite diagonal. We have $\mu_{k}(\mu_{k}(B))=B,\ (\text{the
same for}\ \hat{B})$. Assign a variable to each of the $2n+3$ edges. Assume by
induction that we have associated a rational function to the edges. When doing
a diagonal flip the rational function associated to the removed diagonal
(namely $X$ is replaced by the one $X^{\prime}$ where $XX^{\prime}=ac+bd,\
a,b,c,d\ \text{sides of the neighboring triangles}$. The rational functions
associated to the diagonals does not depend to the sequence connecting two
triangulations. In general the exchange relation corresponding to a flip in
direction $k$ given by matrix mutation $B=(b_{ij})$ is given by

(5.3)
$X_{k}X_{k}^{\prime}=\prod_{b_{ik}>0}x_{i}^{b_{ik}}+\prod_{b_{ik}<0}x_{i}^{-b{ik}}.$

######  Definition 5.1.

A seed is a pair $(\textbf{X},\hat{B})$ where X is set of variables, and
$\hat{B}$ is an exchange matrix.

######  Definition 5.2.

[31] A cluster algebra is a commutative ring contained in the field of
rational functions in $N$ variables over $\mathbb{Q}$, by the generators from
an initial seed, via the iterative mutation process as

  * •

$\underline{X}=\ \\{x_{1},...,x_{N}\\}$ set of $N$ algebraically independent
generators which split into disjoint sets

(5.4) $\displaystyle C$
$\displaystyle=\\{x_{1},...,x_{n}\\},\qquad\text{(cluster variables)}$
$\displaystyle F$ $\displaystyle=\\{x_{n+1},...,x_{N}\\},\qquad\text{(frozen
variables)}$ $\displaystyle X$ $\displaystyle=C\cup D$

  * •

An $N\times n$ matrix $\hat{B}=(b_{ij})$ and its principal part $B$ which is
skew symmetrizable, (i.e. there exists a positive diagonal matrix $D$ such
that $DBD^{-1}$ is skew symmetrizable) called exchange matrix. A seed mutation
in direction $k$ transforms

(5.5) $\displaystyle\mu_{k}$
$\displaystyle:(\underline{X},\hat{B})\longmapsto(\underline{X}^{\prime},\hat{B}^{\prime})$
$\displaystyle\underline{X}^{\prime}$
$\displaystyle=\underline{X}\setminus\\{x_{k}\\}\cup\\{x_{k}^{\prime}\\}$
$\displaystyle\hat{B}^{\prime}$ $\displaystyle=\mu_{k}(\hat{B})$

Matrix mutations preserve rank and symmerizability with the same $D$. A Seed
mutation is an equivalence relation on seeds. If $S$ is an equivalence class
we set $X=X(S)$ to be the union of all clusters of all seeds in $S$. The
cluster algebra $A(S)$ is generated by the clusters in $X$ and the frozen
variables $x_{n+1},...,x_{N}$ and their inverses

(5.6) $A(S)=\mathbb{C}[X(S),x_{n+1}^{\pm 1},...,x_{N}^{\pm 1}].$

$n$ is called the $rank(A)$.

######  Theorem 5.3.

(Laurent Phenomenon) [32] The cluster variables are Laurent polynomials with
integer coefficient (positive conjectural) in the variables
$\\{x_{1},...,x_{N}\\}$.

######  Definition 5.4.

A cluster algebra is of finite type if it has finitely many seeds.

The classification of finite type cluster algebras is the same as Cartan
classification of root systems. In fact if $A=(a_{ij})$ is a Cartan matrix
then because the Coxeter diagram has not any loop it is possible to decompose
the index set into two disjoint subsets

(5.7) $I=I_{+}\coprod I_{-},\ (\text{the Coxeter diagram is bipartite}),$

[cf. [34] loc. cit.] and define the adjacency matrix $B$ by

(5.8) $b_{ij}=\begin{cases}0\qquad\qquad i=j\\\ a_{ij}\qquad\ \ \ \ i\neq j,\
i\in I_{+}\\\ -a_{ij}\qquad\ \ i\neq j,\ i\in I_{-}\end{cases}$

######  Proposition 5.5.

[35, 34] A cluster algebra is of finite type if and only if the exchange
matrix is obtained form a Cartan matrix as (5.9). This criteria is also
equivalent to the inequality $|b_{ij}b_{ji}|\leq 3$ on the exchange matrix
$B=(b_{ij})$.

The type of $A$ is called the cluster type. The definition in (3.13) is
reversible as

(5.9) $a_{ij}=\begin{cases}-|b_{ij}|,\qquad i\neq j\\\ \ 0\qquad\qquad\
i=j\end{cases}$

In this case there exists a skew symmetrizing matrix $D$ such that the matrix
$DAD^{-1}$ is positive definite. Assume $R$ is an irreducible root system,
$\\{\alpha_{1},....\alpha_{n}\\}$ the set of simple roots and the Cartan
matrix $A$. Let $\\{x_{1},...,x_{n}\\}$ be the cluster variables at a seed of
the corresponding cluster algebra. denote by $R_{\geq-1}$ the set of roots
which are either positive or negative simple.

######  Theorem 5.6.

[34, 35] The cluster variables are naturally labeled by the roots in
$R_{\geq-1}$, that is one has a 1-1 assignment

(5.10) $\alpha=c_{1}\alpha_{1}+...+c_{m}\alpha_{n}\ \stackrel{{\scriptstyle
1-1}}{{\longmapsto}}\ \ \
x_{\alpha}=\frac{P_{\alpha}(x_{1},...,x_{n})}{x_{1}^{c_{1}}...x_{n}^{c_{n}}}$

where $P_{\alpha}$ is a polynomial. In this correspondence
$x_{-\alpha_{i}}=x_{i}$.

######  Definition 5.7.

[34, 35] The cluster complex is a simplicial complex on the set of all cluster
variables, where its maximal simplices are clusters. By Theorem 5.6 the
cluster complex can be labeled by the $R_{\geq-1}$ associated to some root
systems.

######  Proposition 5.8.

[34, 35] The dual graph of the the cluster complex is precisely the exchange
graph of the cluster algebra.

######  Proposition 5.9.

[34, 35] The roots that label the cluster variables in a given cluster form a
basis of the root lattice $Q$. The cone spanned by such roots form a complete
simplicial fan in the ambient real vector space $Q_{\mathbb{R}}$ (cluster
fan). The fan is the normal fan of simple $n$-dimensional convex polytope in
the dual space $Q_{\mathbb{R}}^{\vee}$.

A quiver $Q$ is an oriented graph, accordingly with two types of vertices,
’frozen’ and ’mutable’. We ignore edges connecting frozen vertices to each
other. A mutation on a vertex $z$ is defined in 3-steps,

  * •

to each $x\to y\to z$ produce a new $x\to z$.

  * •

reverse all the edges incident to $X$.

  * •

remove oriented 2-cycles.

Mutations are involution, and we can describe cluster exchanges on quivers.

######  Definition 5.10.

A seed is $(Q,\textbf{X})$ where $Q$ is a quiver labeled by a cluster and X is
a set of variables.

A seed mutation at $z$ replaces $Q$ by $\mu_{z}(Q)$ and $z$ by a new cluster
variable $z^{\prime}$ such that

(5.11) $zz^{\prime}=\prod_{z\leftarrow y}y+\prod_{z\rightarrow y}y$

Assign formal variables associated to each vertex of a quiver, the cluster
ring is generated by all the variables in a mutation equivalence class. The
exchange graph of of $A(Q)$ does not depend on the frozen part of $Q$. The
cluster algebras which only differ in the frozen part are said of the same
cluster type, [ see [34], [30], [35]].

######  Theorem 5.11.

(Gabriel’s Theorem) [73, 35] A cluster algebra $A(Q)$ is of finite type iff
the mutable part of its quiver at some seed is an orientation of a Dynkin
diagram.

######  Remark 5.12.

A representation of a quiver is an assignment of $\mathbb{C}$-vector space to
each vertex, and a linear map to an oriented edge. A quiver is of finite type
if it has only a finite number of indecomposable representations.

######  Theorem 5.13.

[40] The coordinate ring of any partial flag variety $SL_{n}\mathbb{C}/P$ has
a natural cluster algebra structure.

######  Example 5.14.

Consider the Grassmanninan $Gr(k,n)$, where have the Plücker embeddings

(5.12) $\displaystyle Gr(k,n)\hookrightarrow\
\mathbb{P}(\bigwedge^{k}\mathbb{C}^{n})$ $\displaystyle
pt=(v_{1},...,v_{k})\longmapsto v_{1}\wedge...\wedge
v_{k},\qquad(v_{i}\in\mathbb{C}^{n})$

By setting $I=\\{1\leq i_{1}\leq i_{2}\leq...\leq i_{k}\leq n\\}$ the $I$-th
Plücker coordinate is given by

(5.13) $\Delta_{I}=\det({v_{i}}_{1},...,{v_{i}}_{k})$

The Plücker coordinates generate $\mathbb{C}[Gr(n,k)]$. One defines the
$(k,n)$-diagrams by putting $n$ marked points $1,2,3,...,n$ clockwise on the
boundary of a disc and oriented strands

(5.14) $S_{i}:\ i\longmapsto i+k\mod n,\qquad(1\leq i\leq n)$

with no oriented lenses, self or alternate crossings, nor triple crossing on a
point [see [17] for details]. We obtain several alternating regions whose
boundaries are oriented clockwise. Denote by $i_{1}...i_{s}$ the region
according to the boundary strands. The dual graph to a $(k,n)$-diagram is the
cluster complex, [cf. [17]].

######  Theorem 5.15.

[68] The homogeneous coordinate ring of any Grassmannian $Gr(k,n)(\mathbb{C})$
has a natural cluster algebra structure.

######  Theorem 5.16.

[68] Let $S$ be a $(k,n)$-diagram, and

(5.15) $X(S)=\\{\Delta_{I(R)}\ |\ R\ \text{alternating region of}\ S\ \\}$

Then $X(S)$ is a cluster, $\mathbb{C}[Gr(n,k)]=\mathbb{C}[X(S)]/I$. The
exchange relations are generalized Plücker relations, that is $I$ is given by
the Plücker relations.

There are generalized minors $\Delta_{\dot{u}w_{j},\dot{v}w_{j}}$ defined as
follows, [30, 35]. Lets write

(5.16) $G_{0}=U_{-}TU,\qquad x=[x]_{-}[x]_{0}][x]_{+}\ \ \text{(}unique)$

($T$ maximal torus and $U$ unipotent) where $G_{0}$ the open subset of $G$
having this decomposition. Define the minor $\Delta_{uw_{i},vw_{i}}$ as the
regular function on $G$ s.t. whose restriction on $\dot{u}G_{0}\dot{v}^{-1}$
is given by

(5.17) $\Delta_{uw_{i},vw_{i}}(x)=([\dot{u}^{-1}x\dot{v}]_{0})^{w_{i}},\qquad
u,v\in W$

where $w_{i}$ is a fundamental weight and the exponent by a weight means the
multiplicative character on the maximal torus $T$ given by (3.10). These
functions appear as the cluster variables in the construction of the
coordinate ring of flag varieties, cf. [35, 30]. The generalized minors are
well defined nowhere vanishing functions on the double Bruhat cells. According
to the Theorem 3.2 write

(5.18) $G=\coprod_{u}BuB=\coprod_{v}B_{-}vB_{-}.$

The double Bruhat cells are defined by

(5.19) $G^{u,v}=BuB\cap B_{-}vB_{-}.$

As a variety $G^{u,v}$ is biregularly isomorphic to a Zariski open subset of
an affine space, [30, 10]. The variety $G^{u,v}$ can be characterized by
certain vanishing conditions of the generalized minors, [[35], proposition
2.8]. The generalized minors also characterize the double Bruhat cells. That
is the double Bruhat cells are exactly defined by the nonvanishing of certain
minors. They are naturally constructed by the mutations initiated from an
original seed variables. In particular they are included in the cluster
variables.

##  6\. The coordinate ring of $\frac{G}{P}[\psi_{H}]$ as a weighted cluster
algebra

In this section we extend some of the facts stated in section 5, for the
coordinate ring of Homogeneous space $G/P$ to the case of weighted homogeneous
space i.e. to $\mathbb{C}\left[\frac{G}{P}\left[\psi_{H}\right]\right]$.

######  Theorem 6.1.

The coordinate ring of $\frac{G}{P}\left[\psi_{H}\right]$ is a weighted graded
homogeneous ring. The field extension
$\mathbb{C}(G/P)\big{/}\mathbb{C}\left(\frac{G}{P}\left[\psi_{H}\right]\right)$
has degree $[\times_{\alpha\in R_{+}}\langle\alpha,H\rangle]$.

###### Proof.

In the weighted case we still have the embedding in the product of weighted
projective spaces as

(6.1) $\frac{G}{P}\big{[}\psi_{H_{1}}\big{]}\varlonghookrightarrow\prod_{j\in
J}\mathbb{P}\left(L(\langle\alpha_{j},H\rangle w_{j})\right)$

and therefore $\mathbb{C}\left[\frac{G}{P}[\psi_{H_{1}}]\right]$ is graded by
$\Pi_{J}[\psi_{H}]=N^{J}[\psi_{H}]$ the monoid of integral dominant weights of
the form $\lambda=\sum_{J}a_{j}\lambda_{j}w_{j}$. Therefore

(6.2) $\mathbb{C}\left[\frac{G}{P}\big{[}\psi_{H}\big{]}\right]\ =\
\bigoplus_{\lambda\in\Pi_{J}[\psi_{H}]}L(\lambda)$

Thus it is generated by

(6.3) $\bigoplus_{j\in J}L(\langle\alpha_{j},H\rangle w_{j})$

where $w_{j}$ are the fundamental weights. We deduce that the degree of the
extension is equal to

$\displaystyle\left[\mathbb{C}(G/P):\mathbb{C}\left(\frac{G}{P}\big{[}\psi_{H}\big{]}\right)\right]$
$\displaystyle=\left|\dfrac{\langle\bigoplus_{j\in
J}L(w_{j})\rangle}{\langle\bigoplus_{j\in J}L(\langle\alpha_{j},H\rangle
w_{j})\rangle}\right|$ $\displaystyle=\prod_{\alpha\in
R(P)_{+}}\langle\alpha,H\rangle$

∎

We extend the notion of quivers to weighted quivers. We do this for quivers of
finite type.

######  Definition 6.2.

[60] A weighted quiver $(Q,w)$ is a quiver $Q$ with a weight function
$w:Q\longrightarrow\mathbb{Z}$. The mutation $\mu_{k}$ at the vertex $x_{k}$
changes the weights at the vertices as follows,

  * •

For each arrow $x_{i}\to x_{k}$ change the weight

(6.4) $\mu_{k}:w_{i}\longmapsto w_{i}+w_{k}$

that is

(6.5) $\mu_{k}(w)(x_{i})=w_{i}+[b_{ik}]_{+}w_{k},\
[b_{ik}]_{+}=\max(b_{ik},0)$

  * •

reverse the sign at vertex $x_{k}$, i.e.

(6.6) $\mu_{k}(w)(x_{k})=-w_{k}.$

The exchange relations are defined as follows. Label the vertices by the
variables $z_{i}=x_{i}+y_{i}\epsilon$, where $x_{i},y_{i}$ are usual variables
and $\epsilon^{2}=0$. Then write the exchange relation as at $k$ by

(6.7) $zz^{\prime}=\prod_{z_{k}\to z_{j}}z_{j}+\prod_{z_{i}\to z_{k}}z_{i}$

and the other variables remain unchanged.

######  Remark 6.3.

In weighted quivers we assume that the vertices are labeled by
$z_{i}=x_{i}+y_{i}\epsilon$ where $\epsilon$ is a formal variable with
$\epsilon^{2}=0$. The variables $y_{i}$ are called odd variables. Actually one
of the ways that weighted quivers are used are in clusters superalgebras,
[57]. As is shown in [57] the variables appearing by mutation at all possible
mutable vertices are Laurent polynomials in the initial variables. The Laurent
phenomenon implies that the denominators are monomials in the variables
$x_{i}$ while the variable $y_{i}$ appear linearly in the numerators, [60],
[58].

The same as in Section 5 the cluster algebra is generated by all the cluster
variables and their iterating mutations in a mutation class $S$. We call this
algebra weighted cluster algebra associated to the weighted quiver $Q[w]$
denoted $A(Q)[w]$.

######  Example 6.4.

[60] We mention periodic quivers with periodic weight functions, where after a
finite sequence of mutations we get back to the original quiver with the same
weight function.

  * •

One can label the vertices of a triangle or a square with weights to be 0, 1,
-1, and orient the edges such that they give period 1 quivers with period 1
weight function.

  * •

The primitive quiver $P_{N}^{(t)}$, $1\leq t\leq N/2$ is the quiver with $n$
vertex and $n$ arrows such that the vertex $x_{i}$ is joined to the vertex
$x_{i+t},\mod(N)$ where the indices are from $1,...,N$. The arrow is from
higher index to the lower one. The exchange matrix is given by

(6.8) $b_{ij}=\begin{cases}-1\qquad j-i=t,\\\ \ 1\qquad\ \ i-j=t\\\ \ 0\qquad\
\ \text{else}\end{cases}$

One simply finds a weight function on $P_{N}^{(t)}$ to be periodic of period
1, [see the ref. for details and a classification].

######  Proposition 6.5.

The coordinate ring $\mathbb{C}\left[\frac{G}{P}[\psi_{H}]\right]$ is a
weighted cluster algebra of finite type. It can be characterized by the
cluster algebra of the weighted quiver obtained from an orientation of the
Dynkin diagram of $G$ and the weight function $w=\psi_{H}$, together with the
mutations defined as in Definition 6.2.

###### Proof.

(sketch) By Definition 3.1, Theorem 5.5 and Theorem 5.11 the cluster algebra
of $\mathbb{C}\left[\frac{G}{P}[\psi_{H}]\right]$ is obtained in the same way
as $\mathbb{C}\left[\frac{G}{P}\right]$ constructed from the mutations on the
Dynkin diagram of $G$, when the nodes of the diagram are assigned by some
weights. Therefore the argument in Theorems 6.1 at the beginning of the
section proves the assertion. The generalized minors are defined on
$\dot{u}G_{0}\dot{v}^{-1}$ by the same formula

(6.9) $\Delta_{uw_{i},vw_{i}}(x)=([\dot{u}^{-1}x\dot{v}]_{0})^{w_{i}},\qquad
u,v\in W,$

The definition shows that if we choose suitable weights at the simple roots
the weight of the minors changes as

(6.10)
$w(\Delta_{uw_{i},vw_{i}}(x_{\alpha}))[\psi_{H}]=w(\Delta_{uw_{i},vw_{i}}(x_{\alpha}))\times
w(x_{\alpha})$

where here $x_{\alpha}$ is thought as different root groups constructed by the
images of representations of $SL_{2}\mathbb{C}$. The generalized minors appear
as cluster variables and the exchange relations of the mutations at any node
are standard determinant identities. The coordinate rings
$\mathbb{C}[G^{u,v}]$ are generated by the weighted generalized minors. For
the description in terms of quivers, according to the constructions in Section
5, and the definition 6.2 we may start from a weight system which assigns a
weight to each node of the Dynkin diagram and do all the possible mutations.
By the Theorems 5.5, and 5.11, the clusters have finite elements. That is we
get back to the original Dynkin diagram with the weight associated to each
simple root (node). The weights of generalized minors is multiplied by the
weights on the coordinate charts. The exchange relations (6.7) appear as
various determinant identities between generalized minors. The sign change of
the weights when making exchange relations by the mutation at a node does not
conflict the weighted system as we obtain Laurent polynomials that generate
the coordinate rings of affine charts. So the Theorem is a consequence of the
above Theorems in the unweighted case. ∎

######  Example 6.6.

  * (1)

Let $\mathcal{A}=\mathbb{C}[SL_{2}\mathbb{C}]$. It is a cluster algebra of
rank 1. Setting the entries of a matrix by $x_{11},x_{12},x_{21},x_{22}$, the
variables $x_{11},x_{22}$ are cluster variables. The ring $\mathcal{A}$ is
generated over $\mathbb{C}[x_{12},x_{21}]$ by the cluster variables with the
exchange relation

(6.11) $x_{11}x_{22}-x_{12}x_{21}=1$

Now consider the homogeneous space $X=SL_{2}/B^{-}$. We can choose the weights
by assigning two positive integers $n>m$ to the torus variables
$x_{11},x_{22}$ respectively. The Dynkin diagram consists of single point with
the weight $n-m>0$. We have just one positive root
$\alpha=\epsilon_{1}-\epsilon_{2}$ and one root coordinate $x_{\alpha}$ with
weight $2(n-m)$. The Weyl group is $S_{2}$. The coordinate $x_{\alpha}^{w}$ in
the other Bruhat cell, for the nontrivial element $w\in W$ has the same weight
as $x_{\alpha}$. We can also regard the Dynkin diagram as a trivial quiver,
with weight $n-m$. The the mutation just changes its sign and back.

  * (2)

Consider $X=SL_{3}/N$ and the Cluster algebra $\mathcal{A}[SL_{3}/N]$, where
$N$ is the maximal upper triangular unipotent subgroup. It is also a cluster
algebra of rank 1. Denote the coordinates on $X$ by
$x_{1},x_{2},x_{3},x_{12},x_{13},x_{23}$. Then we can choose $x_{2},x_{13}$ as
cluster variables and $\mathcal{A}$ is generated by them over
$\mathbb{C}[x_{1},x_{3},x_{12},x_{23}]$ and the exchange relation

(6.12) $x_{2}x_{13}=x_{1}x_{23}+x_{3}x_{12}$

We have two simple roots $\alpha_{1},\alpha_{2}$ and 3 positive roots
$\alpha_{1},\alpha_{2},\alpha_{1}+\alpha_{2}$. The weights can be chosen
independently on the simple roots. We can illustrate this by choosing three
positive weights $w_{1}>w_{2}>w_{3}>0$ on the variables $x_{1},x_{2},x_{3}$.
Then the weights on the other coordinates are
$w_{1}+w_{2},w_{2}+w_{3},w_{3}+w_{1}$. The above exchange relation still holds
with the new weights.

  * (3)

Consider $Gr(2,5)$, the Grassmannian of 2-planes in $\mathbb{C}^{5}$. The
affine cone of $Gr(2,5)$ embedds in $\bigwedge^{2}\mathbb{C}^{5}$. Thus the
coordinates on $Cone(Gr(2,5))$ can be explained via the formal map

(6.13)
$\displaystyle\bigwedge^{2}\begin{pmatrix}x_{1}&x_{2}&x_{3}&x_{4}&x_{5}\\\
y_{1}&y_{2}&y_{3}&y_{4}&y_{5}\end{pmatrix}=$
$\displaystyle\begin{pmatrix}z_{11}&z_{13}&z_{14}&z_{15}\\\
0&z_{23}&z_{24}&z_{25}\\\ 0&0&z_{34}&z_{35}\\\ 0&0&0&z_{45}\end{pmatrix}$

where $z_{ij}=x_{i}y_{j}-x_{j}y_{i}$. The coordinate ring of the cone is
generated by the variables $z_{ij}$ and the relations

(6.14) $P_{ijkl}=z_{ij}z_{kl}-z_{ik}z_{jl}+z_{il}z_{jk}.$

If we assign the weights $w_{i}$ on $x_{i}$ and $y_{i}$. Then the weight on
$z_{ij}$ is $w_{i}+w_{j}$. The affine charts on $Gr(2,5)$ are given by
$z_{ij}\neq 0$. On the weighted Grassmanninan the chart with $z_{ij}\neq 0$ is
the quotient $\mathbb{C}^{6}/\mu_{wt(z_{ij})}$.

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