# Ideal Polytopes for Representations of $GL_{n}(\mathbb{C})$

Teresa Lüdenbach

###### Abstract

00footnotetext:  _Date:_ 25th August 2022 00footnotetext: _Keywords:_ Tropical
geometry, representation theory, polytopes.
This work was supported by the Engineering and Physical Sciences Research
Council [EP/S021590/1].
The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The
London School of Geometry and Number Theory), University College London, and
King’s College London.

In this paper we use the superpotential $\mathcal{W}_{q}$ for the flag variety
$GL_{n}/B$ and particular coordinate systems that we call ‘ideal coordinates
for $\mathbf{i}$’, to construct polytopes $\mathcal{P}^{\mathbf{i}}_{\lambda}$
inside $\mathbb{R}^{R_{+}}$, associated to highest weight representations
$V_{\lambda}$ of $GL_{n}(\mathbb{C})$. Here $\mathbf{i}$ is a reduced
expression of the longest element of the Weyl group and $R_{+}$ is the set of
positive roots of $GL_{n}$. The lattice points of
$\mathcal{P}^{\mathbf{i}}_{\lambda}$ can be used to encode a basis of the
representation $V_{\lambda}$. In particular, for a specific choice of
$\mathbf{i}$, the polytope $\mathcal{P}^{\mathbf{i}}_{\lambda}$ is
unimodularly equivalent to a Gelfand-Tsetlin polytope. The construction of the
polytopes involves tropicalisation of the superpotential, namely
$\mathcal{P}^{\mathbf{i}}_{\lambda}$ is
$\left\\{\mathrm{Trop}\left(\mathcal{W}^{\mathbf{i}}_{t^{\lambda}}\right)\geq
0\right\\}$ written in a tropical version of the ideal coordinates for
$\mathbf{i}$.

Using work of Judd ([13]) we have that there is a unique ‘positive’ critical
point of $\mathcal{W}_{t^{\lambda}}$ over the field of Puiseux series. Its
coordinates, in terms of the ideal coordinates for $\mathbf{i}$, are positive
in the sense of having positive leading term. The remarkable property of our
new polytopes relates to the tropical version of this critical point, which,
for every choice of $\mathbf{i}$, gives a point in $\mathbb{R}^{R_{+}}$ that
lies in the interior of the polytope $\mathcal{P}^{\mathbf{i}}_{\lambda}$. We
prove that this tropical critical point is independent of the reduced
expression $\mathbf{i}$, and that it is given by a pattern called the ‘ideal
filling’ for $\lambda$ that was introduced by Judd.

Finally, combining these results with work of Rietsch ([22]) relating critical
points of the superpotential with Toeplitz matrices, we show that for a
totally positive lower-triangular Toeplitz matrix over the field of Puiseux
series factorised into simple root subgroups, the valuations of the factors
give an ideal filling.

###### Contents

  1. 1 Introduction
  2. 2 Mirror symmetry for G/B applied to representation theory
    1. 2.1 Notation and definitions
    2. 2.2 Landau-Ginzburg models
    3. 2.3 The string coordinates
    4. 2.4 The form of the weight matrix
  3. 3 The ideal coordinates
    1. 3.1 The Chamber Ansatz
    2. 3.2 Chamber Ansatz minors
    3. 3.3 The coordinate change
  4. 4 Givental-type quivers and critical points
    1. 4.1 The Givental superpotential
    2. 4.2 The quiver torus as another toric chart on Z
    3. 4.3 Critical points of the superpotential
  5. 5 The tropical viewpoint
    1. 5.1 The basics of tropicalisation
    2. 5.2 Constructing polytopes
    3. 5.3 Tropical critical points and the weight map
    4. 5.4 Ideal fillings
    5. 5.5 A family of ideal polytopes

###### List of Figures

  1. 1 A Gelfand-Tsetlin polytope
  2. 2 Ideal filling arrangement for $n=4$
  3. Example 3.1.3 (Ansatz arrangements for $\mathbf{i}_{0}$, ${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ in dimension $4$).
  4. 5 Labelled line segments in graphs for computing Chamber Ansatz minors
  5. 6 The graph for $u_{1}$ when $n=4$
  6. 7 The graph for $u^{T}$ when $n=4$
  7. 8 Example family of paths for the proof of Lemma 3.2.4, $u_{1}$ case
  8. 9 Example family of paths for the proof of Lemma 3.2.4, $u^{T}$ case
  9. Proof.
  10. 12 Labelled line segments in graphs in the proof of Lemma 3.3.1
  11. Proof.
  12. 15 The graph for $u_{1}$ when $n=4$, $\boldsymbol{p}$ coordinates
  13. 4.1 The Givental superpotential
  14. 4.2 The quiver torus as another toric chart on Z
  15. 20 Critical point conditions in the quiver
  16. 4.3 Critical points of the superpotential
  17. 23 Quiver decoration when $n=3$, string coordinates
  18. Example 5.3.2 (Dimension 3).
  19. 26 Quiver decoration when $n=3$, ideal coordinates
  20. 27 Superpotential polytope $\mathcal{P}_{\lambda,\boldsymbol{\mu}}$ for $n=3$ and $\lambda=(2,1,-1)$ (ideal coordinates)
  21. 28 Ideal filling for $n=4$
  22. 29 Arrows in the bottom and left hand walls of the quiver
  23. 30 Subquiver for the proof of Lemma 5.4.5
  24. 31 Subquiver for induction, horizontal case
  25. 32 Subquiver for the proof of Lemma 5.4.6
  26. 33 Fillings in dimension $3$

###### List of Tables

  1. 1 Vertices and distinguished points, and their corresponding weights, string coordinates
  2. 2 Vertices and distinguished points, and their corresponding weights, ideal coordinates

##  1 Introduction

Representations of Lie groups are often described in terms of their weights -
the characters arising in the action of a maximal torus. A standard way to
depict these weights is by embedding the character lattice into a real vector
space and viewing the weights as lattice points in their convex hull, the so-
called ‘weight polytope’ of the representation. For an irreducible
representation the weights along the boundary of the weight polytope
(including the highest weight) all have one-dimensional weight spaces. The
weight spaces corresponding to interior points can be higher-dimensional.
Accordingly, a better ‘picture’ of the representation may be given by a
higher-dimensional polytope that projects onto the weight polytope, such that
the lattice points in a fibre parametrise a basis of the corresponding weight
space.

A famous example of such a construction is given by the Gelfand-Tsetlin
polytope of a representation of $GL_{n}$, given first in [10] (e.g. Figure 1).
More recent examples relate to Lusztig’s canonical basis and its combinatorial
and geometric construction ([18], [19]), as well as Kashiwara’s crystal basis
operators ([15]). Of particular interest on the crystal basis side are the
string polytopes introduced by Littelmann in [17]. On the canonical basis side
there is another parametrisation due to Lusztig ([20]). His parametrisation
uses coordinate charts on the Langlands dual flag variety and the notion of
tropicalisation that he introduced.

$x$$y$$z$O

$x$$y$$z$O

Figure 1: A Gelfand-Tsetlin polytope

Building on the work of Lusztig and Kashiwara, Berenstein and Kazhdan in [3],
[4], ‘geometrised’ such polytopes via their theory of geometric crystals.
Their construction includes a function that encodes all of the walls of these
polytopes. This function turns out to agree with the superpotential of a flag
variety, which was later independently constructed by Rietsch in the the
context of mirror symmetry ([22], see also Chhaibi’s work in [5]). Another
example of this are the polytopes constructed using the mirror symmetry of
Grassmannians by Rietsch and Williams in [23], which relate to fundamental
representations of $GL_{n}$ and cluster duality ([8]).

Rietsch showed in [22] that the critical points of the superpotential describe
the spectrum of the quantum cohomology ring of $G/P$ in a presentation given
by Dale Peterson. The idea to consider the critical points of the
superpotential, $\mathcal{W}_{q}$, was transported to the setting of geometric
crystals and polytopes by Judd in [13], working in the case of full flag
varieties $SL_{n}/B$. He considered an analogue, $\mathcal{W}_{t^{\lambda}}$,
of $\mathcal{W}_{q}$ over the field of Puiseux series, associated to a
dominant weight $\lambda$ of $SL_{n}$. He showed that there is a unique
critical point of $\mathcal{W}_{t^{\lambda}}$ with positive leading term
coefficients. By tropicalisation he showed that this special point gives rise
to a canonical point in the interior of the string polytope for the
representation of $SL_{n}$ with highest weight $\lambda$. We will refer to
this point as the tropical critical point of the superpotential.

As part of Judd’s study of these objects, he investigated when the tropical
critical point is integral. To do so, he introduced a combinatorial object
called an ideal filling for a dominant weight $\lambda$ and formed a close
connection to the tropical critical point. Such a ideal filling for $\lambda$
is given by assigning non-negative real numbers $n_{ij}$ to boxes in an upper
triangular form such that $n_{ij}=\max\\{n_{i+1\;j},n_{i\;j-1}\\}$ for
$j-i\geq 2$, together with a constraint on the $n_{ij}$ determined by
$\lambda$. For example if $n=4$ the arrangement looks like Figure 2.

$n_{12}$$n_{13}$$n_{23}$$n_{14}$$n_{24}$$n_{34}$ Figure 2: Ideal filling
arrangement for $n=4$

Since the ideal filling for $\lambda$ is a combinatorial reformulation of the
tropical critical point, we should expect it to also have a polytope in some
special toric chart. Unfortunately, building a polytope in these ‘ideal
filling coordinates’ was not part of Judd’s work. This poses the following
question: can we find a toric chart for which the tropical critical point is
exactly the ideal filling for $\lambda$? In answer, the main result of our
work is the construction of such a coordinate system.

Initially our ‘ideal filling coordinates’ are indexed by a specific choice of
reduced expression for $w_{0}$, the longest element of the Weyl group. Namely
our coordinates correspond to factorising a lower-triangular unipotent matrix
into simple root subgroups according to the reduced expression
$s_{i_{1}}\cdots s_{i_{N}}$ given by

$\mathbf{i}_{0}=(i_{1},\ldots,i_{N}):=(1,2,\ldots,n-1,1,2\ldots,n-2,\ldots,1,2,1)$

where we refer to Sections 2.1 and 3 for the precise definitions. However we
later generalise our ‘ideal coordinates’ to arbitrary reduced expressions.
Remarkably this gives rise to family of polytopes that all have the same
tropical critical point (see Proposition 5.5.6).

As a corollary we have the following theorem which gives an interpretation of
ideal fillings using Toeplitz matrices over the field of Puiseux series:

$\mathcal{K}=\bigcup_{n\in\mathbb{Z}_{>0}}\mathbb{C}((t^{\frac{1}{n}}))$

######  Theorem 1.0.1.

Let $\phi_{i}:SL_{2}\to GL_{n}$ be the homomorphism corresponding to the
$i$-th simple root of $GL_{n}$ and take

$\mathbf{y}_{i}(z)=\phi_{i}\begin{pmatrix}1&0\\\ z&1\end{pmatrix}.$

Let $\mathbf{i}=(i_{1},\ldots,i_{N})$ stand for an arbitrary reduced
expression $s_{i_{1}}\cdots s_{i_{N}}$ for $w_{0}$, where $N=\binom{n}{2}$.
Then we have an ordering on the set of positive roots
$R_{+}=\\{\alpha_{ij}=\epsilon_{i}-\epsilon_{j}\ |\ 1\leq i<j\leq n\\}$ given
by

$\alpha^{\mathbf{i}}_{j}=\begin{cases}\alpha_{i_{1},i_{1}+1}&\text{for
}j=1,\\\ s_{i_{1}}\cdots s_{i_{j-1}}\alpha_{i_{j},i_{j}+1}&\text{for
}j=2,\ldots N.\end{cases}$

Now take $m_{\alpha^{\mathbf{i}}_{1}},\ldots,m_{\alpha^{\mathbf{i}}_{N}}$ to
be Puiseux series with positive leading coefficients and non-negative
valuations $\mu_{\alpha}=\mathrm{Val}_{\mathbf{K}}(m_{\alpha})$ (defined in
Section 5.1 as the exponent of the first non-zero term). If the product
$\mathbf{y}_{i_{1}}\left(m_{\alpha^{\mathbf{i}}_{1}}^{-1}\right)\cdots\mathbf{y}_{i_{N}}\left(m_{\alpha^{\mathbf{i}}_{N}}^{-1}\right)$
is a Toeplitz matrix, then the valuations $\mu_{\alpha}$ form an ideal
filling:

$\mu_{\alpha_{12}}$$\mu_{\alpha_{13}}$$\mu_{\alpha_{1n}}$$\cdots$$\mu_{\alpha_{23}}$$\cdots$$\mu_{\alpha_{2n}}$$\vdots$$\ddots$$\mu_{\alpha_{n-1,n}}$

Moreover every ideal filling with rational entries arises in this way.

This result follows by combining the theorem of Rietsch that the critical
points of the superpotential are given by Toeplitz matrices [22, non-T-
equivariant case of Theorem 4.1], with our result that the factorisations of
the positive critical point lead to ideal fillings (combination of Proposition
5.4.3, Corollary 5.4.8 and Proposition 5.5.6). Additionally, the result
extends from $\mathbb{Q}$-valued to $\mathbb{R}$-valued ideal fillings if we
instead work over the field of generalised Puiseux series (defined in Section
5.1).

Our work is structured as follows: in Section 2 we introduce the mirror to the
flag variety $G/B$ and present the first of two coordinate systems. We do this
since the key to constructing a polytope using the superpotential, is to
express the superpotential in some torus coordinate chart. It will later
transpire that this first system gives rise to string polytopes. The second
coordinate system, the ‘ideal’ coordinates, will be best suited to the
tropical critical point. We begin Section 3 by defining this system for the
reduced expression given by $\mathbf{i}_{0}$ and devote much of the section to
constructing the ideal coordinates from the string coordinates.

Then, following Judd ([13]), in Section 4 we introduce quivers as defined by
Givental in [11]. These will provide a framework within which to neatly hold
the information of the toric charts, highest weight and superpotential. They
will also give a helpful description of the critical point conditions for a
given highest weight.

Finally, in Section 5 we consider everything we have developed up until this
point through the lens of tropical geometry. This is where we will discuss the
polytopes mentioned above and prove, for a given highest weight $\lambda$,
that the ideal filling and tropical critical point coincide. We conclude by
generalising the ideal coordinates to arbitrary reduced expressions and
presenting our new family of polytopes, as well as the conjecture mentioned
above.

######  Acknowledgements.

I wish to thank my supervisor Konni Rietsch for suggesting this idea to me,
and particularly for her helpful explanations and patience throughout.

##  2 Mirror symmetry for $G/B$ applied to representation theory

###  2.1 Notation and definitions

Let $\mathbb{K}$ be a field of characteristic $0$, containing a positive
semifield. Unless otherwise stated we take $G=GL_{n}(\mathbb{K})$ with $B$,
$B_{-}$ the Borel subgroups of upper and lower triangular matrices. Let $U$,
$U_{-}$ be their respective unipotent radicals, that is the subgroups of upper
and lower triangular matrices with all diagonal entries equal to $1$, and let
$T=B\cap B_{-}$ be the diagonal matrices in $G$. The Langlands dual group to
$G$ is denoted $G^{\vee}=GL_{n}(\mathbb{K})$, and may be taken together with
the corresponding subgroups $B^{\vee}$, $B^{\vee}_{-}$, $U^{\vee}$,
$U^{\vee}_{-}$ and $T^{\vee}$ in $G^{\vee}$.

For $i=1,\ldots,n$, we write $\epsilon_{i}$, $\epsilon^{\vee}_{i}$ for the
standard characters and cocharacters of $T$. Let
$X^{*}(T)=\mathrm{Hom}(T,\mathbb{K}^{*})$,
$X_{*}(T)=\mathrm{Hom}(\mathbb{K}^{*},T)$ the respective character and
cocharacter lattices, dually paired in the standard way by

$\langle\ ,\ \rangle:X^{*}(T)\times
X_{*}(T)\to\mathrm{Hom}(\mathbb{K}^{*},\mathbb{K}^{*})\cong\mathbb{Z}.$

We take $\alpha_{ij}=\epsilon_{i}-\epsilon_{j}\in X^{*}(T)$,
$\alpha_{ij}^{\vee}=\epsilon^{\vee}_{i}-\epsilon^{\vee}_{j}\in
X^{*}(T^{\vee})=X_{*}(T)$. Additionally for each $i\in I=\\{1,\ldots,n-1\\}$
we write $\alpha_{i}=\alpha_{i,i+1}$,
$\alpha_{i}^{\vee}=\alpha_{i,i+1}^{\vee}$. Then the roots and positive roots
of $G$ are

$R=\\{\alpha_{ij}\ |\ i\neq j\\}\text{ and }R_{+}=\\{\alpha_{ij}\ |\ i<j\\}$

respectively and the simple roots of $G$ are $\\{\alpha_{i}\ |\ i\in I\\}$.
The Cartan matrix is $A=(a_{ij})$ defined by
$a_{ij}=\langle\alpha_{j},\alpha_{i}^{\vee}\rangle$.

The fundamental weights of $G$ are given by
$\omega_{i}=\epsilon_{1}+\cdots+\epsilon_{i}$. Additionally we denote the set
of dominant integral weights by

${X^{*}(T)}^{+}=\\{\lambda\in X^{*}(T)\ |\
\langle\lambda,\alpha_{ij}^{\vee}\rangle\geq 0\ \forall i<j\\}.$

For $\lambda\in{X^{*}(T)}^{+}$, let $V_{\lambda}$ denote the irreducible
representation with highest weight $\lambda$.

Note that we may identify
$X^{*}(T)\otimes\mathbb{R}=X_{*}(T^{\vee})\otimes\mathbb{R}$ with the dual,
$\mathfrak{h}^{*}_{\mathbb{R}}$, of the Lie algebra of the split real torus of
$G$.

The Weyl group of $G$ is the symmetric group, $W=N_{G}(T)/T=S_{n}$, generated
by the simple reflections $s_{i}$ for $i\in I$. It acts on $X^{*}(T)$ by
permuting the roots and we denote the action of $w\in W$ on $\alpha\in
X^{*}(T)$ by $w\alpha$.

Associated to each simple root $\alpha_{i}$ there is a homomorphism
$\phi_{i}:SL_{2}\to G$, explicitly

$\phi_{i}:\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}1&&&&&\\\ &\ddots&&&&\\\ &&a&b&&\\\
&&c&d&&\\\ &&&&\ddots&\\\ &&&&&1\end{pmatrix}\quad\text{with }a\text{ in
position }(i,i),$

and we have a number of $1$-parameter subgroups of $G$ respectively defined by

$\displaystyle\mathbf{x}_{i}(z)=\phi_{i}\begin{pmatrix}1&z\\\
0&1\end{pmatrix},\quad$
$\displaystyle\mathbf{y}_{i}(z)=\phi_{i}\begin{pmatrix}1&0\\\
z&1\end{pmatrix},$
$\displaystyle\mathbf{x}_{-i}(z)=\phi_{i}\begin{pmatrix}z^{-1}&0\\\
1&z\end{pmatrix},\quad$
$\displaystyle\mathbf{t}_{i}(t)=\phi_{i}\begin{pmatrix}t&0\\\
0&t^{-1}\end{pmatrix}.$

for $z\in\mathbb{K}$, $t\in\mathbb{K}^{*}$ and $i\in I$. The simple
reflections in the Weyl group, $s_{i}\in W$, are given explicitly by
$s_{i}=\bar{s}_{i}T$ where

$\bar{s}_{i}=\mathbf{x}_{i}(-1)\mathbf{y}_{i}(1)\mathbf{x}_{i}(-1)=\phi_{i}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}.$

More generally we may write each $w\in W$ as a product with a minimal number
of factors, $w=s_{i_{1}}\cdots s_{i_{m}}$. We get a representative of $w$ in
$N_{G}(T)$ by taking $\bar{w}=\bar{s}_{i_{1}}\cdots\bar{s}_{i_{m}}$. Here $m$
is called the length of $w$, denoted $l(w)$, and the choice of expression
$s_{i_{1}}\cdots s_{i_{m}}$ is said to be reduced. In particular it is well
known that $\bar{w}$ is independent of this choice. For ease of notation we
will often let $\mathbf{i}=(i_{1},\ldots,i_{m})$ stand for the reduced
expression ${s}_{i_{1}}\cdots{s}_{i_{m}}$.

In a similar way for $G^{\vee}$ we have, for each $i\in I$, a homomorphism
$\phi_{i}^{\vee}:PGL_{2}\to G^{\vee}$ and $\mathbf{x}^{\vee}_{i}(z)$,
$\mathbf{y}^{\vee}_{i}(z)$, $\mathbf{x}^{\vee}_{-i}(z)$,
$\mathbf{t}^{\vee}_{i}(t)$ defined analogously. The Weyl group of $G^{\vee}$
is again the symmetric group and we use the same notation as above.

With this in mind, we make an observation which will be used frequently; given
a reduced expression, say $s_{i_{1}}\cdots s_{i_{m}}$, we can construct
matrices in $G^{\vee}$ which are indexed by $\mathbf{i}=(i_{1},\ldots,i_{m})$.
We do this by taking products of the matrices defined above. An explicit
example is given by the following map:

$\mathbf{x}^{\vee}_{\mathbf{i}}:(\mathbb{K}^{*})^{N}\to U^{\vee}\cap
B_{-}^{\vee}\bar{w}_{0}B_{-}^{\vee},\quad(z_{1},\ldots,z_{N})\mapsto\mathbf{x}^{\vee}_{i_{1}}(z_{1})\cdots\mathbf{x}^{\vee}_{i_{N}}(z_{N}).$

###  2.2 Landau-Ginzburg models

The mirror to the flag variety $G/B$ is a pair $(Z,\mathcal{W})$, called a
Landau-Ginzburg model, where $Z\subset G^{\vee}$ is an affine variety and
$\mathcal{W}:Z\to\mathbb{K}^{*}$ is a holomorphic function called the
superpotential. In order to give a more precise description we first recall
Bruhat decomposition, namely that $G$ may be written as a disjoint union of
Bruhat cells $B\bar{w}B$ (see [24, Theorem 8.3.8]):

$G=\bigsqcup_{w\in W}B\bar{w}B.$

Similarly we may write $G/B$ as

$G/B=\bigsqcup_{w\in
W}B\bar{w}B/B\quad\text{with}\quad\mathrm{dim}(B\bar{w}B/B)=l(w).$

We note that the cells $B\bar{w}B$ do not depend on the choice of
representative $\bar{w}$.

These Bruhat cells give rise to a partial ordering of Weyl group elements,
known as the Bruhat order (see [24, Theorem 8.5.4]); for $v,w\in W$ we say
$v\leq w$ if $B\bar{v}B\subseteq\overline{B\bar{w}B}$. With respect to this
ordering there is a unique maximal element $w_{0}\in W$ and we set
$N=l(w_{0})$.

Additionally, we use the Bruhat order to define open Richardson varieties.
These are given by intersecting opposite Bruhat cells; for $v,w\in W$ such
that $v\leq w$ we have

$\mathcal{R}_{v,w}:=(B_{-}\bar{v}B\cap B\bar{w}B)/B\subset G/B.$

It is well known that $\mathcal{R}_{v,w}$ is smooth, irreducible and has
dimension $l(w)-l(v)$, [16]. On the dual side we have

$\mathcal{R}^{\vee}_{v,w}:=(B^{\vee}_{-}\bar{v}B^{\vee}\cap
B^{\vee}\bar{w}B^{\vee})/B^{\vee}\subset G^{\vee}/B^{\vee}.$

We now return to $(Z,\mathcal{W})$, the Landau-Ginzburg model for $G/B$, and
define the subvariety

$Z:=B_{-}^{\vee}\cap B^{\vee}\bar{w}_{0}B^{\vee}\subset G^{\vee}.$

In order to define the superpotential $\mathcal{W}$, we let
$\chi:U^{\vee}\to\mathbb{K}$ be the sum of above-diagonal elements

$\chi(u):=\sum_{i=1}^{n-1}u_{i\,i+1},\quad u=(u_{ij})\in U^{\vee}.$

Then the superpotential is given by

$\mathcal{W}:Z\to\mathbb{K}^{*},\quad
u_{1}d\bar{w}_{0}u_{2}\mapsto\chi(u_{1})+\chi(u_{2})$

where $u_{1},u_{2}\in U^{\vee}$ and $d\in T^{\vee}$. This map will appear
frequently in subsequent sections.

The motivation for introducing the Landau-Ginzburg model is to study the
representation theory of $G$ using the mirror to $G/B$. It is natural then to
equip $Z$ with highest weight and weight maps. The highest weight map recovers
the original torus factor, $d$, as follows:

$\mathrm{hw}:Z\to T^{\vee},\quad u_{1}d\bar{w}_{0}u_{2}\mapsto d.$

For the weight map we first note that each element $b\in Z$ may be written as
$b=[b]_{-}[b]_{0}$ with $[b]_{-}\in U_{-}^{\vee}$, $[b]_{0}\in T^{\vee}$. Then
the weight map is given by the projection

$\mathrm{wt}:Z\to T^{\vee},\quad b=u_{1}d\bar{w}_{0}u_{2}\mapsto[b]_{0}.$

We will often write the above decomposition of $b$ as $b=[b]_{-}t_{R}$ to
remind us that the torus factor is taken on the right.

###  2.3 The string coordinates

In order to make the connection with representation theory we restrict our
attention to various toric charts $T^{\vee}\times(\mathbb{K}^{*})^{N}\to Z$,
indexed by reduced expressions for $w_{0}$. The first chart we want to
consider is useful for reconstructing the string polytope via the
superpotential. These ‘string coordinates’, which we introduce in this
section, were used by Judd [13] following Chhaibi [5], who was in turn
inspired by the work of Berenstein and Kazhdan [3], [4].

The toric chart in question is defined by the composition of a number of maps
which we write here for overview and then define in detail:

${T^{\vee}\times(\mathbb{K}^{*})^{N}}$${T^{\vee}\times(B_{-}^{\vee}\cap
U^{\vee}\bar{w}_{0}U^{\vee})}$${T^{\vee}\times(U^{\vee}\cap
B_{-}^{\vee}\bar{w}_{0}B_{-}^{\vee})}$${Z}$$\scriptstyle{{\left(id,\,\mathbf{x}_{-\mathbf{i}}^{\vee}\right)}}$$\scriptstyle{\tau}$$\scriptstyle{\Phi}$

If we let $\mathbf{i}=(i_{1},\ldots,i_{N})$ stand for a reduced expression
$s_{i_{1}}\cdots s_{i_{N}}$ for $w_{0}$, then first map constructs a matrix
indexed by $\mathbf{i}$ as follows:

$\mathbf{x}_{-\mathbf{i}}^{\vee}:(\mathbb{K}^{*})^{N}\to B_{-}^{\vee}\cap
U^{\vee}\bar{w}_{0}U^{\vee}\,,\quad(z_{1},\ldots,z_{N})\mapsto\mathbf{x}_{-i_{1}}^{\vee}(z_{1})\cdots\mathbf{x}_{-i_{N}}^{\vee}(z_{N}).$

The second map, $\tau$, may be written as the composition of a twist map
$\eta^{w_{0},e}$ and an involution $\iota$. We present $\tau$ in this way
since the involution will be helpful later. The twist map is defined to be

$\eta^{w_{0},e}:B_{-}^{\vee}\cap U^{\vee}\bar{w}_{0}U^{\vee}\to U^{\vee}\cap
B_{-}^{\vee}\bar{w}_{0}B_{-}^{\vee}\,,\quad
b\mapsto[(\bar{w}_{0}b^{T})^{-1}]_{+}.$

Here $b^{T}$ is the transpose of $b$ and $[g]_{+}$ is given by decomposing
$g=[g]_{-}[g]_{0}[g]_{+}$ such that $[g]_{-}\in U_{-}^{\vee}$, $[g]_{0}\in
T^{\vee}$, $[g]_{+}\in U^{\vee}$. The involution is given by

$\iota:G^{\vee}\to G^{\vee},\quad
g\mapsto(\bar{w}_{0}g^{-1}\bar{w}_{0}^{-1})^{T}.$

We note that this map preserves $U^{\vee}$. It remains to define $\tau$ by
applying the composition $\iota\circ\eta^{w_{0},e}$ to the second factor:

$\tau:T^{\vee}\times(B_{-}^{\vee}\cap U^{\vee}\bar{w}_{0}U^{\vee})\to
T^{\vee}\times(U^{\vee}\cap B_{-}^{\vee}\bar{w}_{0}B_{-}^{\vee}),\
(d,u)\mapsto\left(d,\iota(\eta^{w_{0},e}(u))\right).$

The final map in the definition of the string toric chart is an isomorphism
which allows us to factorise elements of $Z$:

$\Phi:T^{\vee}\times(U^{\vee}\cap B_{-}^{\vee}\bar{w}_{0}B_{-}^{\vee})\to
Z,\quad(d,u_{1})\mapsto u_{1}d\bar{w}_{0}u_{2}.$

Here $u_{2}\in U^{\vee}$ is the unique element such that
$u_{1}d\bar{w}_{0}u_{2}\in Z$. Then, the string toric chart on $Z$
(corresponding to $\mathbf{i}$) is defined to be

$\varphi_{\mathbf{i}}:T^{\vee}\times(\mathbb{K}^{*})^{N}\to
Z,\quad\varphi_{\mathbf{i}}(d,\boldsymbol{z})=\Phi\circ\tau\left(d,\mathbf{x}_{-\mathbf{i}}^{\vee}(\boldsymbol{z})\right)=\Phi\left(d,\iota\left(\eta^{w_{0},e}\left(\mathbf{x}_{-\mathbf{i}}^{\vee}(\boldsymbol{z})\right)\right)\right).$
(1)

Later in this work we will need the specific toric chart corresponding to

$\mathbf{i}_{0}=(i_{1},\ldots,i_{N}):=(1,2,\ldots,n-1,1,2\ldots,n-2,\ldots,1,2,1)$

so, unless otherwise stated, from now on we will take

$u:=\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}(\boldsymbol{z}),\quad
u_{1}:=\iota(\eta^{w_{0},e}(u)),\quad b:=u_{1}d\bar{w}_{0}u_{2}.$

With this notation the composition of maps defining
$\varphi_{\mathbf{i}_{0}}$, the string toric chart corresponding to
$\mathbf{i}_{0}$, may be visualised as follows:

${T^{\vee}\times(\mathbb{K}^{*})^{N}}$${T^{\vee}\times(B_{-}^{\vee}\cap
U^{\vee}\bar{w}_{0}U^{\vee})}$${T^{\vee}\times(U^{\vee}\cap
B_{-}^{\vee}\bar{w}_{0}B_{-}^{\vee})}$${Z}$${(d,\boldsymbol{z})}$${(d,u)}$${(d,u_{1})}$${b}$$\scriptstyle{{\left(id,\,\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}\right)}}$$\scriptstyle{\tau}$$\scriptstyle{\Phi}$

######  Example 2.3.1 (Dimension 3).

The reduced expression is $\mathbf{i}_{0}=(1,2,1)$. We will start with
$\left(d,(z_{1},z_{2},z_{3})\right)$ and apply the sequence of maps defined
above.

Applying $\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}$ to $(z_{1},z_{2},z_{3})$ gives
the matrix $u$:

$u=\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}(z_{1},z_{2},z_{3})=\begin{pmatrix}\frac{1}{z_{1}z_{3}}&&\\\
\frac{1}{z_{3}}+\frac{z_{1}}{z_{2}}&\frac{z_{1}z_{3}}{z_{2}}&\\\
1&z_{3}&z_{2}\end{pmatrix},\quad\bar{w}_{0}=\begin{pmatrix}&&1\\\ &-1&\\\
1&&\end{pmatrix}.$

We recall the definition of second map:
$\tau\left(d,(z_{1},z_{2},z_{3})\right)=\left(d,\iota\circ\eta^{w_{0},e}(u)\right)$.
To see this in action we first evaluate $\eta^{w_{0},e}(u)$ and then apply
$\iota$ to obtain the matrix $u_{1}$:

$\displaystyle\eta^{w_{0},e}(u)$
$\displaystyle=\left[\left(\bar{w}_{0}\begin{pmatrix}\frac{1}{z_{1}z_{3}}&\frac{z_{1}}{z_{2}}+\frac{1}{z_{3}}&1\\\
&\frac{z_{1}z_{3}}{z_{2}}&z_{3}\\\
&&z_{2}\end{pmatrix}\right)^{-1}\right]_{+}=\left[\begin{pmatrix}1&z_{1}+\frac{z_{2}}{z_{3}}&z_{1}z_{3}\\\
-\frac{1}{z_{1}}&-\frac{z_{2}}{z_{1}z_{3}}&\\\
\frac{1}{z_{2}}&&\end{pmatrix}\right]_{+}$
$\displaystyle=\begin{pmatrix}1&z_{1}+\frac{z_{2}}{z_{3}}&z_{1}z_{3}\\\
0&1&z_{3}\\\ 0&0&1\end{pmatrix}$
$u_{1}=\iota\left(\eta^{w_{0},e}(\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}(\boldsymbol{z}))\right)=\left(\bar{w}_{0}\begin{pmatrix}1&z_{1}+\frac{z_{2}}{z_{3}}&z_{1}z_{3}\\\
0&1&z_{3}\\\
0&0&1\end{pmatrix}^{-1}\bar{w}_{0}^{-1}\right)^{T}=\begin{pmatrix}1&z_{3}&z_{2}\\\
&1&z_{1}+\frac{z_{2}}{z_{3}}\\\ &&1\end{pmatrix}$

Of note, we can factorise this matrix using a different reduced expression;
$\mathbf{i}^{\prime}_{0}=(2,1,2)$. We obtain

$\tau\left(d,\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}\left(z_{1},z_{2},z_{3}\right)\right)=\left(d,u_{1}\right)=\left(d,\mathbf{x}_{\mathbf{i}^{\prime}_{0}}^{\vee}\left(z_{1},z_{3},\frac{z_{2}}{z_{3}}\right)\right).$

It remains to apply the map $\Phi$. We recall that $u_{2}\in U^{\vee}$ will be
the unique element such that $b=u_{1}d\bar{w}_{0}u_{2}\in Z$.

$\displaystyle b$
$\displaystyle=\Phi\left(d,u_{1}\right)=u_{1}d\bar{w}_{0}u_{2}$
$\displaystyle=\begin{pmatrix}1&z_{3}&z_{2}\\\ &1&z_{1}+\frac{z_{2}}{z_{3}}\\\
&&1\end{pmatrix}\begin{pmatrix}d_{1}&&\\\ &d_{2}&\\\
&&d_{3}\end{pmatrix}\begin{pmatrix}&&1\\\ &-1&\\\
1&&\end{pmatrix}\begin{pmatrix}1&\frac{d_{2}}{d_{3}}\frac{z_{3}}{z_{2}}&\frac{d_{1}}{d_{3}}\frac{1}{z_{1}z_{3}}\\\
&1&\frac{d_{1}}{d_{2}}\left(\frac{1}{z_{3}}+\frac{z_{2}}{z_{1}z_{3}^{2}}\right)\\\
&&1\end{pmatrix}$ $\displaystyle=\begin{pmatrix}d_{3}z_{2}&&\\\
d_{3}\left(z_{1}+\frac{z_{2}}{z_{3}}\right)&d_{2}\frac{z_{1}z_{3}}{z_{2}}&\\\
d_{3}&d_{2}\frac{z_{3}}{z_{2}}&d_{1}\frac{1}{z_{1}z_{3}}\end{pmatrix}$

From our construction of $b$ we see that the superpotential is

$\mathcal{W}(d,\boldsymbol{z})=z_{1}+\frac{z_{2}}{z_{3}}+z_{3}+\frac{d_{1}}{d_{2}}\left(\frac{1}{z_{3}}+\frac{z_{2}}{z_{1}z_{3}^{2}}\right)+\frac{d_{2}}{d_{3}}\frac{z_{3}}{z_{2}}$

and the weight matrix is

$\mathrm{wt}(b)=\begin{pmatrix}d_{3}z_{2}&&\\\
&d_{2}\frac{z_{1}z_{3}}{z_{2}}&\\\ &&d_{1}\frac{1}{z_{1}z_{3}}\end{pmatrix}.$
(2)

###  2.4 The form of the weight matrix

In this section we generalise the formula for the weight matrix
$\mathrm{wt}(b)=t_{R}$ in terms of the string coordinates,
$(d,\boldsymbol{z})$, as in the above example, Equation (2). We begin with the
two factorisations of $b$ that we have already seen:

$b=u_{1}d\bar{w}_{0}u_{2}=[b]_{-}t_{R}\quad\text{where}\quad[b]_{-}\in
U^{\vee}_{-}.$

Now recalling the involution $\iota$, we see this acts on elements $b\in Z$ as

$\iota(b)=\iota(u_{1})\bar{w}_{0}d^{-1}\iota(u_{2}).$

Defining $\tilde{u}_{i}:=\iota(u_{i})$ and $\tilde{b}:=\iota(b)$ gives
$\tilde{b}=\tilde{u}_{1}\bar{w}_{0}d^{-1}\tilde{u}_{2}$. Moreover we can write

$\tilde{b}=\left[\tilde{b}\right]_{-}\tilde{t}_{R}\quad\text{where}\quad\left[\tilde{b}\right]_{-}\in
U^{\vee}_{-},\ \tilde{t}_{R}\in T^{\vee}.$

Then

$\tilde{b}=\iota(b)=\left(\bar{w}_{0}b^{-1}\bar{w}_{0}^{-1}\right)^{T}=\left(\bar{w}_{0}\left([b]_{-}t_{R}\right)^{-1}\bar{w}_{0}^{-1}\right)^{T}=\left(\bar{w}_{0}[b]_{-}^{-1}\bar{w}_{0}^{-1}\right)^{T}\bar{w}_{0}t_{R}^{-1}\bar{w}_{0}^{-1}.$

Thus $\tilde{t}_{R}=\bar{w}_{0}t_{R}^{-1}\bar{w}_{0}^{-1}$ or equivalently,
$t_{R}=\bar{w}_{0}\tilde{t}_{R}^{-1}\bar{w}_{0}^{-1}$.

######  Proposition 2.4.1.

Let $N=\binom{n}{2}$ and recall the definition
$u:=\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}(\boldsymbol{z})$ where

$\mathbf{i}_{0}=(i_{1},\ldots,i_{N}):=(1,2,\ldots,n-1,1,2\ldots,n-2,\ldots,1,2,1).$

Then $t_{R}=\bar{w}_{0}d[u]_{0}\bar{w}_{0}^{-1}$.

The proof of Proposition 2.4.1 will require the following lemma:

######  Lemma 2.4.2.

Let $\mathbf{i}=(i_{1},\ldots,i_{N})$ stand for a reduced expression
$s_{i_{1}}\cdots s_{i_{N}}$ for $w_{0}$ and take
$\boldsymbol{z}\in\mathbb{K}^{N}$. If
$A=\mathbf{x}_{-i_{1}}^{\vee}(z_{1})\cdots\mathbf{x}_{-i_{N}}^{\vee}(z_{N})$
and $I$ denotes the identity matrix, then

$\left[\left(\bar{w}_{0}A^{T}\right)^{-1}\right]_{0}=I.$

###### Proof of Proposition 2.4.1.

Since $t_{R}=\bar{w}_{0}\tilde{t}_{R}^{-1}\bar{w}_{0}^{-1}$ we will in fact
prove that $\tilde{t}_{R}=d^{-1}[u]_{0}^{-1}$. Recalling the definition

$u_{1}:=\iota(\eta^{w_{0},e}(u))$

we see that

$\tilde{u}_{1}:=\iota(u_{1})=\eta^{w_{0},e}(u)=\left[\left(\bar{w}_{0}u^{T}\right)^{-1}\right]_{+}.$

We have

$\left(u^{T}\right)^{-1}=\left(\bar{w}_{0}u^{T}\right)^{-1}\bar{w}_{0}=\left[\left(\bar{w}_{0}u^{T}\right)^{-1}\right]_{-}\left[\left(\bar{w}_{0}u^{T}\right)^{-1}\right]_{0}\tilde{u}_{1}\bar{w}_{0}.$

In particular by Lemma 2.4.2 we see that

$\left(u^{T}\right)^{-1}\in
U^{\vee}_{-}\tilde{u}_{1}\bar{w}_{0}\quad\Rightarrow\quad
u^{-1}\in\bar{w}_{0}^{-1}\tilde{u}_{1}^{T}U^{\vee}_{+}.$

Thus if $\lambda\in{X^{*}(T)}^{+}$ is a dominant integral weight and we denote
a corresponding highest weight vector by $v^{+}_{\lambda}\in V_{\lambda}$,
then

$u^{-1}\cdot v^{+}_{\lambda}=\bar{w}_{0}^{-1}\tilde{u}_{1}^{T}\cdot
v^{+}_{\lambda}.$

This expression allows for two computations of the coefficient of the highest
weight vector in $u^{-1}\cdot v^{+}_{\lambda}$. Firstly, since
$u=[u]_{0}[u]_{-}$ we have

$\left\langle u^{-1}\cdot
v^{+}_{\lambda},v^{+}_{\lambda}\right\rangle=\lambda\left([u]_{0}^{-1}\right).$

Secondly, we rewrite $\bar{w}_{0}^{-1}\tilde{u}_{1}^{T}$ using
$\tilde{b}=\tilde{u}_{1}\bar{w}_{0}d^{-1}\tilde{u}_{2}=\left[\tilde{b}\right]_{-}\tilde{t}_{R}$;

$\bar{w}_{0}^{-1}\tilde{u}_{1}^{T}=\bar{w}_{0}^{-1}\left(\tilde{b}\tilde{u}_{2}^{-1}d\bar{w}_{0}^{-1}\right)^{T}=\bar{w}_{0}^{-1}\bar{w}_{0}d^{T}\left(\tilde{u}_{2}^{-1}\right)^{T}\tilde{b}^{T}=d\left(\tilde{u}_{2}^{-1}\right)^{T}\tilde{t}_{R}\left[\tilde{b}\right]_{-}^{T}.$

Then we see that the result follows from the second computation;

$\displaystyle\lambda\left([u]_{0}^{-1}\right)=\left\langle u^{-1}\cdot
v^{+}_{\lambda},v^{+}_{\lambda}\right\rangle=\left\langle\bar{w}_{0}^{-1}\tilde{u}_{1}^{T}\cdot
v^{+}_{\lambda},\ v^{+}_{\lambda}\right\rangle$
$\displaystyle=\big{\langle}d\left(\tilde{u}_{2}^{-1}\right)^{T}\tilde{t}_{R}\left[\tilde{b}\right]_{-}^{T}\cdot
v^{+}_{\lambda},\ v^{+}_{\lambda}\big{\rangle}$
$\displaystyle=\lambda(\tilde{t}_{R})\lambda(d)\big{\langle}\left(\tilde{u}_{2}^{-1}\right)^{T}\left[\tilde{b}\right]_{-}^{T}\cdot
v^{+}_{\lambda},\
v^{+}_{\lambda}\big{\rangle}=\lambda(\tilde{t}_{R})\lambda(d).$

∎

###### Proof of Lemma 2.4.2.

Taking the decomposition

$\left(\bar{w}_{0}A^{T}\right)^{-1}=\left[\left(\bar{w}_{0}A^{T}\right)^{-1}\right]_{-}\left[\left(\bar{w}_{0}A^{T}\right)^{-1}\right]_{0}\left[\left(\bar{w}_{0}A^{T}\right)^{-1}\right]_{+}$

and defining

$U:=\left[\left(\bar{w}_{0}A^{T}\right)^{-1}\right]_{+}^{-1}\in U^{\vee}_{+}\
,\quad D:=\left[\left(\bar{w}_{0}A^{T}\right)^{-1}\right]_{0}^{-1}\in
T^{\vee}\ ,\quad
L:=\left[\left(\bar{w}_{0}A^{T}\right)^{-1}\right]_{-}^{-1}\in U^{\vee}_{-}$

we see that

$\left(\bar{w}_{0}A^{T}\right)^{-1}=L^{-1}D^{-1}U^{-1}\quad\Rightarrow\quad
A^{T}=\bar{w}_{0}^{-1}UDL.$

We will show that $D=I$.

We recall

$\mathbf{x}_{-i}^{\vee}(z)^{T}=\phi^{\vee}_{i}\begin{pmatrix}z^{-1}&1\\\
0&z\end{pmatrix}\ ,\quad\bar{s}_{i}^{-1}=\phi_{i}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}.$

For any fundamental representation
$V_{\omega_{i}}=\bigwedge^{i}\mathbb{C}^{n}$, applying $\bar{w}_{0}^{-1}$ to a
lowest weight vector $v^{-}_{\omega_{i}}=v_{n-i+1}\wedge\cdots\wedge v_{n}$
gives the corresponding highest weight vector
$v^{+}_{\omega_{i}}=v_{1}\wedge\cdots\wedge v_{i}$, so

$\left\langle\bar{w}_{0}^{-1}\cdot v^{-}_{\omega_{i}},\
v^{+}_{\omega_{i}}\right\rangle=1.$

Similarly applying
$A^{T}=\mathbf{x}_{-i_{N}}^{\vee}(z_{N})^{T}\cdots\mathbf{x}_{-i_{1}}^{\vee}(z_{1})^{T}$
gives

$\left\langle A^{T}\cdot v^{-}_{\omega_{i}},\
v^{+}_{\omega_{i}}\right\rangle=\left\langle\mathbf{x}_{-i_{N}}^{\vee}(z_{N})^{T}\cdots\mathbf{x}_{-i_{1}}^{\vee}(z_{1})^{T}\cdot
v^{-}_{\omega_{i}},\ v^{+}_{\omega_{i}}\right\rangle=1.$

We may also evaluate the coefficient of the highest weight vector in
$A^{T}\cdot v^{-}_{\lambda}$ using the expression $A^{T}=\bar{w}_{0}^{-1}UDL$;

$\displaystyle\left\langle A^{T}\cdot v^{-}_{\omega_{i}},\
v^{+}_{\omega_{i}}\right\rangle=\left\langle\bar{w}_{0}^{-1}UDL\cdot
v^{-}_{\omega_{i}},\
v^{+}_{\omega_{i}}\right\rangle=\left\langle\bar{w}_{0}^{-1}UD\cdot
v^{-}_{\omega_{i}},\ v^{+}_{\omega_{i}}\right\rangle$
$\displaystyle=\omega_{i}(D)\left\langle\bar{w}_{0}^{-1}U\cdot
v^{-}_{\omega_{i}},\ v^{+}_{\omega_{i}}\right\rangle$
$\displaystyle=\omega_{i}(D)\left\langle\bar{w}_{0}^{-1}\cdot
v^{-}_{\omega_{i}},\ v^{+}_{\omega_{i}}\right\rangle=\omega_{i}(D).$

Thus $\omega_{i}(D)=1$ for all $i$, so $D=I$. ∎

Now that we better understand the weight matrix $\mathrm{wt}(b)=t_{R}$, we
complete this section by expressing it directly in terms of
$(d,\boldsymbol{z})$.

######  Corollary 2.4.3.

The weight matrix $t_{R}$, given in terms of the string coordinates
$(d,\boldsymbol{z})$, is the diagonal matrix with entries

$\left(t_{R}\right)_{n-j+1,n-j+1}=\frac{d_{j}\prod_{\begin{subarray}{c}1\leq
m\leq N\\\ i_{m}=j-1\end{subarray}}z_{m}}{\prod_{\begin{subarray}{c}1\leq
m\leq N\\\ i_{m}=j\end{subarray}}z_{m}},\quad j=1,\ldots,n.$ (3)

###### Proof.

We recall the matrix
$u=\mathbf{x}_{-i_{1}}^{\vee}(z_{1})\cdots\mathbf{x}_{-i_{N}}^{\vee}(z_{N})$
where

$\mathbf{i}_{0}=(i_{1},\ldots,i_{N}):=(1,2,\ldots,n-1,1,2\ldots,n-2,\ldots,1,2,1)$

and

$\mathbf{x}_{-i}^{\vee}(z)=\phi_{i}\begin{pmatrix}z^{-1}&0\\\
1&z\end{pmatrix}.$

We see that

$\displaystyle\left([u]_{0}\right)_{jk}$
$\displaystyle=\begin{cases}\prod_{m=1}^{N}\left(\mathbf{x}_{-i_{m}}^{\vee}(z_{m})\right)_{jj}&\text{if}\
j=k\\\ 0&\text{if}\ j\neq k\end{cases}$
$\displaystyle=\begin{cases}\frac{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j-1\end{subarray}}z_{m}}{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j\end{subarray}}z_{m}}&\text{if}\ j=k\\\ 0&\text{if}\ j\neq
k\end{cases}$ (4)

noting that $\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=0\end{subarray}}z_{m}=1=\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=n\end{subarray}}z_{m}$. Thus since
$t_{R}=\bar{w}_{0}d[u]_{0}\bar{w}_{0}^{-1}$ we see that it is the diagonal
matrix with entries

$\left(t_{R}\right)_{n-j+1,n-j+1}=\frac{d_{j}\prod_{\begin{subarray}{c}1\leq
m\leq N\\\ i_{m}=j-1\end{subarray}}z_{m}}{\prod_{\begin{subarray}{c}1\leq
m\leq N\\\ i_{m}=j\end{subarray}}z_{m}},\quad j=1,\ldots,n.$

∎

##  3 The ideal coordinates

Our preferred coordinate system on $Z$, which we will call the ideal
coordinate system, is far more natural than the string toric chart since it is
easier to define. We begin by recalling the reduced expression

$\mathbf{i}_{0}=(i_{1},\ldots,i_{N}):=(1,2,\ldots,n-1,1,2\ldots,n-2,\ldots,1,2,1)$

and consider the map

$\left(\mathbb{K}^{*}\right)^{N}\times T^{\vee}\to
Z\,,\quad\left((m_{1},\ldots,m_{N}),t_{R}\right)\mapsto\mathbf{y}_{i_{1}}^{\vee}\left(\frac{1}{m_{1}}\right)\cdots\mathbf{y}_{i_{N}}^{\vee}\left(\frac{1}{m_{N}}\right)t_{R}.$

We recall that $Z$ has two projections to $T^{\vee}$, given by the highest
weight and weight maps. In the previous coordinate system the highest weight
map was obvious due to the form of $b=u_{1}d\bar{w}_{0}u_{2}$, whereas the
weight required more effort to compute. In this new system the weight is much
more straightforward.

We could consider this system with coordinates $(\boldsymbol{m},t_{R})$, but
instead, for ease of later application, we wish to work with coordinates
$(d,\boldsymbol{m})$ which we call the ideal coordinates111 The choice to work
with the inverted coordinates $\frac{1}{m_{i}}$ is motivated by the main
theorem of Section 4, Theorem 4.3.1, which also gives rise to the name for
this coordinate system. :

$\psi:T^{\vee}\times\left(\mathbb{K}^{*}\right)^{N}\to
Z\,,\quad\left(d,(m_{1},\ldots,m_{N})\right)\mapsto\mathbf{y}_{i_{1}}^{\vee}\left(\frac{1}{m_{1}}\right)\cdots\mathbf{y}_{i_{N}}^{\vee}\left(\frac{1}{m_{N}}\right)t_{R}(d,\boldsymbol{m}).$

Here $t_{R}(d,\boldsymbol{m})$ is the weight matrix given now in terms of the
coordinates $(d,\boldsymbol{m})$. An explicit description of the map

$T^{\vee}\times\left(\mathbb{K}^{*}\right)^{N}\to
T^{\vee}\,,\quad\left(d,\boldsymbol{m}\right)\mapsto t_{R}(d,\boldsymbol{m})$

will be given shortly.

First, we present the main theorem of this section. For a given highest weight
matrix, $d$, this theorem describes the coordinate change from the string
coordinates to the ideal coordinates, allowing us to move freely between the
two systems:

######  Theorem 3.0.1.

To change from the string coordinates, $(d,\boldsymbol{z})$, to the ideal
coordinates, $(d,\boldsymbol{m})$, we first let

$s_{k}:=\sum_{j=1}^{k-1}(n-j)$

then for $k=1,\ldots,n-1$, $a=1,\ldots,n-k$ we have

$m_{s_{k}+a}=\begin{cases}z_{1+s_{n-a}}&\text{if }k=1,\\\
\frac{z_{k+s_{n-k-a+1}}}{z_{k-1+s_{n-k-a+1}}}&\text{otherwise}.\end{cases}$

The string and ideal coordinate systems are related by repeated application of
a theorem known as the Chamber Ansatz, which we will discuss in Section 3.1.
We will then develop our understanding of the relation between the string and
ideal coordinates in Sections 3.2 and 3.3, culminating in the proof of Theorem
3.0.1. To complete the current section we present an example followed by a
further application of this theorem.

######  Example 3.0.2.

In dimension 3 the the coordinate change is

$m_{1}=z_{3},\quad m_{2}=z_{1},\quad m_{3}=\frac{z_{2}}{z_{1}}\quad\qquad
z_{1}=m_{2},\quad z_{2}=m_{2}m_{3},\quad z_{3}=m_{1}.$

In Example 2.3.1 we saw that the matrix $b$ was given by

$b=\Phi\left(d,\mathbf{x}_{\mathbf{i}^{\prime}_{0}}^{\vee}\left(z_{1},z_{3},\frac{z_{2}}{z_{3}}\right)\right)=\begin{pmatrix}d_{3}z_{2}&&\\\
d_{3}\left(z_{1}+\frac{z_{2}}{z_{3}}\right)&d_{2}\frac{z_{1}z_{3}}{z_{2}}&\\\
d_{3}&d_{2}\frac{z_{3}}{z_{2}}&d_{1}\frac{1}{z_{1}z_{3}}\end{pmatrix}.$

With the new coordinates we have

$b=\begin{pmatrix}d_{3}m_{2}m_{3}&&\\\
d_{3}\left(m_{2}+\frac{m_{2}m_{3}}{m_{1}}\right)&d_{2}\frac{m_{1}}{m_{3}}&\\\
d_{3}&d_{2}\frac{m_{1}}{m_{2}m_{3}}&d_{1}\frac{1}{m_{1}m_{2}}\end{pmatrix}=\mathbf{y}_{\mathbf{i}_{0}}^{\vee}\left(\frac{1}{m_{1}},\frac{1}{m_{2}},\frac{1}{m_{3}}\right)\begin{pmatrix}d_{3}m_{2}m_{3}&&\\\
&d_{2}\frac{m_{1}}{m_{3}}&\\\ &&d_{1}\frac{1}{m_{1}m_{2}}\end{pmatrix}.$

Additionally the superpotential is now given by

$\mathcal{W}(d,\boldsymbol{m})=m_{1}+m_{2}+\frac{m_{2}m_{3}}{m_{1}}+\frac{d_{2}}{d_{3}}\frac{m_{1}}{m_{2}m_{3}}+\frac{d_{1}}{d_{2}}\left(\frac{m_{3}}{{m_{1}}^{2}}+\frac{1}{m_{1}}\right).$

We again also give the weight matrix:

$\mathrm{wt}(b)=\begin{pmatrix}d_{3}m_{2}m_{3}&&\\\
&d_{2}\frac{m_{1}}{m_{3}}&\\\ &&d_{1}\frac{1}{m_{1}m_{2}}\end{pmatrix}.$

We now present an application of Theorem 3.0.1 on the weight matrix, namely we
describe $\mathrm{wt}(b)=t_{R}$ in terms of the ideal coordinates.

######  Corollary 3.0.3.

The weight matrix $t_{R}$, given in terms of the ideal coordinates
$(d,\boldsymbol{m})$, is the diagonal matrix with entries

$\left(t_{R}\right)_{n-j+1,n-j+1}=\frac{d_{j}\prod\limits_{k=1,\ldots,j-1}m_{s_{k}+(j-k)}}{\prod\limits_{r=1,\ldots,n-j}m_{s_{j}+r}},\quad
j=1,\ldots,n,\ \text{with}\ r=k-j.$ (5)

###### Proof.

Recalling Corollary 2.4.3, we see that we need to show

$\frac{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j-1\end{subarray}}z_{m}}{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j\end{subarray}}z_{m}}=\frac{\prod\limits_{k=1,\ldots,j-1}m_{s_{k}+(j-k)}}{\prod\limits_{r=1,\ldots,n-j}m_{s_{j}+r}},\quad\text{where}\
r=k-j.$ (6)

The denominator of the left hand side of (6) is

$\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j\end{subarray}}z_{m}=\prod_{\begin{subarray}{c}m=j+s_{r}\\\
r=1,\ldots,n-j\end{subarray}}z_{m}=\prod_{r=1,\ldots,n-j}z_{j+s_{n-j-r+1}}.$

Similarly the numerator is

$\displaystyle\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j-1\end{subarray}}z_{m}$ $\displaystyle=\begin{cases}1&\text{if }j=1\\\
\prod\limits_{\begin{subarray}{c}m=j-1+s_{r}\\\
r=1,\ldots,n-j+1\end{subarray}}z_{m}&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}1&\text{if }j=1\\\
\prod\limits_{r=0,\ldots,n-j}z_{j-1+s_{n-j-r+1}}&\text{otherwise}.\end{cases}$

Consequently if $j=1$ the left hand side of (6) becomes

$\frac{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j-1\end{subarray}}z_{m}}{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j\end{subarray}}z_{m}}=\frac{1}{\prod\limits_{r=1,\ldots,n-j}z_{j+s_{n-j-r+1}}}=\frac{1}{\prod\limits_{r=1,\ldots,n-j}m_{s_{j}+r}}$

as desired.

If $j\geq 2$ then the left hand side of (6) is

$\frac{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j-1\end{subarray}}z_{m}}{\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=j\end{subarray}}z_{m}}=\frac{\prod\limits_{r=0,\ldots,n-j}z_{j-1+s_{n-j-r+1}}}{\prod\limits_{r=1,\ldots,n-j}z_{j+s_{n-j-r+1}}}\\\
=z_{j-1+s_{n-j+1}}\prod_{r=1,\ldots,n-j}\frac{1}{m_{s_{j}+r}}.$

It remains to show that

$\prod\limits_{k=1,\ldots,j-1}m_{s_{k}+(j-k)}=z_{j-1+s_{n-j+1}}.$

Indeed from the coordinate change formula in Theorem 3.0.1, for $j\geq 2$ we
see

$m_{s_{k}+(j-k)}=\frac{z_{k+s_{n-k-(j-k)+1}}}{z_{k-1+s_{n-k-(j-k)+1}}}=\frac{z_{k+s_{n-j+1}}}{z_{k-1+s_{n-j+1}}}.$

So the product becomes telescopic and, as desired, we obtain

$\prod\limits_{k=1,\ldots,j-1}m_{s_{k}+(j-k)}=z_{1+s_{n-j+1}}\prod\limits_{k=2,\ldots,j-1}\frac{z_{k+s_{n-j+1}}}{z_{k-1+s_{n-j+1}}}=z_{j-1+s_{n-j+1}}.$

∎

###  3.1 The Chamber Ansatz

In order to prove Theorem 3.0.1 we require a sequence of lemmas, the first of
which (Lemma 3.1.4) makes use of the afore mentioned Chamber Ansatz. In this
section we introduce the Chamber Ansatz and then state and prove Lemma 3.1.4.

######  Definition 3.1.1.

Let $J=\\{j_{1}<\cdots<j_{l}\\}\subseteq[1,n]$ and
$K=\\{k_{1}<\cdots<k_{l}\\}\subseteq[1,n]$. The pair $(J,K)$ is called
admissible if $j_{s}\leq k_{s}$ for $s=1,\ldots,l$.

For such an admissible pair $(J,K)$, we denote by $\Delta^{J}_{K}$ the
$l\times l$ minor with row and column sets defined by $J$ and $K$
respectively.

We now state a specific case of the Generalised Chamber Ansatz presented by
Marsh and Rietsch in [21, Theorem 7.1]:

######  Theorem 3.1.2 (Chamber Ansatz).

Let $B=z\bar{w}_{0}B^{\vee}_{+}\in\mathcal{R}^{\vee}_{e,w_{0}}$, where $z\in
U^{\vee}_{+}$. Let $\mathbf{w}=(w_{(0)},w_{(1)},\ldots,w_{(n)})$ be a sequence
of partial products for $w_{0}$ defined by its sequence of factors

$\left(w_{(1)},w_{(1)}^{-1}w_{(2)},\ldots,w_{(N-1)}^{-1}w_{(N)}\right)=(s_{i_{1}},\ldots,s_{i_{N}}).$

Then there is an element

$g=\mathbf{y}_{i_{1}}(t_{1})\mathbf{y}_{i_{2}}(t_{2})\cdots\mathbf{y}_{i_{N}}(t_{N})\in
U^{\vee}_{-}\cap B^{\vee}_{+}\bar{w}_{0}B^{\vee}_{+}$

such that $B=gB^{\vee}_{+}$. Moreover for $k=1,\ldots,N$ we have

$t_{k}=\frac{\prod_{j\neq
i_{k}}\Delta^{\omega^{\vee}_{j}}_{w_{(k)}\omega^{\vee}_{j}}(z)^{-a_{j,i_{k}}}}{\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(k)}\omega^{\vee}_{i_{k}}}(z)\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(k-1)}\omega^{\vee}_{i_{k}}}(z)}.$

Each $\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(k)}\omega^{\vee}_{i_{k}}}$, where
$\omega^{\vee}_{i_{k}}$ ranges through the set of fundamental weights, is
called a (standard) chamber minor. As above, it is given by the $i_{k}\times
i_{k}$ minor with $\omega^{\vee}_{i_{k}}$ encoding the row set and
$w_{(k)}\omega^{\vee}_{i_{k}}$ encoding the column set. We note that these row
and column sets form admissible pairs.

Much of the information in the Chamber Ansatz may be read from an associated
pseudoline arrangement; it may be viewed as a singular braid diagram and is
called an ansatz arrangement. In dimension $n$, for the case we are
considering, the ansatz arrangement consists of $n$ pseudolines. These are
numbered from bottom to top on the left side of the arrangement.

Each factor $g_{k}=\mathbf{y}_{i_{k}}^{\vee}(t_{k})$ of $g$ gives rise to a
crossing of the pseudolines at level $i_{k}$. We label each chamber in the
diagram with the labels of the strands passing below it and associate to the
chamber with label $S$ the flag minor $\Delta^{\left[1,|S|\right]}_{S}$.

If $A_{k}$, $B_{k}$, $C_{k}$ and $D_{k}$ are the minors corresponding to the
chambers surrounding the $k$-th singular point, with $A_{k}$ and $D_{k}$ above
and below it and $B_{k}$ and $C_{k}$ to the left and right, then the Chamber
Ansatz gives

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize{$D_{k}$}}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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t_{k}=\frac{A_{k}D_{k}}{B_{k}C_{k}}.$

Let $N=\binom{n}{2}$ and again take

$\begin{gathered}u:=\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}(\boldsymbol{z}),\quad
u_{1}:=\iota(\eta^{w_{0},e}(u)),\quad b:=u_{1}d\bar{w}_{0}u_{2},\\\
\mathbf{i}_{0}=(i_{1},\ldots,i_{N}):=(1,2,\ldots,n-1,1,2\ldots,n-2,\ldots,1,2,1).\end{gathered}$

We will also need

$\mathbf{i}^{\prime}_{0}=(i^{\prime}_{1},\ldots,i^{\prime}_{N}):=(n-1,n-2,\ldots,1,n-1,n-2,\ldots,2,\ldots,n-1,n-2,n-1)$

and we use the superscript ‘$\mathrm{op}$’ to denote taking such an expression
in reverse, for example

${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}:=(i^{\prime}_{N},\ldots,i^{\prime}_{1})=(n-1,n-2,n-1,\ldots,n-3,n-2,n-1).$

######  Example 3.1.3 (Ansatz arrangements for $\mathbf{i}_{0}$,
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ in dimension $4$).

Since $n=4$, we have $N=6$. For $\mathbf{i}_{0}=(1,2,3,1,2,1)$, the sequence
of partial products for $w_{0}$ is given by

$\mathbf{w}=(w_{(0)},w_{(1)},\ldots,w_{(6)})=(e,s_{1},s_{1}s_{2},s_{1}s_{2}s_{3},s_{1}s_{2}s_{3}s_{1},s_{1}s_{2}s_{3}s_{1}s_{2},s_{1}s_{2}s_{3}s_{1}s_{2}s_{1}).$

The ansatz arrangement for $\mathbf{i}_{0}$ is given in Figure 4.

For ${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}=(3,2,3,1,2,3)$, the sequence of
partial products for $w_{0}$ is

$\mathbf{w}=(w_{(0)},w_{(1)},\ldots,w_{(6)})=(e,s_{3},s_{3}s_{2},s_{3}s_{2}s_{3},s_{3}s_{2}s_{3}s_{1},s_{3}s_{2}s_{3}s_{1}s_{2},s_{3}s_{2}s_{3}s_{1}s_{2}s_{3}).$

The ansatz arrangement for ${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ is given
in Figure 4.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$1$$2$$3$$4$$123$$234$$12$$23$$34$$1$$2$$3$$4$
Figure 3: The ansatz arrangement for $\mathbf{i}_{0}$ in dimension 4

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$1$$2$$3$$4$$123$$124$$134$$234$$12$$14$$34$$1$$4$
Figure 4: The ansatz arrangement for ${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$
in dimension 4

We are now ready to state the first lemma needed for the proof of Theorem
3.0.1:

######  Lemma 3.1.4.

We can factorise $u_{1}$ and $b$ as follows:

$\displaystyle u_{1}$
$\displaystyle=\mathbf{x}_{i^{\prime}_{1}}^{\vee}(p_{1})\cdots\mathbf{x}_{i^{\prime}_{N}}^{\vee}(p_{N}),$
(7) $\displaystyle b$
$\displaystyle=\mathbf{y}_{i_{1}}^{\vee}\left(\frac{1}{m_{1}}\right)\cdots\mathbf{y}_{i_{N}}^{\vee}\left(\frac{1}{m_{N}}\right)t_{R}.$
(8)

The $p_{i}$ and $m_{i}$ are given by the Chamber Ansatz in terms of chamber
minors:

$\displaystyle p_{N-k+1}$ $\displaystyle=\frac{\prod_{j\neq
i^{\prime}_{N-k+1}}\Delta^{\omega^{\vee}_{j}}_{w_{(k)}\omega^{\vee}_{j}}(u^{T})^{-a_{j,i^{\prime}_{N-k+1}}}}{\Delta^{\omega^{\vee}_{i^{\prime}_{N-k+1}}}_{w_{(k)}\omega^{\vee}_{i^{\prime}_{N-k+1}}}(u^{T})\Delta^{\omega^{\vee}_{i^{\prime}_{N-k+1}}}_{w_{(k-1)}\omega^{\vee}_{i^{\prime}_{N-k+1}}}(u^{T})},\quad
k=1,\ldots,N,$ (9) $\displaystyle\frac{1}{m_{k}}$
$\displaystyle=\frac{\prod_{j\neq
i_{k}}\Delta^{\omega^{\vee}_{j}}_{w_{(k)}\omega^{\vee}_{j}}(u_{1})^{-a_{j,i_{k}}}}{\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(k)}\omega^{\vee}_{i_{k}}}(u_{1})\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(k-1)}\omega^{\vee}_{i_{k}}}(u_{1})},\quad
k=1,\ldots,N.$ (10)

###### Proof.

We will use the Chamber Ansatz to prove that $u_{1}$ and $b$ have the
factorisations given in (7) and (8) respectively.

To show (7) we first note that if
$x=\mathbf{x}_{\mathbf{i}}^{\vee}(z_{1},\ldots,z_{N})$ then
$x^{T}=\mathbf{y}_{\mathbf{i}^{\mathrm{op}}}^{\vee}(z_{N},\ldots,z_{1})$. Thus
to apply the Chamber Ansatz we need a matrix $A\in U^{\vee}_{+}$ such that
$u_{1}^{T}B^{\vee}_{+}=A\bar{w}_{0}B^{\vee}_{+}$. We will extract this matrix
from the definition of $u_{1}$:

$u_{1}=\iota(\eta^{w_{0},e}(u))=\iota\left([(\bar{w}_{0}u^{T})^{-1}]_{+}\right).$

After applying the involution $\iota$ we see that

$B^{\vee}_{-}\left(\bar{w}_{0}u^{T}\right)^{-1}=B^{\vee}_{-}\eta^{w_{0},e}(u)=B^{\vee}_{-}\iota(u_{1})=B^{\vee}_{-}\left(\bar{w}_{0}u_{1}^{-1}\bar{w}_{0}^{-1}\right)^{T}=B^{\vee}_{-}\left(\bar{w}_{0}u_{1}^{T}\bar{w}_{0}^{-1}\right)^{-1}.$

Taking the inverse gives

$\bar{w}_{0}u_{1}^{T}\bar{w}_{0}^{-1}B^{\vee}_{-}=\bar{w}_{0}u^{T}B^{\vee}_{-}\quad\Rightarrow\quad
u_{1}^{T}\bar{w}_{0}^{-1}B^{\vee}_{-}=u^{T}B^{\vee}_{-}.$

Then using the relation $\bar{w}_{0}B^{\vee}_{+}\bar{w}_{0}^{-1}=B^{\vee}_{-}$
we obtain the desired form:

$u_{1}^{T}\bar{w}_{0}^{-1}\bar{w}_{0}B^{\vee}_{+}\bar{w}_{0}^{-1}=u^{T}\bar{w}_{0}B^{\vee}_{+}\bar{w}_{0}^{-1}\quad\Rightarrow\quad
u_{1}^{T}B^{\vee}_{+}=u^{T}\bar{w}_{0}B^{\vee}_{+}.$

We take the reduced expression for $w_{0}$ defined by
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ and let
$\mathbf{w}=(w_{(0)},w_{(1)},\ldots,w_{(N)})$ be the sequence of partial
products for $w_{0}$ given by its sequence of factors

$\left(w_{(1)},w_{(1)}^{-1}w_{(2)},\ldots,w_{(N-1)}^{-1}w_{(N)}\right)=(s_{i^{\prime}_{N}},\ldots,s_{i^{\prime}_{1}}).$

Then by the Chamber Ansatz we have
$u_{1}^{T}B^{\vee}_{+}=u^{T}\bar{w}_{0}B^{\vee}_{+}=\mathbf{y}_{i^{\prime}_{N}}^{\vee}(p_{N})\cdots\mathbf{y}_{i^{\prime}_{1}}^{\vee}(p_{1})B^{\vee}_{+}$
with

$p_{N-k+1}=\frac{\prod_{j\neq
i^{\prime}_{N-k+1}}\Delta^{\omega^{\vee}_{j}}_{w_{(k)}\omega^{\vee}_{j}}(u^{T})^{-a_{j,i^{\prime}_{N-k+1}}}}{\Delta^{\omega^{\vee}_{i^{\prime}_{N-k+1}}}_{w_{(k)}\omega^{\vee}_{i^{\prime}_{N-k+1}}}(u^{T})\Delta^{\omega^{\vee}_{i^{\prime}_{N-k+1}}}_{w_{(k-1)}\omega^{\vee}_{i^{\prime}_{N-k+1}}}(u^{T})},\quad
k=1,\ldots,N$

which is exactly in the form of (9). Moreover since both $u_{1}^{T}\in
U^{\vee}_{-}$ and
$\mathbf{y}_{i^{\prime}_{N}}^{\vee}(t^{\prime}_{1})\cdots\mathbf{y}_{i^{\prime}_{1}}^{\vee}(t^{\prime}_{N})\in
U^{\vee}_{-}$ this determines $u_{1}$ completely;
$u_{1}=\mathbf{x}_{i^{\prime}_{1}}^{\vee}(p_{1})\cdots\mathbf{x}_{i^{\prime}_{N}}^{\vee}(p_{N})$
as desired.

To give the factorisation in (8) we note that by definition
$bB^{\vee}_{+}=u_{1}\bar{w}_{0}B^{\vee}_{+}$ with $u_{1}\in U^{\vee}_{+}$. For
this second application of the Chamber Ansatz we let $w_{0}$ be described by
$\mathbf{i}_{0}$. We again take $\mathbf{w}=(w_{(0)},w_{(1)},\ldots,w_{(N)})$
to be the respective sequence of partial products for $w_{0}$ defined by its
sequence of factors

$\left(w_{(1)},w_{(1)}^{-1}w_{(2)},\ldots,w_{(N-1)}^{-1}w_{(N)}\right)=(s_{i_{1}},\ldots,s_{i_{N}}).$

Then by the Chamber Ansatz we have
$bB^{\vee}_{+}=u_{1}\bar{w}_{0}B^{\vee}_{+}=\mathbf{y}_{i_{1}}^{\vee}\left(\frac{1}{m_{1}}\right)\cdots\mathbf{y}_{i_{N}}^{\vee}\left(\frac{1}{m_{N}}\right)B^{\vee}_{+}$
with

$\frac{1}{m_{k}}=\frac{\prod_{j\neq
i_{k}}\Delta^{\omega^{\vee}_{j}}_{w_{(k)}\omega^{\vee}_{j}}(u_{1})^{-a_{j,i_{k}}}}{\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(k)}\omega^{\vee}_{i_{k}}}(u_{1})\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(k-1)}\omega^{\vee}_{i_{k}}}(u_{1})},\quad
k=1,\ldots,N$

which proves (10). ∎

###  3.2 Chamber Ansatz minors

We wish to further describe the coordinate changes defined by our two
applications of the Chamber Ansatz. In this section we show they are monomial
by considering the required minors of $u^{T}$ and $u_{1}$. Similar to how the
ansatz arrangement tells us which quotients of minors to take when applying
Chamber Ansatz, we may use a planar acyclic directed graph to easily compute
these minors, and in particular to confirm that they are all monomial (see [9,
Proposition 4.2], generalising [2, Theorem 2.4.4]). Note that the following
description is slightly different to that given by Fomin and Zelevinsky in
[9], since we do not need the same level of generality.

Let $\mathbf{i}=(i_{1},\ldots,i_{N})$ define some reduced expression for
$w_{0}$ and consider

$\mathbf{x}_{\mathbf{i}}^{\vee}(\boldsymbol{z})=\mathbf{x}_{i_{1}}^{\vee}(z_{1})\cdots\mathbf{x}_{i_{N}}^{\vee}(z_{N}).$

For particular choices of admissible pairs $(J,K)$, we wish to compute the
minors

$\Delta^{J}_{K}\left(\mathbf{x}_{\mathbf{i}}^{\vee}(\boldsymbol{z})\right)\text{
or }\
\Delta^{J}_{K}\left(\mathbf{x}_{-\mathbf{i}}^{\vee}(\boldsymbol{z})^{T}\right).$

In the second case it will be helpful to express
$\mathbf{x}_{-\mathbf{i}}^{\vee}(\boldsymbol{z})^{T}$ as a product of matrices
$\mathbf{x}_{i}$ and $\mathbf{t}_{i}$. To do this we notice

$\mathbf{x}_{-i}^{\vee}(z)^{T}=\phi_{i}^{\vee}\begin{pmatrix}z^{-1}&1\\\
0&z\end{pmatrix}=\phi_{i}^{\vee}\left(\begin{pmatrix}z^{-1}&0\\\
0&z\end{pmatrix}\begin{pmatrix}1&z\\\
0&1\end{pmatrix}\right)=\mathbf{t}_{i}^{\vee}(z^{-1})\mathbf{x}_{i}^{\vee}(z).$

In particular we may rewrite
$\mathbf{x}_{-\mathbf{i}}^{\vee}(\boldsymbol{z})^{T}$ as follows:

$\mathbf{x}_{-\mathbf{i}}^{\vee}(\boldsymbol{z})^{T}=\mathbf{x}_{-i_{N}}^{\vee}(z_{N})^{T}\cdots\mathbf{x}_{-i_{1}}^{\vee}(z_{1})^{T}=\mathbf{t}_{i_{N}}^{\vee}(z_{N}^{-1})\mathbf{x}_{i_{N}}^{\vee}(z_{N})\cdots\mathbf{t}_{i_{1}}^{\vee}(z_{1}^{-1})\mathbf{x}_{i_{1}}^{\vee}(z_{1}).$

To construct the graph, $\Gamma$, corresponding to a matrix
$x=\mathbf{x}_{\mathbf{i}}^{\vee}(\boldsymbol{z})$ or
$\mathbf{x}_{-\mathbf{i}}^{\vee}(\boldsymbol{z})^{T}$ we begin with $n$
parallel horizontal lines. We add vertices to the ends of each line and number
them from bottom to top on both sides. Then for each factor
$\mathbf{t}_{i_{k}}^{\vee}(z_{k}^{-1})$, $\mathbf{x}_{i_{k}}^{\vee}(z_{k})$ of
$x$ we include a labelled line segment and vertices at height $i_{k}$ defined
in Figure 5.

$\bullet$$\bullet$$\bullet$$\bullet$$\frac{1}{z_{k}}$$z_{k}$For factors
$\mathbf{t}_{i_{k}}^{\vee}(z_{k}^{-1})$

$\bullet$$\bullet$$z_{k}$For factors $\mathbf{x}_{i_{k}}^{\vee}(z_{k})$

Figure 5: Labelled line segments in graphs for computing Chamber Ansatz minors

Each line segment is viewed as a labelled edge of $\Gamma$, oriented left to
right. For an edge $e$, the labelling, called the weight of $e$ and denoted
$w(e)$, is given by the diagrams above and taken to be $1$ if left
unspecified. The weight $w(\pi)$ of an oriented path $\pi$ is defined to be
the product of weights $w(e)$ taken over all edges $e$ in $\pi$.

The set of vertices of the graph $\Gamma$ is given by the endpoints of all
line segments. Those vertices appearing as the leftmost (resp. rightmost)
endpoints of the horizontal lines are the sources (resp. sinks) of $\Gamma$.

With this notation, [9, Proposition 4.2] becomes the following:

######  Theorem 3.2.1.

For an admissible pair $(J,K)$ of size $l$

$\Delta^{J}_{K}\left(\mathbf{x}_{\mathbf{i}^{\prime}}^{\vee}(\boldsymbol{z})\right)=\sum_{\pi_{1},\ldots,\pi_{l}}w(\pi_{1})\cdots
w(\pi_{l})$

where the sum is taken over all families of l vertex-disjoint paths
$\\{\pi_{1},\ldots,\pi_{l}\\}$ connecting the sources labelled by $J$ with the
sinks labelled by $K$.

To prove that the minors of $u^{T}$, $u_{1}$ appearing in our applications of
the Chamber Ansatz are monomial we must show that in each case there is only
one possible family of paths $\\{\pi_{1},\ldots,\pi_{l}\\}$. Before showing
this we give two examples to clarify the above construction.

######  Example 3.2.2.

We take $n=4$, so $N=6$, and we wish to compute the minors
$\Delta^{J}_{K}\left(u_{1}\right)$ where

$u_{1}=\mathbf{x}_{\mathbf{i}^{\prime}_{0}}^{\vee}\left(z_{1},z_{4},z_{6},\frac{z_{2}}{z_{4}},\frac{z_{5}}{z_{6}},\frac{z_{3}}{z_{5}}\right)=\begin{pmatrix}1&z_{6}&z_{5}&z_{3}\\\
&1&z_{4}+\frac{z_{5}}{z_{6}}&z_{2}+\frac{z_{3}z_{4}}{z_{5}}+\frac{z_{3}}{z_{6}}\\\
&&1&z_{1}+\frac{z_{2}}{z_{4}}+\frac{z_{3}}{z_{5}}\\\ &&&1\end{pmatrix}$

with $\mathbf{i}^{\prime}_{0}=(3,2,1,3,2,3)$. The graph for $u_{1}$ is given
in Figure 6.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$4$$3$$2$$1$$z_{1}$$z_{4}$$z_{6}$$\frac{z_{2}}{z_{4}}$$\frac{z_{5}}{z_{6}}$$\frac{z_{3}}{z_{5}}$
Figure 6: The graph for $u_{1}$ when $n=4$

Computing the minor $\Delta^{1}_{4}(u_{1})$ directly from the matrix $u_{1}$
is trivial. If we use the graph instead, we see that it is given by the weight
of one path with three diagonal edges:

$\Delta^{1}_{4}(u_{1})=z_{6}\frac{z_{5}}{z_{6}}\frac{z_{3}}{z_{5}}=z_{3}.$

The minor $\Delta^{\\{1,2,3\\}}_{\\{2,3,4\\}}(u_{1})$ takes far more effort to
compute directly, but using the graph makes the computation simple. This minor
is the product of weights of three paths, each with one diagonal edge. We
highlight the paths with parentheses:

$\Delta^{\\{1,2,3\\}}_{\\{2,3,4\\}}(u_{1})=(z_{6})(z_{4})(z_{1})=z_{1}z_{4}z_{6}.$

######  Example 3.2.3.

Again we take $n=4$, so $N=6$ and we wish to compute the minors
$\Delta^{J}_{K}\left(u^{T}\right)$ where

$u^{T}=\left(\mathbf{x}_{-\mathbf{i}_{0}}^{\vee}(z_{1},\ldots,z_{6})\right)^{T}=\begin{pmatrix}\frac{1}{z_{1}z_{4}z_{6}}&\frac{z_{1}z_{4}(z_{4}z_{6}+z_{5})+z_{2}z_{5}}{z_{2}z_{4}z_{5}z_{6}}&\frac{z_{3}z_{5}+z_{6}(z_{2}z_{5}+z_{3}z_{4})}{z_{3}z_{5}z_{6}}&1\\\
&\frac{z_{1}z_{4}z_{6}}{z_{2}z_{5}}&\frac{z_{6}(z_{2}z_{5}+z_{3}z_{4})}{z_{3}z_{5}}&z_{6}\\\
&&\frac{z_{2}z_{5}}{z_{3}}&z_{5}\\\ &&&z_{3}\end{pmatrix}$

with $\mathbf{i}_{0}=(1,2,3,1,2,1)$. The graph for

$u^{T}=\mathbf{t}_{1}^{\vee}(z_{6}^{-1})\mathbf{x}_{1}^{\vee}(z_{6})\mathbf{t}_{2}^{\vee}(z_{5}^{-1})\mathbf{x}_{2}^{\vee}(z_{5})\mathbf{t}_{1}^{\vee}(z_{4}^{-1})\mathbf{x}_{1}^{\vee}(z_{4})\mathbf{t}_{3}^{\vee}(z_{3}^{-1})\mathbf{x}_{3}^{\vee}(z_{3})\mathbf{t}_{2}^{\vee}(z_{2}^{-1})\mathbf{x}_{2}^{\vee}(z_{2})\mathbf{t}_{1}^{\vee}(z_{1}^{-1})\mathbf{x}_{1}^{\vee}(z_{1})$

is given in Figure 7.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$4$$3$$2$$1$$\frac{1}{z_{6}}$$z_{6}$$z_{6}$$\frac{1}{z_{5}}$$z_{5}$$z_{5}$$\frac{1}{z_{4}}$$z_{4}$$z_{4}$$\frac{1}{z_{3}}$$z_{3}$$z_{3}$$\frac{1}{z_{2}}$$z_{2}$$z_{2}$$\frac{1}{z_{1}}$$z_{1}$$z_{1}$
Figure 7: The graph for $u^{T}$ when $n=4$

We again give example computations, with parentheses highlighting products of
multiple paths:

$\displaystyle\Delta^{\\{1,2\\}}_{\\{1,4\\}}(u^{T})=\left(\frac{1}{z_{6}}\frac{1}{z_{4}}\frac{1}{z_{1}}\right)\left(z_{6}\frac{1}{z_{5}}z_{5}\frac{1}{z_{3}}z_{3}\right)=\frac{1}{z_{1}z_{4}},$
$\displaystyle\Delta^{\\{1,2\\}}_{\\{3,4\\}}(u^{T})=\left(\frac{1}{z_{6}}\frac{1}{z_{4}}z_{4}\frac{1}{z_{2}}z_{2}\right)\left(z_{6}\frac{1}{z_{5}}z_{5}\frac{1}{z_{3}}z_{3}\right)=1.$

######  Lemma 3.2.4.

All minors in both applications of the Chamber Ansatz in the proof of Lemma
3.1.4 are monomial and consequently the resulting coordinate changes are
monomial.

###### Proof.

In each application of the Chamber Ansatz, the relevant minors are those flag
minors with column sets given by the chamber labels of the corresponding
ansatz arrangements.

######  Claim.

Let

$\begin{gathered}\mathbf{i}_{0}=(i_{1},\ldots,i_{N}):=(1,2,\ldots,n-1,1,2\ldots,n-2,\ldots,1,2,1),\\\
\mathbf{i}^{\prime}_{0}=(i^{\prime}_{1},\ldots,i^{\prime}_{N}):=(n-1,n-2,\ldots,1,n-1,n-2,\ldots,2,\ldots,n-1,n-2,n-1),\\\
{\mathbf{i}^{\prime}_{0}}^{\text{op}}:=(i^{\prime}_{N},\ldots,i^{\prime}_{1}).\end{gathered}$

Then

  1. 1.

Chamber labels of the ansatz arrangement for $\mathbf{i}_{0}$ are of the form
$\\{a,\ldots,b\\}$.

  2. 2.

Chamber labels of the ansatz arrangement for
${\mathbf{i}^{\prime}_{0}}^{\text{op}}$ are of the form $\\{1,\ldots,a\\}$,
$\\{b,\ldots,n\\}$ or $\\{1,\ldots,a\\}\cup\\{b,\ldots,n\\}$.

Note that flag minors of $u_{1}$ and $u^{T}$ correspond to chamber labels of
the ansatz arrangement for $\mathbf{i}_{0}$ and
${\mathbf{i}^{\prime}_{0}}^{\text{op}}$ respectively.

###### Proof of Claim.

We may construct the reduced expression
${\mathbf{i}^{\prime}_{0}}^{\text{op}}$ from $\mathbf{i}_{0}$ in two steps;
first replace each $i_{k}$ in $\mathbf{i}_{0}$ with $n-i_{k}$ (this gives
${\mathbf{i}^{\prime}_{0}}$) and then reverse the order. In terms of the
ansatz arrangement, both of these operations result in reflections. Viewing
the ansatz arrangement for $\mathbf{i}_{0}$ in the the plane, with the origin
at the centre of the arrangement, we see that the first step above reflects
the ansatz arrangement in the horizontal axis. Note that this causes each
chamber label $S$ to change in the following way

$S\mapsto\\{1,\ldots,n\\}\setminus S.$

In the second step we reverse the order of the reduced expression which gives
a refection of the arrangement in the vertical axis and in particular there is
no further change to the chamber labels. It follows that the two statements in
the claim are equivalent and so we will consider only the $\mathbf{i}_{0}$
case.

Given the ansatz arrangement for $\mathbf{i}_{0}$ in dimension $n$, if we
ignore the first $n-1$ crossings and the top pseudoline after this point (i.e.
we remove the 1-string), the remaining graph has the form of the ansatz
arrangement in dimension $n-1$, with labelling $2,\ldots n$ rather than
$1,\ldots,n-1$. This is a consequence of the form of $\mathbf{i}_{0}$, namely
that the reduced expression $\mathbf{i}_{0}$ in dimension $n-1$ is given by
the last $\binom{n-1}{2}$ entries of the expression $\mathbf{i}_{0}$ in
dimension $n$.

Since, by definition of the ansatz arrangement, the leftmost chamber labels
are always given by sets of consecutive integers, it follows by induction that
all chamber labels are of this form. ∎

We now use the graphs for $u_{1}$ and $u^{T}$ to see that the relevant flag
minors are all monomial, namely by using Theorem 3.2.1 and showing that there
is only one possible family of paths in each case.

  1. 1.

Minors of $u_{1}$:

    1. (a)

Column sets of the form $\\{1,\ldots,b\\}$: Since $u_{1}\in U^{\vee}_{+}$
these minors always equal $1$. We can see this from the graph for $u_{1}$
since the paths must be horizontal and the lack of non-trivial torus factors
means that all horizontal edges have weight $1$.

    2. (b)

Column sets of the form $\\{a,\ldots,b\\}$ with $a>1$: Note that there is only
one edge connecting the bottom two horizontal lines. After travelling up this
edge there is only one possible path to the third line and so on. Thus there
is only one path from the source $1$ to the sink $a$.

In order for the paths in our family to remain vertex disjoint, the path from
the source $2$ to the sink $a+1$ must take the first opportunity to travel
upwards and indeed every possible opportunity to travel upwards until it
reaches the line at height $a+1$. This imposes the same restriction on the
path from the source $3$ to sink $a+2$ and so on for all paths in this family.
In particular there is only one possible family of paths, thus these minors
are monomial. For example see Figure 8.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$4$$3$$2$$1$
Figure 8: Example family of paths for the proof of Lemma 3.2.4, $u_{1}$ case

  2. 2.

Minors of $u^{T}$:

    1. (a)

Column sets of the form $\\{1,\ldots,a\\}$: These minors are monomial since
they correspond to horizontal paths in the graph for $u^{T}$.

    2. (b)

Column sets of the form $\\{b,\ldots,n\\}$: We first note that there is only
one edge from the line at height $n-1$ to the $n$-th horizontal line. Before
this point there is only one edge from the line at height $n-2$ to the line at
height $n-1$. Working backwards in this way we see there is only one possible
path from each source which ends at the sink $n$, and in particular only one
such path from the source $n-b+1$.

Similarly, in order to have vertex distinct paths there is now only one
possible way to reach the sink $n-1$ from the source $n-b$. Continuing in this
way we see that there is only one possible family of paths and so these minors
are monomial. For example see Figure 9.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$4$$3$$2$$1$
Figure 9: Example family of paths for the proof of Lemma 3.2.4, $u^{T}$ case

    3. (c)

Column sets of the form $\\{1,\ldots,a\\}\cup\\{b,\ldots,n\\}$: We see that
these minors are all monomial by combining the previous two cases.

∎

###  3.3 The coordinate change

In this section we address two final lemmas needed for the proof of Theorem
3.0.1, both detailing coordinate changes. We then recall and prove this
theorem.

######  Lemma 3.3.1.

For $k=1,\ldots,n-1$, $a=1,\ldots,n-k$ we have

$p_{s_{k}+a}=\begin{cases}z_{1+s_{a}}&\text{if }k=1\\\
\frac{z_{k+s_{a}}}{z_{k-1+s_{a+1}}}&\text{otherwise}\end{cases}\qquad\text{where}\
\ s_{k}:=\sum_{j=1}^{k-1}(n-j).$

###### Proof.

In the proof of Lemma 3.1.4 we used the Chamber Ansatz with
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ to show that
$u_{1}=\mathbf{x}_{i^{\prime}_{1}}^{\vee}(p_{1})\cdots\mathbf{x}_{i^{\prime}_{N}}^{\vee}(p_{N})$
where

$p_{k}=\frac{\prod_{j\neq
i^{\prime}_{k}}\Delta^{\omega^{\vee}_{j}}_{w_{(N-k+1)}\omega^{\vee}_{j}}(u^{T})^{-a_{j,i^{\prime}_{k}}}}{\Delta^{\omega^{\vee}_{i_{k}}}_{w_{(N-k+1)}\omega^{\vee}_{i^{\prime}_{k}}}(u^{T})\Delta^{\omega^{\vee}_{i^{\prime}_{k}}}_{w_{(N-k)}\omega^{\vee}_{i^{\prime}_{k}}}(u^{T})},\quad
k=1,\ldots,N.$

Since this expression is quite unpleasant, we instead work diagrammatically.
In particular, we will use the ansatz arrangement for
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ and the graph for $u^{T}$.

To give a visual aid we recall Figure 4, the dimension $4$ example of the
ansatz arrangement for
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}=(i^{\prime}_{N},\ldots,i^{\prime}_{1})=(3,2,3,1,2,3)$:

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$1$$2$$3$$4$$123$$124$$134$$234$$12$$14$$34$$1$$4$

We now define a new labelling of the chambers of the ansatz arrangement for
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ in terms of pairs $(k,a)$. This is
motivated by two facts:

  1. 1.

Any integer $1\leq m\leq N:=\binom{n}{2}$ may be written as

$m=s_{k}+a$

for some unique pair $(k,a)$, with $k=1,\ldots,n-1$, $a=1,\ldots,n-k$.

  2. 2.

For such a pair $(k,a)$, the label of the chamber to the _left_ of the
$\left(N-s_{k}-a+1\right)$-th crossing is given by

$\begin{cases}\\{1,\ldots,k\\}&\text{if }k+a=n,\\\
\\{1,\ldots,k\\}\cup\\{k+a+1,\ldots,n\\}&\text{if }k+a\neq n.\end{cases}$

It follows that we assign the pair $(k,a)$ to the chamber on the left of the
$\left(N-s_{k}-a+1\right)$-th crossing. The rightmost chambers are labelled
consistently, taking $k=0$. We leave the chambers above and below the
pseudoline arrangement unlabelled.

Continuing with our dimension $4$ example, the chamber pairs $(k,a)$ are shown
in Figure 11. In general we consider the $(N-s_{k}-a+1)$-th crossing. The
pairs $(k,a)$ for each surrounding chamber are given, where they are defined,
by the diagram in Figure 11.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$1$$2$$3$$4$$(3,1)$$(2,1)$$(1,1)$$(0,1)$$(2,2)$$(1,2)$$(0,2)$$(1,3)$$(0,3)$
Figure 10: The ansatz arrangement for
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ in dimension 4 with $(k,a)$
labelling

$\bullet$$(k-1,a+1)$$(k,a)$$(k-1,a)$$(k,a-1)$$\left(N-s_{k}-a+1\right)$-th
crossing Figure 11: Labelling of chamber pairs $(k,a)$ for
${\mathbf{i}^{\prime}_{0}}^{\mathrm{op}}$ ansatz arrangement

With this notation we return to the coordinate change given by the Chamber
Ansatz. It requires us to compute the minors corresponding to the chamber
labels, which we will do in terms of the pairs $(k,a)$. To help us with this
we recall Figure 7, namely that in dimension $4$ the graph for

$u^{T}=\mathbf{t}_{1}^{\vee}(z_{6}^{-1})\mathbf{x}_{1}^{\vee}(z_{6})\mathbf{t}_{2}^{\vee}(z_{5}^{-1})\mathbf{x}_{2}^{\vee}(z_{5})\mathbf{t}_{1}^{\vee}(z_{4}^{-1})\mathbf{x}_{1}^{\vee}(z_{4})\mathbf{t}_{3}^{\vee}(z_{3}^{-1})\mathbf{x}_{3}^{\vee}(z_{3})\mathbf{t}_{2}^{\vee}(z_{2}^{-1})\mathbf{x}_{2}^{\vee}(z_{2})\mathbf{t}_{1}^{\vee}(z_{1}^{-1})\mathbf{x}_{1}^{\vee}(z_{1})$

is given by

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$4$$3$$2$$1$$\frac{1}{z_{6}}$$z_{6}$$z_{6}$$\frac{1}{z_{5}}$$z_{5}$$z_{5}$$\frac{1}{z_{4}}$$z_{4}$$z_{4}$$\frac{1}{z_{3}}$$z_{3}$$z_{3}$$\frac{1}{z_{2}}$$z_{2}$$z_{2}$$\frac{1}{z_{1}}$$z_{1}$$z_{1}$

To compute the minors corresponding to the chamber labels, we first use (4)
from Section 2.4 to see that for $k=1,\ldots,n-1$ we have

$\Delta^{\\{1,\ldots,k\\}}_{\\{1,\ldots,k\\}}(u^{T})=\left(\prod_{\begin{subarray}{c}1\leq
m\leq N\\\
i_{m}\in\\{1,\ldots,k\\}\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod_{\begin{subarray}{c}1\leq
m\leq N\\\
i_{m}\in\\{1,\ldots,k-1\\}\end{subarray}}z_{m}\right)=\prod_{\begin{subarray}{c}1\leq
m\leq N\\\ i_{m}=k\end{subarray}}\frac{1}{z_{m}}.$

Note that if $a=1$ then the chamber in the ansatz arrangement above the
relevant crossings has label $\\{1,\ldots,n\\}$, with corresponding minor

$\Delta^{\\{1,\ldots,n\\}}_{\\{1,\ldots,n\\}}(u^{T})=\left(\prod_{\begin{subarray}{c}1\leq
m\leq N\\\
i_{m}\in\\{1,\ldots,n-1\\}\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod_{\begin{subarray}{c}1\leq
m\leq N\\\ i_{m}\in\\{1,\ldots,n-1\\}\end{subarray}}z_{m}\right)=1.$

For the remaining minors, using the graph for $u^{T}$ and Theorem 3.2.1 we see
that

$\Delta^{\\{1,\ldots,n-a\\}}_{\\{1,\ldots,k\\}\cup\\{k+a+1,\ldots,n\\}}(u^{T})=\Delta^{\\{1,\ldots,k\\}}_{\\{1,\ldots,k\\}}(u^{T})\Delta^{\\{k+1,\ldots,n-a\\}}_{\\{k+a+1,\ldots,n\\}}(u^{T}).$

The minors $\Delta^{\\{k+1,\ldots,n-a\\}}_{\\{k+a+1,\ldots,n\\}}(u^{T})$
correspond to paths in the graph for $u^{T}$ which form ‘staircases’ so their
weights have contributions from both horizontal and diagonal edges.

The proof of Lemma 3.2.4 implies that on each path there are no horizontal
edges with non-trivial weight after the last diagonal edge has been traversed.
Additionally there is only one horizontal edge with non-trivial weight between
each diagonal ‘step’, namely the edge directly preceding the diagonal in
Figure 12.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\frac{1}{z_{m}}$$z_{m}$$z_{m}$
Figure 12: Labelled line segments in graphs in the proof of Lemma 3.3.1

In particular, on each path the contributions from the diagonal edges and the
horizontal edges directly preceding them will always cancel. So, roughly
speaking, for each path we only need to consider the contributions from the
horizontal edges which occur well before the first diagonal ‘step’; that is,
if the first diagonal edge in the path occurs with weight $z_{p}$ then the
only factors in the evaluation of the minor will be $\frac{1}{z_{m}}$,
${z_{m}}$ with $m>p$ (note that the ordering of the $i_{m}$ is reversed in the
graph of $u^{T}$ due to taking the transpose):

$\displaystyle\Delta^{\\{k+1,\ldots,n-a\\}}_{\\{k+a+1,\ldots,n\\}}(u^{T})$
$\displaystyle=\left(\prod_{\begin{subarray}{c}m>i_{m}+\sum_{j=1}^{a-1}(n-j)\\\
i_{m}\in\\{k+1,\ldots,n-a\\}\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod_{\begin{subarray}{c}m>i_{m}+\sum_{j=1}^{a-1}(n-j)\\\
i_{m}\in\\{k,\ldots,n-a-1\\}\end{subarray}}z_{m}\right)$
$\displaystyle=\left(\prod_{\begin{subarray}{c}m>n-a+s_{a}\\\
i_{m}=n-a\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod_{\begin{subarray}{c}m>k+s_{a}\\\
i_{m}=k\end{subarray}}z_{m}\right).$

Noticing that the last occurrence of $i_{m}=n-a$ in the reduced expression
$\mathbf{i}_{0}$, is exactly when $m=n-a+s_{a}$, we obtain

$\Delta^{\\{k+1,\ldots,n-a\\}}_{\\{k+a+1,\ldots,n\\}}(u^{T})=\prod_{\begin{subarray}{c}m>k+s_{a}\\\
i_{m}=k\end{subarray}}z_{m}.$

Additionally we see that this minor is equal to $1$ if $k=0$, since $i_{m}\geq
1$ by definition.

Combining the above we obtain

$\displaystyle\Delta^{\\{1,\ldots,n-a\\}}_{\\{1,\ldots,k\\}\cup\\{k+a+1,\ldots,n\\}}(u^{T})$
$\displaystyle=\Delta^{\\{1,\ldots,k\\}}_{\\{1,\ldots,k\\}}(u^{T})\Delta^{\\{k+1,\ldots,n-a\\}}_{\\{k+a+1,\ldots,n\\}}(u^{T})$
$\displaystyle=\left(\prod_{\begin{subarray}{c}1\leq m\leq N\\\
i_{m}=k\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod_{\begin{subarray}{c}m>k+s_{a}\\\
i_{m}=k\end{subarray}}z_{m}\right)=\prod_{\begin{subarray}{c}m\leq k+s_{a}\\\
i_{m}=k\end{subarray}}\frac{1}{z_{m}}.$

We now use the Chamber Ansatz to compute the $p_{s_{k}+a}$ coordinates,
considering separately the case when $a=1$.

For $k=1,\ldots,n-1$, $a=2,\ldots,n-k$ we have

$p_{s_{k}+a}=\frac{\left(\prod\limits_{\begin{subarray}{c}m\leq k+s_{a-1}\\\
i_{m}=k\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod\limits_{\begin{subarray}{c}m\leq
k-1+s_{a+1}\\\
i_{m}=k-1\end{subarray}}\frac{1}{z_{m}}\right)}{\left(\prod\limits_{\begin{subarray}{c}m\leq
k+s_{a}\\\
i_{m}=k\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod\limits_{\begin{subarray}{c}m\leq
k-1+s_{a}\\\
i_{m}=k-1\end{subarray}}\frac{1}{z_{m}}\right)}=\frac{\prod\limits_{\begin{subarray}{c}k+s_{a-1}<m\leq
k+s_{a}\\\
i_{m}=k\end{subarray}}z_{m}}{\prod\limits_{\begin{subarray}{c}k-1+s_{a}<m\leq
k-1+s_{a+1}\\\ i_{m}=k-1\end{subarray}}z_{m}},$ (11)

We note that each of the products on the last line has exactly one term,
namely when $m$ is equal to the upper bound. Thus we obtain

$p_{s_{k}+a}=\begin{cases}z_{k+s_{a}}&\text{if }k=1,\\\
\frac{z_{k+s_{a}}}{z_{k-1+s_{a+1}}}&\text{otherwise}.\end{cases}$

We apply a similar argument in the case when $a=1$. For $k=1,\ldots,n-1$ the
minor corresponding to the chamber above the pseudoline arrangement is equal
to $1$ so we have

$\displaystyle p_{s_{k}+a}=\frac{\prod\limits_{\begin{subarray}{c}m\leq
k-1+s_{a+1}\\\
i_{m}=k-1\end{subarray}}\frac{1}{z_{m}}}{\left(\prod\limits_{\begin{subarray}{c}m\leq
k+s_{a}\\\
i_{m}=k\end{subarray}}\frac{1}{z_{m}}\right)\left(\prod\limits_{\begin{subarray}{c}m\leq
k-1+s_{a}\\\ i_{m}=k-1\end{subarray}}\frac{1}{z_{m}}\right)}$
$\displaystyle=\frac{\prod\limits_{\begin{subarray}{c}m\leq k\\\
i_{m}=k\end{subarray}}z_{m}}{\prod\limits_{\begin{subarray}{c}k-1<m\leq
k-1+(n-1)\\\ i_{m}=k-1\end{subarray}}z_{m}}$
$\displaystyle=\begin{cases}z_{k}&\text{if }k=1\\\
\frac{z_{k}}{z_{k-1+(n-1)}}&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}z_{k+s_{a}}&\text{if }k=1\\\
\frac{z_{k+s_{a}}}{z_{k-1+s_{a+1}}}&\text{otherwise}\end{cases}$

since $s_{a+1}=s_{2}=n-1$ and $s_{a}=s_{1}=0$.

Note that the need to consider the $a=1$ case separately is highlighted in
(11); if $a=1$ the numerator would be

$\prod\limits_{\begin{subarray}{c}k+s_{0}<m\leq k+s_{1}\\\
i_{m}=k\end{subarray}}z_{m}=\prod\limits_{\begin{subarray}{c}k<m\leq k\\\
i_{m}=k\end{subarray}}z_{m}=1$

whereas in fact we should have $z_{k}=z_{k+s_{a}}$ in the numerator.

Combining the above we obtain the desired coordinate change:

$p_{s_{k}+a}=\begin{cases}z_{k+s_{a}}&\text{if }k=1,\\\
\frac{z_{k+s_{a}}}{z_{k-1+s_{a+1}}}&\text{otherwise}.\end{cases}$

∎

######  Lemma 3.3.2.

For $k=1,\ldots,n-1$, $a=1,\ldots,n-k$ we have

$\frac{1}{m_{s_{k}+a}}=\begin{cases}\prod\limits_{\begin{subarray}{c}r=1,\ldots,k\end{subarray}}\frac{1}{p_{s_{r+1}-a+1}}&\text{if
}k=1\\\
\frac{\prod\limits_{\begin{subarray}{c}r=1,\ldots,k-1\end{subarray}}p_{s_{r+1}-a}}{\prod\limits_{\begin{subarray}{c}r=1,\ldots,k\end{subarray}}p_{s_{r+1}-a+1}}&\text{otherwise}\end{cases}\qquad\text{where}\
\ s_{k}:=\sum_{j=1}^{k-1}(n-j).$

###### Proof.

In the proof of Lemma 3.1.4 we used the Chamber Ansatz with $\mathbf{i}_{0}$
to show that
$bB^{\vee}_{+}=\mathbf{y}_{i_{1}}\left(\frac{1}{m_{1}}\right)\cdots\mathbf{y}_{i_{N}}\left(\frac{1}{m_{N}}\right)B^{\vee}_{+}$
where

$\frac{1}{m_{k}}=\frac{\prod_{j\neq
i_{k}}\Delta^{\omega_{j}}_{w_{(k)}\omega_{j}}(u_{1})^{-a_{j,i_{k}}}}{\Delta^{\omega_{i_{k}}}_{w_{(k)}\omega_{i_{k}}}(u_{1})\Delta^{\omega_{i_{k}}}_{w_{(k-1)}\omega_{i_{k}}}(u_{1})},\quad
k=1,\ldots,N.$

Similar to the proof of the first coordinate change, since this expression is
unpleasant we again work diagrammatically. In particular, we will use the
ansatz arrangement for $\mathbf{i}^{\prime}_{0}$ and the graph for $u_{1}$.

As before, to give a visual aid we recall Figure 4, the dimension $4$ example
of the ansatz arrangement for $\mathbf{i}_{0}=(1,2,3,1,2,1)$:

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$1$$2$$3$$4$$123$$234$$12$$23$$34$$1$$2$$3$$4$

Similar to the $u^{T}$ case, we give new labelling of the chambers of the
ansatz arrangement in terms of pairs $(k,a)$. This time, for $k=1,\ldots,n-1$,
$a=1,\ldots,n-k$, the label of the chamber to the _right_ of the
$(s_{k}+a)$-th crossing is given by

$\\{k+1,\ldots,k+a\\}.$

It is then natural to assign the pair $(k,a)$ to the chamber on the right of
the $(s_{k}+a)$-th crossing. The leftmost chambers will be labelled
consistently, taking $k=0$. Again we leave the chambers above and below the
pseudoline arrangement unlabelled.

Continuing our dimension $4$ example, the chamber pairs $(k,a)$ are shown in
Figure 14. In general, the pairs $(k,a)$ for each chamber surrounding the
$(s_{k}+a)$-th crossing, where there are defined, are given by the diagram in
Figure 14.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$1$$2$$3$$4$$(0,3)$$(1,3)$$(0,2)$$(1,2)$$(2,2)$$(0,1)$$(1,1)$$(2,1)$$(3,1)$
Figure 13: The ansatz arrangement for $\mathbf{i}_{0}$ in dimension 4 with
$(k,a)$ labelling

$\bullet$$(k,a-1)$$(k-1,a)$$(k,a)$$(k-1,a+1)$$(s_{k}+a)$-th crossing Figure
14: Labelling of chamber pairs $(k,a)$ for $\mathbf{i}_{0}$ ansatz arrangement

Before applying the Chamber Ansatz we again compute minors in terms of the
pairs $(k,a)$. We have just seen that in dimension $4$ the matrix $u_{1}$ is
given by

$u_{1}=\mathbf{x}_{\mathbf{i}^{\prime}_{0}}^{\vee}\left(z_{1},z_{4},z_{6},\frac{z_{2}}{z_{4}},\frac{z_{5}}{z_{6}},\frac{z_{3}}{z_{5}}\right).$

This is quite messy however, so for ease of computation we will continue to
work for with the $p_{j}$ coordinates;
$u_{1}=\mathbf{x}_{\mathbf{i}^{\prime}_{0}}^{\vee}(p_{1},\ldots,p_{6})$. The
corresponding graph for $u_{1}$ is given in Figure 15.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$4$$3$$2$$1$$4$$3$$2$$1$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$p_{5}$$p_{6}$
Figure 15: The graph for $u_{1}$ when $n=4$, $\boldsymbol{p}$ coordinates

From the proof of Lemma 3.2.4 we recall that there is only one family of
vertex disjoint paths in the graph for $u_{1}$ from the set of sources
$\\{1,\ldots,a\\}$ to the set of sinks $\\{k+1,\ldots,k+a\\}$. Within this
family, the weight of the path from $b\in\\{1,\ldots,a\\}$ to $k+b$ is given
by

$\prod_{\begin{subarray}{c}m=(n-b)+\sum_{j=2}^{r}(n-j)\\\
r=1,\ldots,k\end{subarray}}p_{m}=\prod_{\begin{subarray}{c}r=1,\ldots,k\end{subarray}}p_{s_{r+1}-b+1}.$

Taking the product over this family, that is, the product of the weights of
the paths from $\\{1,\ldots,a\\}$ to $\\{k+1,\ldots,k+a\\}$, we obtain the
desired minor

$\Delta^{\\{1,\ldots,a\\}}_{\\{k+1,\ldots,k+a\\}}(u_{1})=\prod_{\begin{subarray}{c}r=1,\ldots,k\\\
b=1,\ldots,a\end{subarray}}p_{s_{r+1}-b+1}.$

Note that since $u_{1}\in U^{\vee}_{+}$, the minors corresponding to the
leftmost chambers and the chamber above the pseudoline arrangement are all
equal to $1$.

We now use the Chamber Ansatz to compute the $\frac{1}{m_{j}}$ coordinates.
For $k=1,\ldots,n-1$, $a=1,\ldots,n-k$ we have

$\displaystyle\frac{1}{m_{s_{k}+a}}$
$\displaystyle=\frac{\left(\prod\limits_{\begin{subarray}{c}r=1,\ldots,k-1\\\
b=1,\ldots,a+1\end{subarray}}p_{s_{r+1}-b+1}\right)\left(\prod\limits_{\begin{subarray}{c}r=1,\ldots,k\\\
b=1,\ldots,a-1\end{subarray}}p_{s_{r+1}-b+1}\right)}{\left(\prod\limits_{\begin{subarray}{c}r=1,\ldots,k-1\\\
b=1,\ldots,a\end{subarray}}p_{s_{r+1}-b+1}\right)\left(\prod\limits_{\begin{subarray}{c}r=1,\ldots,k\\\
b=1,\ldots,a\end{subarray}}p_{s_{r+1}-b+1}\right)}$
$\displaystyle=\begin{cases}\prod\limits_{\begin{subarray}{c}r=1,\ldots,k\end{subarray}}\frac{1}{p_{s_{r+1}-a+1}}&\text{if
}k=1,\\\
\frac{\prod\limits_{\begin{subarray}{c}r=1,\ldots,k-1\end{subarray}}p_{s_{r+1}-a}}{\prod\limits_{\begin{subarray}{c}r=1,\ldots,k\end{subarray}}p_{s_{r+1}-a+1}}&\text{otherwise.}\end{cases}$
(12)

∎

We now ready to prove Theorem 3.0.1, the statement of which we recall here:

######  Theorem.

For $k=1,\ldots,n-1$, $a=1,\ldots,n-k$ we have

$m_{s_{k}+a}=\begin{cases}z_{1+s_{n-a}}&\text{if }k=1\\\
\frac{z_{k+s_{n-k-a+1}}}{z_{k-1+s_{n-k-a+1}}}&\text{otherwise}\end{cases}\qquad\text{where}\
\ s_{k}:=\sum_{j=1}^{k-1}(n-j).$

###### Proof.

To obtain the $\frac{1}{m_{j}}$ coordinates in terms of the $z_{j}$, we need
to compose the two coordinate transformations from Lemmas 3.3.1 and 3.3.2. To
do this we write the factors $p_{s_{r+1}-a}$, $p_{s_{r+1}-a+1}$ of the
products in (12) in the form $p_{s_{k^{\prime}}+{a^{\prime}}}$ for a suitable
pairs $(k^{\prime},a^{\prime})$;

  1. 1.

Firstly, given a pair $(k,a)$ with $k>1$, for each $r=1,\ldots,k-1$, we wish
to find the pair $(k^{\prime},a^{\prime})$, $k^{\prime}\in\\{1,\ldots,n-1\\}$,
$a^{\prime}\in\\{1,\ldots,n-k^{\prime}\\}$ such that

$s_{k^{\prime}}+a^{\prime}=s_{r+1}-a.$

Indeed since

$s_{r+1}-a=s_{r}+n-r-a$

we take $k^{\prime}=r$, $a^{\prime}=n-r-a$, noting that $k^{\prime}$ and
$a^{\prime}$ satisfy the necessary conditions

$\displaystyle k^{\prime}$
$\displaystyle=r\in\\{1,\ldots,k-1\\}\subseteq\\{1,\ldots,n-1\\},$
$\displaystyle a^{\prime}$
$\displaystyle=n-r-a=n-k^{\prime}-a\in\\{k-k^{\prime},\ldots,n-k^{\prime}-1\\}\subseteq\\{1,\ldots,n-k^{\prime}\\}.$

  2. 2.

Similarly given a pair $(k,a)$ for any $k$, i.e. $1\leq k\leq n-1$, for each
$r=1,\ldots,k$, we wish to find these pairs $(k^{\prime},a^{\prime})$,
$k^{\prime}\in\\{1,\ldots,n-1\\}$, $a^{\prime}\in\\{1,\ldots,n-k^{\prime}\\}$
such that

$s_{k^{\prime}}+a^{\prime}=s_{r+1}-a+1=s_{r}+n-r-a+1.$

We take $k^{\prime}=r$, $a^{\prime}=n-r-a+1$, again noting that $k^{\prime}$
and $a^{\prime}$ satisfy the necessary conditions

$\displaystyle k^{\prime}$
$\displaystyle=r\in\\{1,\ldots,k\\}\subseteq\\{1,\ldots,n-1\\},$
$\displaystyle a^{\prime}$
$\displaystyle=n-r-a+1=n-k^{\prime}-a+1\in\\{k-k^{\prime}+1,\ldots,n-k^{\prime}\\}\subseteq\\{1,\ldots,n-k^{\prime}\\}.$

In order to evaluate the products in (12) we need to consider what happens to
the pair $(k^{\prime},a^{\prime})$ when we increase $r$ by $1$.

  1. 1.

In the first case: for $r\in\\{1,\ldots,k-2\\}$

$s_{r+2}-a=s_{r+1}+n-(r+1)-a=s_{k^{\prime}}+a^{\prime}+n-k^{\prime}-1=s_{k^{\prime}+1}+a^{\prime}-1.$

Note that $k^{\prime}+1$ and $a^{\prime}-1$ satisfy the necessary conditions

$\displaystyle k^{\prime}+1$
$\displaystyle=r+1\in\\{2,\ldots,k-1\\}\subseteq\\{1,\ldots,n-1\\},$
$\displaystyle a^{\prime}-1$
$\displaystyle=n-r-a-1\in\\{k-(k^{\prime}+1),\ldots,n-(k^{\prime}+1)-1\\}\subseteq\\{1,\ldots,n-(k^{\prime}+1)\\}.$

  2. 2.

Similarly in the second case: for $r\in\\{1,\ldots,k-1\\}$

$s_{r+2}-a+1=s_{r+1}+n-(r+1)-a+1=s_{k^{\prime}}+a^{\prime}+n-k^{\prime}-1=s_{k^{\prime}+1}+a^{\prime}-1.$

Again we note that $k^{\prime}+1$ and $a^{\prime}-1$ satisfy the necessary
conditions

$\displaystyle k^{\prime}+1$
$\displaystyle=r+1\in\\{2,\ldots,k\\}\subseteq\\{1,\ldots,n-1\\},$
$\displaystyle a^{\prime}-1$
$\displaystyle=n-r-a\in\\{k-(k^{\prime}+1)+1,\ldots,n-(k^{\prime}+1)\\}\subseteq\\{1,\ldots,n-(k^{\prime}+1)\\}.$

Now recalling

$p_{s_{k}+a}=\begin{cases}z_{k+s_{a}}&\text{if }k=1\\\
\frac{z_{k+s_{a}}}{z_{k-1+s_{a+1}}}&\text{otherwise}\end{cases}$

we have, for $r>1$ (so that $k^{\prime}>1$)

$\displaystyle p_{s_{r+1}-a}p_{s_{r+2}-a}$
$\displaystyle=p_{s_{k^{\prime}}+a^{\prime}}p_{s_{k^{\prime}+1}+a^{\prime}-1}$
$\displaystyle\quad\text{where }k^{\prime}=r,\ a^{\prime}=n-r-a$
$\displaystyle=\frac{z_{k^{\prime}+s_{a^{\prime}}}}{z_{k^{\prime}-1+s_{a^{\prime}+1}}}\frac{z_{k^{\prime}+1+s_{a^{\prime}-1}}}{z_{k^{\prime}+s_{a^{\prime}}}}$
$\displaystyle=\frac{z_{k^{\prime}+1+s_{a^{\prime}-1}}}{z_{k^{\prime}-1+s_{a^{\prime}+1}}}.$

Consequently the following product becomes telescopic, so we have

$\prod\limits_{\begin{subarray}{c}r=1,\ldots,k-1\end{subarray}}p_{s_{r+1}-a}=z_{k-1+s_{n-(k-1)-a}}.$

Similarly in the second case

$\displaystyle p_{s_{r+1}-a+1}p_{s_{r+2}-a+1}$
$\displaystyle=p_{s_{k^{\prime}}+a^{\prime}}p_{s_{k^{\prime}+1}+a^{\prime}-1}$
$\displaystyle\quad\text{where }k^{\prime}=r-1,\ a^{\prime}=n-r-a+1$
$\displaystyle=\frac{z_{k^{\prime}+1+s_{a^{\prime}-1}}}{z_{k^{\prime}-1+s_{a^{\prime}+1}}}.$

Again we have a telescopic product, giving

$\prod\limits_{\begin{subarray}{c}r=1,\ldots,k\end{subarray}}p_{s_{r+1}-a+1}=z_{k+s_{n-k-a+1}}.$

Thus we have

$\frac{1}{m_{s_{k}+a}}=\begin{cases}\frac{1}{z_{k+s_{n-k-a+1}}}&\text{if
}k=1\\\
\frac{z_{k-1+s_{n-k-a+1}}}{z_{k+s_{n-k-a+1}}}&\text{otherwise}\end{cases}$

which gives the desired coordinate change: for $k=1,\ldots,n-1$,
$a=1,\ldots,n-k$

$m_{s_{k}+a}=\begin{cases}z_{1+s_{n-a}}&\text{if }k=1\\\
\frac{z_{k+s_{n-k-a+1}}}{z_{k-1+s_{n-k-a+1}}}&\text{otherwise.}\end{cases}$

∎

##  4 Givental-type quivers and critical points

In this section we recall an earlier Landau-Ginzburg model for the full flag
variety, defined on a torus by Givental [11]. We relate our tori from Sections
2 and 3, as well as the superpotential defined in Section 2.2, to Givental’s
torus and his formulation of the superpotential. We then use this to start
describing the critical points of the superpotential. The key result in this
section is Theorem 4.3.1, describing the $\boldsymbol{m}$ coordinates of such
a critical point. This formula was conjectured by Konstanze Rietsch and
checked in a particular case by Zainab Al-Sultani ([1]).

###  4.1 The Givental superpotential

In this section we recall Givental’s construction from [11]. We begin by
considering a quiver, which consists of $n(n+1)/2$ vertices in lower
triangular form together with arrows going up and left. We label the vertices
with $v_{ij}$ for $1\leq j\leq i\leq n$ in the same way as we would for matrix
entries and denote the set of such vertices by $\mathcal{V}$. The vertices
$v_{ii}$ are star vertices and all others are dot vertices. We denote the sets
of star and dot vertices respectively by

$\mathcal{V}^{*}=\left\\{v_{ii}\ |\ 1\leq i\leq
n\right\\},\quad\mathcal{V}^{\bullet}=\left\\{v_{ij}\ |\ 1\leq j<i\leq
n\right\\}.$

The set of arrows of the quiver is denoted
$\mathcal{A}=\mathcal{A}_{\mathrm{v}}\cup\mathcal{A}_{\mathrm{h}}$. The
vertical arrows $a_{ij}\in\mathcal{A}_{\mathrm{v}}$ are labelled such that
$h(a_{ij})=v_{ij}$ where $h(a)\in\mathcal{V}$ denotes the head of the arrow
$a$. The horizontal arrows $b_{ij}\in\mathcal{A}_{\mathrm{h}}$ are labelled
such that $t(b_{ij})=v_{ij}$ where $t(a)\in\mathcal{V}$ denotes the tail of
the arrow $a$. For example when $n=4$ the quiver is given in Figure 17.

$\bullet$$\bullet$$\bullet$$\boldsymbol{*}$$v_{11}$$\bullet$$\bullet$$\boldsymbol{*}$$v_{22}$$\bullet$$\boldsymbol{*}$$v_{33}$$\boldsymbol{*}$$v_{44}$$v_{21}$$v_{31}$$v_{41}$$v_{32}$$v_{42}$$v_{43}$$a_{11}$$a_{21}$$a_{31}$$b_{22}$$b_{32}$$b_{42}$$a_{22}$$a_{32}$$a_{33}$$b_{33}$$b_{43}$$b_{44}$
Figure 16: Quiver when $n=4$

$a_{2}$$a_{3}$$a_{4}$$a_{1}$ Figure 17: Subquiver for box relations

We consider three tori which are defined in terms of the quiver, of which we
introduce the first two now. The first torus is
$(\mathbb{K}^{*})^{\mathcal{V}}$ with coordinates $x_{v}$ for
$v\in\mathcal{V}$, we call this the vertex torus. The second torus,
$\bar{\mathcal{M}}\subset(\mathbb{K}^{*})^{N-n+1}$, corresponds to the arrows
of the quiver and so will be called the arrow torus. It is given by

$\bar{\mathcal{M}}:=\left\\{(r_{a})_{a\in\mathcal{A}}\in(\mathbb{K}^{*})^{\mathcal{A}}\
|\ r_{a_{1}}r_{a_{2}}=r_{a_{3}}r_{a_{4}}\text{ when
}a_{1},a_{2},a_{3},a_{4}\text{ form a square as in Figure \ref{fig quiver box
relations}}\right\\}.$

These two tori are related by the following surjection, given coordinate-wise:

$(\mathbb{K}^{*})^{\mathcal{V}}\to\bar{\mathcal{M}},\quad
r_{a}=\frac{x_{h(a)}}{x_{t(a)}}.$ (13)

Note that we get a point in the preimage of
$\boldsymbol{r}_{\mathcal{A}}:=(r_{a})_{a\in\mathcal{A}}\in\bar{\mathcal{M}}$
by first setting $x_{v_{nn}}=1$. Then for $v\neq v_{nn}$ we take
$x_{v}=\prod_{a\in P_{v}}r_{a}$ where $P_{v}$ is any path from $v_{nn}$ to
$v$. This map is well-defined since the preimage of $(1)_{a\in\mathcal{A}}$ is
the set $\\{(c)_{v\in\mathcal{V}}\ |\ c\in\mathbb{K}^{*}\\}$.

An analogy of this surjection is the map

$T^{\vee}\to(\mathbb{K}^{*})^{n-1}$

given by the simple roots $\alpha^{\vee}_{1},\ldots,\alpha^{\vee}_{n-1}$ of
$G^{\vee}$.

It will be more convenient for our purposes to use arrow coordinates rather
than vertex coordinates. If we were to work with $SL_{n}$ then the arrow
torus, $\bar{\mathcal{M}}$, would suffice, however we wish to work with
$GL_{n}$ and so need to keep track of which fibre of the map (13) we are in.
Consequently we now introduce our third torus, $\mathcal{M}$, which we call
the quiver torus:

$\mathcal{M}:=\left\\{(\boldsymbol{x}_{\mathcal{V}^{*}},\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}})\in(\mathbb{K}^{*})^{\mathcal{V}^{*}}\times(\mathbb{K}^{*})^{\mathcal{A}_{\mathrm{v}}}\right\\}\hookrightarrow(\mathbb{K}^{*})^{\mathcal{V}}\times\bar{\mathcal{M}}.$

The quiver torus is isomorphic to the vertex torus,
$(\mathbb{K}^{*})^{\mathcal{V}}$, and we will work with these tori
interchangeably. This isomorphism is a consequence of the following
observations:

We note that if we choose the star vertex coordinate $x_{v_{nn}}$, then we are
taking a particular lift of $\boldsymbol{r}_{\mathcal{A}}\in\bar{\mathcal{M}}$
such that in this fibre the map (13) restricts to an isomorphism. Moreover, if
we describe the coordinates of the star vertices and vertical arrows, then the
coordinates of the horizontal arrows, and thus also of the dot vertices, are
given uniquely using (13) and the box relations; namely we use the fact that

$r_{b_{i+1,i+1}}=\frac{x_{v_{ii}}}{x_{v_{i+1,i+1}}}\frac{1}{r_{a_{ii}}},\quad
i=1,\ldots,n-1$ (14)

together with the relations $r_{a_{1}}r_{a_{2}}=r_{a_{3}}r_{a_{4}}$ when the
arrows $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$ form a square as in Figure 17.

We now recall the definition of Givental’s superpotential. On the vertex torus
this can be defined as

$\mathcal{F}:(\mathbb{K}^{*})^{\mathcal{V}}\to\mathbb{K},\quad\boldsymbol{x}_{\mathcal{V}}\mapsto\sum_{a\in\mathcal{A}}\frac{x_{h(a)}}{x_{t(a)}}.$

This factors naturally through the arrow torus via (13) and the following map:

$\bar{\mathcal{F}}:\bar{\mathcal{M}}\to\mathbb{K},\quad\boldsymbol{r}_{\mathcal{A}}\mapsto\sum_{a\in\mathcal{A}}r_{a}.$

We can now define the highest weight and weight maps on the vertex torus. The
highest weight map is given by

$\kappa:(\mathbb{K}^{*})^{\mathcal{V}}\to
T^{\vee},\quad\boldsymbol{x}_{\mathcal{V}}\mapsto(x_{v_{ii}})_{i=1,\ldots,n}.$

The weight map [12] on the vertex torus is defined in two steps; firstly, for
$i=1,\ldots,n$ we let
$\mathcal{D}_{i}:=\\{v_{i,1},v_{i+1,2},\ldots,v_{n,n-i+1}\\}$ be the $i$-th
diagonal and let

$\Xi_{i}:=\prod_{v\in\mathcal{D}_{i}}x_{v}\quad\text{with}\quad\Xi_{n+1}:=1.$
(15)

Then the weight map is given by

$\gamma:(\mathbb{K}^{*})^{\mathcal{V}}\to
T^{\vee},\quad\boldsymbol{x}_{\mathcal{V}}\mapsto(t_{i})_{i=1,\ldots,n}\quad\text{where}\quad
t_{i}=\frac{\Xi_{i}}{\Xi_{i+1}}.$ (16)

Unlike the superpotential $\mathcal{F}$, the maps $\kappa$ and $\gamma$ do not
directly factor through $\bar{\mathcal{M}}$ but thanks to the map (13) we have
commutative diagrams:

${(\mathbb{K}^{*})^{\mathcal{V}}}$${T^{\vee}}$${\bar{\mathcal{M}}}$${(\mathbb{K}^{*})^{n-1}}$$\scriptstyle{\kappa}$$\scriptstyle{(\alpha^{\vee}_{1},\ldots,\alpha^{\vee}_{n-1})}$$\scriptstyle{q}$
${(\mathbb{K}^{*})^{\mathcal{V}}}$${T^{\vee}}$${\bar{\mathcal{M}}}$${(\mathbb{K}^{*})^{n-1}}$$\scriptstyle{\gamma}$$\scriptstyle{(\alpha^{\vee}_{1},\ldots,\alpha^{\vee}_{n-1})}$

The maps along the bottom making the diagrams commute exist and are unique.
For example, the map on $\bar{\mathcal{M}}$ corresponding to $\kappa$ is the
map called $q$ defined by Givental as follows:

$q:\bar{\mathcal{M}}\to(\mathbb{K}^{*})^{n-1},\quad\boldsymbol{r}_{\mathcal{A}}\mapsto\left(r_{a_{11}}r_{b_{22}},\,\ldots,\,r_{a_{n-1,n-1}}r_{b_{nn}}\right).$

######  Remark 4.1.1.

In Givental’s version of (Fano) mirror symmetry, the arrow torus
$\bar{\mathcal{M}}$ (taken over $\mathbb{C}$), is viewed as a family of
varieties via the map $q$. Each fibre $\bar{\mathcal{M}}_{\boldsymbol{q}}$,
being a torus, comes equipped with a natural holomorphic volume form
$\omega_{\boldsymbol{q}}$. He proves a version of mirror symmetry relating the
A-model connection, built out of Gromov-Witten invariants of the flag variety,
to period integrals $S(\boldsymbol{q})$ on the family
$(\bar{\mathcal{M}}_{\boldsymbol{q}},\bar{\mathcal{F}}_{\boldsymbol{q}},\omega_{\boldsymbol{q}})$,
defined using the superpotential:

$S(\boldsymbol{q}):=\int_{\Gamma}e^{\bar{\mathcal{F}}_{\boldsymbol{q}}}\omega_{\boldsymbol{q}}.$

###  4.2 The quiver torus as another toric chart on $Z$

In this section we recall a map $\mathcal{M}\to Z$ from [22] which allows us
to relate the Givental superpotential on the arrow torus, as well as the
highest weight and weight maps on the vertex torus, to their analogues on $Z$.
In particular our toric charts from the previous sections factor through this
map.

A nice way to describe how the string and ideal toric charts factor through
$\mathcal{M}$ is by decorating the arrows and vertices of the quiver. Indeed,
using our previous work relating the string and ideal coordinate systems, we
may easily describe a monomial map from the torus
$T^{\vee}\times(\mathbb{K}^{*})^{N}$ of string coordinates to the quiver torus
$\mathcal{M}$. Composing this with the map $\mathcal{M}\to Z$ will recover our
string toric chart. We then use the coordinate change given in Theorem 3.0.1
to decorate the quiver with the ideal coordinates.

In order to make sense of the way the string toric chart factors through
$\mathcal{M}$, we choose an ordering of the $N$ vertical arrow coordinates.
Starting at the lower left corner of the quiver and moving up each column in
succession we obtain

$\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}}:=\left(r_{a_{n-1,1}},r_{a_{n-2,1}},\ldots,r_{a_{1,1}},r_{a_{n-1,2}},r_{a_{n-2,2}},r_{a_{2,2}},\ldots,r_{a_{n-1,n-2}},r_{a_{n-2,n-2}},r_{a_{n-1,n-1}}\right).$

Similarly, we give an ordering of the star vertex coordinates:

$\boldsymbol{x}_{\mathcal{V}^{*}}:=\left(x_{v_{11}},x_{v_{22}},\ldots,x_{v_{nn}}\right).$

Then for the reduced expression

$\mathbf{i}^{\prime}_{0}=(i^{\prime}_{1},\ldots,i^{\prime}_{N}):=(n-1,n-2,\ldots,1,n-1,n-2,\ldots,2,\ldots,n-1,n-2,n-1)$

we have a map $\theta_{\mathbf{i}^{\prime}_{0}}:\mathcal{M}\to Z$ given by

$\left(\boldsymbol{x}_{\mathcal{V}^{*}},\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}}\right)\mapsto\Phi\left(\kappa_{|_{\mathcal{V}^{*}}}\left(\boldsymbol{x}_{\mathcal{V}^{*}}\right),\mathbf{x}^{\vee}_{\mathbf{i}^{\prime}_{0}}\left(\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}}\right)\right)=\mathbf{x}^{\vee}_{\mathbf{i}^{\prime}_{0}}\left(\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}}\right)\kappa_{|_{\mathcal{V}^{*}}}\left(\boldsymbol{x}_{\mathcal{V}^{*}}\right)\bar{w}_{0}u_{2}$

where $\kappa_{|_{\mathcal{V}^{*}}}$ is the restriction of $\tilde{\kappa}$ to
the star vertex coordinates, $u_{2}\in U^{\vee}$ is the unique element such
that
$\theta_{\mathbf{i}^{\prime}_{0}}\left(\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}}\right)\in
Z$ and $\mathbf{x}^{\vee}_{\mathbf{i}}$, $\Phi$ are the maps from Section 2.3.

Next we recall the following definitions, also from Section 2.3:

$b:=u_{1}d\bar{w}_{0}u_{2}\in Z,\quad u_{1}:=\iota(\eta^{w_{0},e}(u)),\quad
u:=\mathbf{x}^{\vee}_{-\mathbf{i}_{0}}(\boldsymbol{z}),$

together with the fact that by Lemmas 3.1.4 and 3.3.1 we may factorise $u_{1}$
as

$u_{1}=\mathbf{x}_{i^{\prime}_{1}}^{\vee}(p_{1})\cdots\mathbf{x}_{i^{\prime}_{N}}^{\vee}(p_{N})\\\
$

where, for $k=1,\ldots,n-1$, $a=1,\ldots,n-k$, we have

$p_{s_{k}+a}=\begin{cases}z_{1+s_{a}}&\text{if }k=1\\\
\frac{z_{k+s_{a}}}{z_{k-1+s_{a+1}}}&\text{otherwise}\end{cases}\qquad\text{where}\
\ s_{k}:=\sum_{j=1}^{k-1}(n-j).$

Thus taking $\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}}=(p_{1},\ldots,p_{N})$
we see that
$\mathbf{x}^{\vee}_{\mathbf{i}^{\prime}_{0}}\left(\boldsymbol{r}_{\mathcal{A}_{\mathrm{v}}}\right)=u_{1}$
and letting the star vertex coordinates be given by

$x_{v_{ii}}=d_{i},\quad i=1,\ldots,n$

we obtain
$\kappa_{|_{\mathcal{V}^{*}}}\left(\boldsymbol{x}_{\mathcal{V}^{*}}\right)=d$.
This quiver decoration in dimension $4$ is given in Figure 19.

Due to the relations in the quiver, namely (14) and the box relations, this
decoration extends to all the vertices and arrows of the quiver. In
particular, we can extend $\theta_{\mathbf{i}^{\prime}_{0}}$ to a map

$\bar{\theta}_{\mathbf{i}^{\prime}_{0}}:(\mathbb{K}^{*})^{\mathcal{V}}\times\bar{\mathcal{M}}\to
Z$

given by first taking the projection onto $\mathcal{M}$ and then applying
$\theta_{\mathbf{i}^{\prime}_{0}}$. Thus decorating the quiver in this way and
then applying the map $\bar{\theta}_{\mathbf{i}^{\prime}_{0}}$ (or
$\theta_{\mathbf{i}^{\prime}_{0}}$) gives the string toric chart factored
through $(\mathbb{K}^{*})^{\mathcal{V}}\times\bar{\mathcal{M}}$ (respectively
$\mathcal{M}$), that is

$T^{\vee}\times(\mathbb{K}^{*})^{N}\to(\mathbb{K}^{*})^{\mathcal{V}}\times\bar{\mathcal{M}}\to
Z.$

We summarise the results from this section:

######  Lemma 4.2.1 ([22, Theorem 9.2 and Lemma 9.3]).

With the above notation, we have the following:

$\mathcal{W}\circ\bar{\theta}_{\mathbf{i}^{\prime}_{0}}=\mathcal{F}\circ\mathrm{pr},\quad\mathrm{hw}\circ\bar{\theta}_{\mathbf{i}^{\prime}_{0}}=\kappa\circ\mathrm{pr},\quad\mathrm{wt}\circ\bar{\theta}_{\mathbf{i}^{\prime}_{0}}=\gamma\circ\mathrm{pr}.$

where $\mathrm{pr}$ is the projection of
$(\mathbb{K}^{*})^{\mathcal{V}}\times\bar{\mathcal{M}}$ onto the first factor.

$\bullet$$\bullet$$\bullet$$\boldsymbol{*}$$d_{1}$$\bullet$$\bullet$$\boldsymbol{*}$$d_{2}$$\bullet$$\boldsymbol{*}$$d_{3}$$\boldsymbol{*}$$d_{4}$$z_{6}$$z_{4}$$z_{1}$$\frac{z_{5}}{z_{6}}$$\frac{z_{2}}{z_{4}}$$\frac{z_{3}}{z_{5}}$$\frac{d_{1}}{d_{2}}\frac{1}{z_{6}}$$\frac{d_{1}}{d_{2}}\frac{z_{5}}{z_{4}z_{6}^{2}}$$\frac{d_{1}}{d_{2}}\frac{z_{2}z_{5}}{z_{1}z_{4}^{2}z_{6}^{2}}$$\frac{d_{2}}{d_{3}}\frac{z_{6}}{z_{5}}$$\frac{d_{2}}{d_{3}}\frac{z_{3}z_{4}z_{6}}{z_{2}z_{5}^{2}}$$\frac{d_{3}}{d_{4}}\frac{z_{5}}{z_{3}}$
Figure 18: Quiver decoration when $n=4$,
string coordinates

$\bullet$$\bullet$$\bullet$$\boldsymbol{*}$$d_{1}$$\bullet$$\bullet$$\boldsymbol{*}$$d_{2}$$\bullet$$\boldsymbol{*}$$d_{3}$$\boldsymbol{*}$$d_{4}$$m_{3}$$m_{2}$$m_{1}$$\frac{m_{3}m_{5}}{m_{2}}$$\frac{m_{2}m_{4}}{m_{1}}$$\frac{m_{3}m_{5}m_{6}}{m_{2}m_{4}}$$\frac{d_{1}}{d_{2}}\frac{m_{4}m_{5}}{{m_{1}}^{2}m_{2}}$$\frac{d_{2}}{d_{3}}\frac{m_{1}m_{6}}{m_{2}{m_{4}}^{2}}$$\frac{d_{3}}{d_{4}}\frac{m_{2}m_{4}}{m_{3}m_{5}m_{6}}$$\frac{d_{1}}{d_{2}}\frac{m_{4}}{{m_{1}}^{2}}$$\frac{d_{2}}{d_{3}}\frac{m_{1}}{m_{2}m_{4}}$$\frac{d_{1}}{d_{2}}\frac{1}{m_{1}}$
Figure 19: Quiver decoration when $n=4$,
ideal coordinates

To decorate the quiver with the ideal coordinates, we begin by recalling
Theorem 3.0.1, namely that for $k=1,\ldots,n-1$, $a=1,\ldots,n-k$ we have

$m_{s_{k}+a}=\begin{cases}z_{1+s_{n-a}}&\text{if }k=1,\\\
\frac{z_{k+s_{n-k-a+1}}}{z_{k-1+s_{n-k-a+1}}}&\text{otherwise}.\end{cases}$

In the above quiver decoration we have
$\boldsymbol{r}_{\mathcal{A}}=(p_{1},\ldots,p_{N})$, that is

$r_{a_{ij}}=p_{s_{j}+n-i}=\begin{cases}z_{1+s_{n-i}}&\text{if }j=1,\\\
\frac{z_{j+s_{n-i}}}{z_{j-1+s_{n-i+1}}}&\text{otherwise}.\end{cases}$

We notice that if $j=1$ then

$r_{a_{i1}}=p_{s_{1}+n-i}=z_{1+s_{n-i}}=m_{s_{1}+i}.$

We also want to write $r_{a_{ij}}$ in terms of the $m_{s_{k}+a}$ coordinates
when $j>2$. To do this we first note that for $k=2$ we have

$m_{s_{k}+a}m_{s_{k-1}+a+1}=\frac{z_{k+s_{n-k-a+1}}}{z_{k-1+s_{n-k-a+1}}}z_{k-1+s_{n-(k-1)-(a+1)+1}}=z_{k+s_{n-k-a+1}}.$

Similarly for $j>2$ we have

$m_{s_{k}+a}m_{s_{k-1}+a+1}=\frac{z_{k+s_{n-k-a+1}}}{z_{k-1+s_{n-k-a+1}}}\frac{z_{k-1+s_{n-k-a+1}}}{z_{k-2+s_{n-k-a+1}}}\\\
=\frac{z_{k+s_{n-k-a+1}}}{z_{k-2+s_{n-k-a+1}}}.$

Thus the following product is telescopic:

$\prod_{r=0}^{k-1}m_{s_{k-r}+a+r}=z_{k+s_{n-k-a+1}}.$

Now taking quotients of these products allows us to write the vertical arrow
coordinates, $r_{a_{ij}}=p_{s_{j}+n-i}$, in terms of the $m_{j}$; for $j>2$ we
have

$r_{a_{ij}}=p_{s_{j}+n-i}=\frac{z_{j+s_{n-i}}}{z_{j-1+s_{n-i+1}}}=\frac{z_{j+s_{n-j-(i-j+1)+1}}}{z_{j-1+s_{n-(j-1)-(i-j+1)+1}}}=\frac{\prod_{r=0}^{j-1}m_{s_{j-r}+i-j+1+r}}{\prod_{r=0}^{j-2}m_{s_{j-1-r}+i-j+1+r}}.$

This doesn’t seem particularly helpful at first glance, however it leads to an
iterative description of the new quiver decoration which will be very useful
in the proof of the main theorem in Section 4, Theorem 4.3.1. Denoting the
numerator of $r_{a_{ij}}$ by $n(r_{a_{ij}})$, for $j=1,\ldots,n-1$ we have

$r_{a_{i,j+1}}=\frac{\prod_{r=0}^{j}m_{s_{j+1-r}+i-j+r}}{\prod_{r=0}^{j-1}m_{s_{j-r}+i-j+r}}=m_{s_{j+1}+i-j}\frac{\prod_{r=0}^{j-1}m_{s_{j-r}+i+1-j+r}}{\prod_{r=0}^{j-1}m_{s_{j-r}+i-j+r}}=m_{s_{j+1}+i-j}\frac{n(r_{a_{ij}})}{n(r_{a_{i-1,j}})}.\\\
$ (17)

For example, in dimension $4$ the quiver decoration is given in Figure 19.

###  4.3 Critical points of the superpotential

We begin by recalling the highest weight map on the vertex torus:

$\kappa:(\mathbb{K}^{*})^{\mathcal{V}}\to
T^{\vee},\quad\boldsymbol{x}_{\mathcal{V}}\mapsto\left(x_{v_{ii}}\right)_{i=1,\ldots,n}.$

Now in the fibre over $d\in T^{\vee}$ we have

$x_{v_{ii}}=d_{i}\quad\text{for}\ i=1,\ldots,n.$

The remaining $x_{v}$ for $v\in\mathcal{V}^{\bullet}$ form a coordinate system
on this fibre. In particular we can use these coordinates to compute critical
points of the superpotential, as follows:

$x_{v}\frac{\partial\mathcal{F}}{\partial
x_{v}}=\sum_{a:h(a)=v}\frac{x_{h(a)}}{x_{t(a)}}-\sum_{a:t(a)=v}\frac{x_{h(a)}}{x_{t(a)}}.$

Thus the critical point conditions are

$\sum_{a:h(a)=v}\frac{x_{h(a)}}{x_{t(a)}}=\sum_{a:t(a)=v}\frac{x_{h(a)}}{x_{t(a)}}\quad\text{for}\
v\in\mathcal{V}^{\bullet}.$

Since we favour working with arrow coordinates, we rewrite these equations:

$\sum_{a:h(a)=v}r_{a}=\sum_{a:t(a)=v}r_{a}\quad\text{for}\
v\in\mathcal{V}^{\bullet}.$ (18)

This means we can now use the arrow coordinates in the quiver, and thus the
ideal coordinates, to give simple descriptions of both the superpotential and
the defining equations of its critical points. In fact, using the quiver
decoration in terms of the ideal coordinates, we can take this a step further:

######  Theorem 4.3.1.

If the critical point conditions hold at every dot vertex
$v\in\mathcal{V}^{\bullet}$, then the sum of the outgoing arrow coordinates at
each dot vertex $v_{ik}$ is given in terms of the ideal coordinates by

$\varpi(v_{ik}):=\sum_{a:t(a)=v_{ik}}r_{a}=m_{s_{k}+i-k}.$

###### Proof.

By construction of the quiver labelling there is only one outgoing arrow at
each $v_{i1}$ for $i=2,\ldots,n$, namely $a_{i-1,1}$, and indeed
$r_{a_{i-1,1}}=m_{i-1}=m_{s_{1}+i-1}$. Since we have the desired property for
the vertices $v_{i1}$, $i=2,\ldots,n-1$, we proceed by an inductive argument
increasing both vertex subscripts simultaneously.

We consider the subquiver given in Figure 20 and suppose the sum of outgoing
arrows at $v_{ik}$ is $m_{s_{k}+i-k}$. Then by the critical point condition at
this vertex we have $m_{s_{k}+i-k}=r_{a_{ik}}+r_{b_{i,k+1}}$.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$a_{i-1,k}$$a_{ik}$$a_{i,k+1}$$b_{i+1,k+1}$$b_{i,k+1}$$b_{ik}$$v_{i+1,k}$$v_{ik}$$v_{i+1,k+1}$$v_{i,k+1}$
Figure 20: Critical point conditions in the quiver

By the definition of the arrow coordinates we have

$r_{a_{i-1,k}}=m_{s_{k}+i-k}\frac{n(r_{a_{i-1,k-1}})}{n(r_{a_{i-2,k-1}})},\quad
r_{a_{i,k+1}}=m_{s_{k+1}+i-k}\frac{n(r_{a_{ik}})}{n(r_{a_{i-1,k}})},\quad
r_{a_{ik}}=m_{s_{k}+i+1-k}\frac{n(r_{a_{i,k-1}})}{n(r_{a_{i-1,k-1}})}.$

In particular we see that

$\frac{r_{a_{i,k+1}}}{r_{a_{ik}}}=m_{s_{k+1}+i-k}\frac{n(r_{a_{i-1,k-1}})}{n(r_{a_{i-1,k}})}.$
(19)

The sum of outgoing arrows at the vertex $v_{i+1,k+1}$ is given by

$\displaystyle\varpi(v_{i+1,k+1}):=\sum_{a:t(a)=v_{i+1,k+1}}r_{a}$
$\displaystyle=r_{a_{i,k+1}}+r_{b_{i+1,k+1}}$
$\displaystyle=r_{a_{i,k+1}}+\frac{r_{a_{i,k+1}}r_{b_{i,k+1}}}{r_{a_{ik}}}$
$\displaystyle\begin{aligned} \text{by the box relation}\\\
r_{a_{ik}}r_{b_{i+1,k+1}}=r_{a_{i,k+1}}r_{b_{i,k+1}}\end{aligned}$
$\displaystyle=r_{a_{i,k+1}}+\frac{r_{a_{i,k+1}}}{r_{a_{ik}}}(m_{s_{k}+i-k}-r_{a_{ik}})$
by the inductive hypothesis
$\displaystyle=\frac{r_{a_{i,k+1}}}{r_{a_{ik}}}m_{s_{k}+i-k}$
$\displaystyle=\frac{m_{s_{k+1}+i-k}n(r_{a_{i-1,k-1}})}{n(r_{a_{i-1,k}})}m_{s_{k}+i-k}$
$\displaystyle\text{using}\ (\ref{eqn quotient of arrow coords, crit point in
quiver})$ $\displaystyle=m_{s_{k+1}+i-k}$ $\displaystyle\text{by definition
of}\ n(r_{a_{i-1,k}})$ $\displaystyle=m_{s_{k+1}+(i+1)-(k+1)}.$

∎

######  Remark 4.3.2.

At first glance, in the above theorem we seem to have lost the information
about the highest weight element $d\in T^{\vee}$. However at a critical point
this information can be partially recovered from the
$\boldsymbol{m}$-coordinates; considering the dot vertices on the bottom wall
of the quiver, for each $j=1,\ldots,n-1$ we have

$\displaystyle r_{b_{n,j+1}}$ $\displaystyle=\varpi(v_{nj})$ by the critical
point conditions $\displaystyle=m_{s_{j}+n-j}$ $\displaystyle\text{by Theorem
\ref{thm crit points, sum at vertex is nu_i}}.$

By the quiver decoration, each arrow coordinate $r_{b_{n,j+1}}$ is given by

$r_{b_{n,j+1}}=\frac{d_{j}}{d_{j+1}}\frac{\prod_{i\in
I_{j_{1}}}m_{i}}{\prod_{i\in I_{j_{2}}}m_{i}}$

for some multisets $I_{j_{1}},I_{j_{2}}$ of the integers $1,\ldots,N$. Thus we
have

$\alpha^{\vee}_{j}(d)=\frac{d_{j}}{d_{j+1}}=m_{s_{j}+n-j}\frac{\prod_{i\in
I_{j_{2}}}m_{i}}{\prod_{i\in I_{j_{1}}}m_{i}},\quad j=1,\ldots,n-1.$

We complete this section by tying together the quiver, critical points and our
interest in the form of the weight matrix from Sections 2.4 and 3. Namely it
is natural to ask what happens to the weight matrix at critical points.

######  Proposition 4.3.3.

At a critical point in the fibre over $d\in T^{\vee}$, the weight matrix is an
$n\times n$ matrix $\mathrm{diag}(c,\ldots,c)$ where

$c^{n}=\prod_{i=1}^{n}d_{i}.$

In order to prove this we require the following lemma:

######  Lemma 4.3.4 ([13, Lemma 5.9]).

Suppose we have a quiver like the one given in Figure 22. We attach a variable
$r_{a}$ to each arrow such that the box relations
$r_{a_{1}}r_{a_{2}}=r_{a_{3}}r_{a_{4}}$ hold whenever
$a_{1},a_{2},a_{3},a_{4}$ form a square (see Figure 22) and the critical point
conditions hold at each black vertex. For each box $B_{j}$, $1\leq j\leq t$,
let $O_{j}=r_{a_{1}}r_{a_{3}}$ and $I_{j}=r_{a_{2}}r_{a_{4}}$. Additionally
let $K_{t}=\prod r_{a}$ where the product is over a (any) path from $v_{t}$ to
$v_{0}$. Then we have

$\prod_{j=1}^{t}O_{j}\frac{r_{a_{\mathrm{out}}}}{r_{a_{\mathrm{in}}}}=K_{t}\quad\text{and}\quad\prod_{j=1}^{t}I_{j}\frac{r_{a_{\mathrm{in}}}}{r_{a_{\mathrm{out}}}}=K_{t}.$

Note this agrees with $\prod_{j=1}^{t}O_{j}\prod_{j=1}^{t}I_{j}=K_{t}^{2}$.

$\boldsymbol{\circ}$$v_{-1}$$\bullet$$v_{0}$$\bullet$$\bullet$$\ddots$$\bullet$$\bullet$$v_{t}$$\boldsymbol{\circ}$$v_{t+1}$$\boldsymbol{\circ}$$\boldsymbol{\circ}$$\boldsymbol{\circ}$$\boldsymbol{\circ}$$\boldsymbol{\circ}$$\boldsymbol{\circ}$$B_{1}$$B_{2}$$B_{t}$$a_{\mathrm{out}}$$a_{\mathrm{in}}$
Figure 21: Diagonal subquiver

$a_{2}$$a_{3}$$a_{4}$$a_{1}$ Figure 22: Arrows forming a box

###### Proof of Proposition 4.3.3.

We consider the set of all arrows with either head or tail on the $i$th
diagonal $\mathcal{D}_{i}$. These arrows form a subquiver like in Figure 22,
with $v_{t}=v_{n,n-i+1}$. Moreover we have

$K_{t}=\frac{x_{v_{0}}}{x_{v_{t}}},\quad
r_{a_{\mathrm{in}}}=\frac{x_{v_{t}}}{x_{v_{t+1}}},\quad
r_{a_{\mathrm{out}}}=\frac{x_{v_{-1}}}{x_{v_{0}}},\quad\text{so}\quad\frac{1}{K_{t}}\frac{r_{a_{\mathrm{out}}}}{r_{a_{\mathrm{in}}}}=\frac{x_{v_{-1}}x_{v_{t+1}}}{x_{v_{0}}^{2}}.$

Thus, using Lemma 4.3.4, we see that

$\frac{t_{i-1}}{t_{i}}=\frac{\Xi_{i+1}\Xi_{i-1}}{\Xi_{i}^{2}}=\frac{x_{v_{-1}}x_{v_{t+1}}}{x_{v_{0}}^{2}}\prod
O_{j}=\frac{1}{K_{t}}\prod_{j=1}^{t}O_{j}\frac{r_{a_{\mathrm{out}}}}{r_{a_{\mathrm{in}}}}=1.$

So at a critical point, the weight matrix
$\mathrm{diag}(t_{1},t_{2},\ldots,t_{n})$ is given by
$\mathrm{diag}(c,c,\ldots,c)$ for some $c$. By taking the determinant we
obtain

$\prod_{i=1}^{n}t_{i}=\prod_{i=1}^{n}\frac{\Xi_{i}}{\Xi_{i+1}}=\frac{\Xi_{1}}{\Xi_{n+1}}=\Xi_{1}$

recalling $\Xi_{n+1}=1$ by definition. This gives the desired value of $c$ as
follows:

$c^{n}=\prod_{i=1}^{n}t_{i}=\Xi_{1}=\prod_{i=1}^{n}d_{i}.$

∎

##  5 The tropical viewpoint

In this section we recall how, by tropicalisation, we can use the
superpotential to obtain polytopes associated to a given highest weight. These
polytopes depend on the choice of positive toric chart. The goal in this
section is to describe the polytope we get from the ideal coordinates.
Additionally we show that for each choice of highest weight, the associated
critical point of the superpotential gives rise to a point inside this
polytope, which is Judd’s tropical critical point [13] and has a beautiful
description in terms of so called ideal fillings.

###  5.1 The basics of tropicalisation

In this section we explain the concept of tropicalisation, following an
original construction due to Lusztig [20]. In order to do this we work over
the field of Generalised Puiseux series, which we will denote by $\mathbf{K}$.

A generalised Puiseux series in a variable $t$ is a series with an exponent
set $(\mu_{k})=(\mu_{0},\mu_{1},\mu_{2},\ldots)\in\mathbb{R}$ which is
strictly monotone and either finite, or countable and tending to infinity.
That is,

$(\mu_{k})\in\mathrm{MonSeq}=\left\\{A\subset\mathbb{R}\ |\
\mathrm{Cardinality}(A\cap\mathbb{R}_{\leq x})<\infty\ \text{for arbitrarily
large}\ x\in\mathbb{R}\right\\}.$

Thus we have

$\mathbf{K}=\left\\{c(t)=\sum_{(\mu_{k})\in\mathrm{MonSeq}}c_{\mu_{k}}t^{\mu_{k}}\
|\ c_{\mu_{k}}\in\mathbb{C}\right\\}.$

The positive part of the field $\mathbf{K}$ is given by

$\mathbf{K}_{>0}:=\left\\{c(t)\in\mathbf{K}\ |\
c_{\mu_{0}}\in\mathbb{R}_{>0}\right\\}$

where we may assume that the lowest order term has a non-zero coefficient.

Given a torus $\mathcal{T}$, we note that we may identify
$\mathcal{T}(\mathbf{K})=\mathrm{Hom}(M,\mathbf{K}^{*})$, where we write $M$
for the character group of $\mathcal{T}$, $X^{*}(\mathcal{T})$, viewed as an
abstract group and written additively. The positive part of
$\mathcal{T}(\mathbf{K})$ is defined by those homomorphisms which take values
in $\mathbf{K}_{>0}$, namely
$\mathcal{T}(\mathbf{K}_{>0})=\mathrm{Hom}(M,\mathbf{K}_{>0})$. For $v\in M$
and $h\in\mathcal{T}(\mathbf{K})$ we will write $\chi^{v}(h)$ for the
associated evaluation $h(v)$ in $\mathbf{K}^{*}$. We call $\chi^{v}$ the
character associated to $v$.

We call a $\mathbf{K}$-linear combination of the characters $\chi^{v}$, a
Laurent polynomial on $\mathcal{T}$. In addition, a Laurent polynomial is said
to be positive if the coefficients of the characters lie in $\mathbf{K}_{>0}$.
Now let $\mathcal{T}^{(1)}$, $\mathcal{T}^{(2)}$ be two tori over
$\mathbf{K}$. We say that a rational map

$\psi:\mathcal{T}^{(1)}\dashrightarrow\mathcal{T}^{(2)}$

is a positive rational map if, for any character $\chi$ of
$\mathcal{T}^{(2)}$, the composition
$\chi\circ\psi:\mathcal{T}^{(1)}\to\mathbf{K}$ is given by a quotient of
positive Laurent polynomials on $\mathcal{T}^{(1)}$.

We now define the tropicalisation of these positive rational maps. Roughly
speaking, it captures what happens to the leading term exponents. In order to
define tropicalisation we use the natural valuation on $\mathbf{K}$ given by

$\mathrm{Val}_{\mathbf{K}}:\mathbf{K}\to\mathbb{R}\cup\\{\infty\\},\quad\mathrm{Val}_{\mathbf{K}}\left(c(t)\right)=\begin{cases}\mu_{0}&\text{if}\
c(t)=\sum_{(\mu_{k})\in\mathrm{MonSeq}}c_{\mu_{k}}t^{\mu_{k}}\neq 0,\\\
\infty&\text{if}\ c(t)=0.\end{cases}$

We define an equivalence relation $\sim$ on $\mathcal{T}(\mathbf{K}_{>0})$
using this valuation: we say $h\sim h^{\prime}$ if and only if
$\mathrm{Val}_{\mathbf{K}}(\chi(h))=\mathrm{Val}_{\mathbf{K}}(\chi(h^{\prime}))$
for all characters $\chi$ of $\mathcal{T}$. Then the tropicalisation of the
torus $\mathcal{T}$ is defined to be

$\mathrm{Trop}(\mathcal{T}):=T(\mathbf{K}_{>0})/\sim.$

This set inherits the structure of an abelian group from the group structure
of $\mathcal{\mathcal{T}}(\mathbf{K}_{>0})$, we denote this as addition.

In practical terms, when $\mathcal{T}=(\mathbf{K}^{*})^{r}$, so that
$\mathcal{T}(\mathbf{K}_{>0})=(\mathbf{K}_{>0})^{r}$, the valuation
$\mathrm{Val}_{\mathbf{K}}$ on each coordinate gives an identification

$\mathrm{Trop}(\mathcal{T})\to\mathbb{R}^{r},\quad\left[\left(c_{1}(t),\ldots,c_{r}(t)\right)\right]\mapsto\left(\mathrm{Val}_{\mathbf{K}}\left(c_{1}(t)\right),\ldots,\mathrm{Val}_{\mathbf{K}}\left(c_{r}(t)\right)\right).$

To state this in a coordinate-free way, if $\mathcal{T}$ is a torus with
cocharacter lattice $N:=X_{*}(\mathcal{T})$, then $\mathrm{Trop}(\mathcal{T})$
is identified with $N_{\mathbb{R}}=N\otimes\mathbb{R}$, see for example [14].
We note that $N_{\mathbb{R}}$ is also identified with the Lie algebra of the
torus taken over $\mathbb{R}$, for example
$\mathrm{Trop}(T^{\vee})=\mathfrak{h}^{*}_{\mathbb{R}}$ (c.f. Section 2.1).

We make the convention that if the coordinates of our torus
$\mathcal{T}\cong(\mathbf{K}^{*})^{r}$ are labelled by Roman letters, then the
corresponding coordinates on $\mathrm{Trop}(\mathcal{T})\cong\mathbb{R}^{r}$
are labelled by the associated Greek letters. In addition, by
$(\mathbf{K}^{*})^{r}_{\boldsymbol{b}}$ we mean $(\mathbf{K}^{*})^{r}$ with
coordinates $(b_{1},\ldots,b_{r})$, and similarly for
$\mathbb{R}^{N}_{\boldsymbol{\zeta}}$, etc.

Suppose that $\mathcal{T}^{(1)}$, $\mathcal{T}^{(2)}$ are two tori over
$\mathbf{K}$ and $\psi:\mathcal{T}^{(1)}\dashrightarrow\mathcal{T}^{(2)}$ is a
positive rational map. The map

$\psi(\mathbf{K}_{>0}):\mathcal{T}^{(1)}(\mathbf{K}_{>0})\to\mathcal{T}^{(2)}(\mathbf{K}_{>0})$

is well-defined and compatible with the equivalence relation $\sim$ (using the
positivity of the leading terms). The tropicalisation $\mathrm{Trop}(\psi)$ is
then defined to be the resulting map

$\mathrm{Trop}(\psi):\mathrm{Trop}(\mathcal{T}^{(1)})\to\mathrm{Trop}(\mathcal{T}^{(2)})$

between equivalence classes. It is piecewise-linear with respect to the linear
structures on the $\mathrm{Trop}(\mathcal{T}^{(i)})$.

In the case of a variety $X$ with a ‘positive atlas’ consisting of torus
charts related by positive birational maps (see [7], [4]), there is a well-
defined positive part $X(\mathbf{K}_{>0})$ and tropical version
$\mathrm{Trop}(X)$, which comes with a tropical atlas whose tropical charts
$\mathrm{Trop}(X)\to\mathbb{R}^{r}$ are related by piecewise-linear maps.
$\mathrm{Trop}(X)$ in this more general setting is a space with a piecewise-
linear structure.

######  Example 5.1.1.

Let $\mathcal{T}=(\mathbf{K}^{*})^{2}_{\boldsymbol{b}}$ and consider the
following map222 This is the superpotential for $\mathbb{CP}^{2}$ (see [6]). :

$\psi:\mathcal{T}\to\mathbf{K},\quad\psi(b_{1},b_{2})=b_{1}+b_{2}+\frac{t^{3}}{b_{1}b_{2}}.$

We may consider $\psi$ as a positive birational map
$\mathcal{T}\dashrightarrow\mathbf{K}^{*}$, and the corresponding map
$\mathrm{Trop}(\psi):\mathrm{Trop}(\mathcal{T})\to\mathbb{R}$ is given in
terms of the natural coordinates $\beta_{1},\beta_{2}$ on
$\mathrm{Trop}(\mathcal{T})\cong\mathbb{R}^{2}$ by

$\mathrm{Trop}(\psi)(\beta_{1},\beta_{2})=\min\\{\beta_{1},\beta_{2},3-\beta_{1}-\beta_{2}\\}.$

In practice, we may think of tropicalisation as replacing addition by $\min$
and replacing multiplication by addition.

###  5.2 Constructing polytopes

In this section we return to the Landau-Ginzburg model for $G/B$, defined in
Section 2.2 as the pair $(Z,\mathcal{W})$. Working now over the field of
generalised Puiseux series, $\mathbf{K}$, there is a well-defined notion of
the totally positive part of $Z(\mathbf{K})$, denoted by $Z(\mathbf{K}_{>0})$.
It is defined, for a given torus chart on $Z(\mathbf{K})$, by the subset where
the characters take values in $\mathbf{K}_{>0}$. Moreover, each of the string,
ideal and quiver torus charts mentioned in previous sections, gives an
isomorphism

$T^{\vee}(\mathbf{K}_{>0})\times(\mathbf{K}_{>0})^{N}\xrightarrow{\sim}Z(\mathbf{K}_{>0})$
(20)

where we consider $T^{\vee}(\mathbf{K}_{>0})$ to be the highest weight torus.

We will now restrict our attention to a fibre of the highest weight map (see
Section 2.2). To do so, we observe that since a dominant integral weight
$\lambda\in X^{*}(T)^{+}$ is a cocharacter of $T^{\vee}$, we can define
$t^{\lambda}\in T^{\vee}(\mathbf{K}_{>0})$ via the condition
$\chi(t^{\lambda})=t^{\langle\lambda,\chi\rangle}$ for $\chi\in
X^{*}(T^{\vee})$. Extending $\mathbb{R}$-bilinearly to the perfect pairing

$\langle\ ,\ \rangle:X^{*}(T)_{\mathbb{R}}\times
X^{*}(T^{\vee})_{\mathbb{R}}\to\mathbb{R},$

we have that $t^{\lambda}$ is well defined for all $\lambda\in
X^{*}(T)_{\mathbb{R}}$, by the same formula. We therefore do not require that
$\lambda$ be integral, though we continue to be interested in those $\lambda$
which are dominant, that is $\lambda\in X^{*}(T)^{+}_{\mathbb{R}}$. This
allows us, for a dominant weight $\lambda\in X^{*}(T)^{+}_{\mathbb{R}}$, to
define

$Z_{t^{\lambda}}(\mathbf{K}):=\\{b\in Z(\mathbf{K})\ |\
\mathrm{hw}(b)=t^{\lambda}\\}.$

We denote the restriction of the superpotential to this fibre by

$\mathcal{W}_{t^{\lambda}}:Z_{t^{\lambda}}(\mathbf{K})\to\mathbf{K}.$

For a fixed element $t^{\lambda}\in T^{\vee}(\mathbf{K}_{>0})$ of the highest
weight torus, the isomorphisms (20) for the string and ideal toric charts
restrict to

$\displaystyle\phi_{t^{\lambda},\boldsymbol{z}}$
$\displaystyle:(\mathbf{K}_{>0})^{N}\to Z_{t^{\lambda}}(\mathbf{K}_{>0}),$
$\displaystyle\phi_{t^{\lambda},\boldsymbol{m}}$
$\displaystyle:(\mathbf{K}_{>0})^{N}\to Z_{t^{\lambda}}(\mathbf{K}_{>0})$

respectively, with coordinates denoted by $(z_{1},\ldots,z_{N})$ and
$(m_{1},\ldots,m_{N})$. These toric charts may be considered as defining a
positive atlas for $Z_{t^{\lambda}}(\mathbf{K}_{>0})$. We denote the
respective compositions of $\phi_{t^{\lambda},\boldsymbol{z}}$ and
$\phi_{t^{\lambda},\boldsymbol{m}}$ with the superpotential
$\mathcal{W}_{t^{\lambda}}$, by

$\displaystyle\mathcal{W}_{t^{\lambda},\boldsymbol{z}}$
$\displaystyle:(\mathbf{K}_{>0})^{N}\to\mathbf{K}_{>0},$
$\displaystyle\mathcal{W}_{t^{\lambda},\boldsymbol{m}}$
$\displaystyle:(\mathbf{K}_{>0})^{N}\to\mathbf{K}_{>0}$

and observe that both are positive rational maps. We denote their
tropicalisations respectively by

$\displaystyle\mathrm{Trop}\left(\mathcal{W}_{t^{\lambda},\boldsymbol{z}}\right)$
$\displaystyle:\mathbb{R}^{N}_{\boldsymbol{\zeta}}\to\mathbb{R},$
$\displaystyle\mathrm{Trop}\left(\mathcal{W}_{t^{\lambda},\boldsymbol{m}}\right)$
$\displaystyle:\mathbb{R}^{N}_{\boldsymbol{\mu}}\to\mathbb{R}.$

We may associate convex polytopes to our tropical superpotentials, defined as
follows:

$\displaystyle\mathcal{P}_{\lambda,\boldsymbol{\zeta}}$
$\displaystyle:=\left\\{\boldsymbol{\alpha}\in\mathbb{R}^{N}_{\boldsymbol{\zeta}}\
|\
\mathrm{Trop}\left(\mathcal{W}_{t^{\lambda},\boldsymbol{z}}\right)(\boldsymbol{\alpha})\geq
0\right\\},$ $\displaystyle\mathcal{P}_{\lambda,\boldsymbol{\mu}}$
$\displaystyle:=\left\\{\boldsymbol{\alpha}\in\mathbb{R}^{N}_{\boldsymbol{\mu}}\
|\
\mathrm{Trop}\left(\mathcal{W}_{t^{\lambda},\boldsymbol{m}}\right)(\boldsymbol{\alpha})\geq
0\right\\}.$

To motivate the definition of these polytopes, first recall the string toric
chart, $\varphi_{\mathbf{i}}$, for an arbitrary reduced expression
$\mathbf{i}$, defined by (1) in Section 2.3. Using this we have
generalisations

$\displaystyle\phi^{\mathbf{i}}_{t^{\lambda},\boldsymbol{z}}$
$\displaystyle:(\mathbf{K}_{>0})^{N}\to Z_{t^{\lambda}}(\mathbf{K}_{>0}),$
$\displaystyle\mathcal{W}^{\mathbf{i}}_{t^{\lambda},\boldsymbol{z}}$
$\displaystyle:(\mathbf{K}_{>0})^{N}\to\mathbf{K}_{>0}$

of the maps above, such that
$\phi_{t^{\lambda},\boldsymbol{z}}=\phi^{\mathbf{i}_{0}}_{t^{\lambda},\boldsymbol{z}}$
and
$\mathcal{W}_{t^{\lambda},\boldsymbol{z}}=\mathcal{W}^{\mathbf{i}_{0}}_{t^{\lambda},\boldsymbol{z}}$.
With this notation we have the following theorem:

######  Theorem 5.2.1 ([13, Theorem 4.1]).

Consider a general reduced expression $\mathbf{i}$ for $\bar{w}_{0}$, and the
superpotential for $GL_{n}/B$ written in the associated string coordinates,
namely $\mathcal{W}^{\mathbf{i}}_{t^{\lambda},\boldsymbol{z}}$. Then the
polytope

$\mathcal{P}^{\mathbf{i}}_{\lambda,\boldsymbol{\zeta}}=\left\\{\boldsymbol{\alpha}\in\mathbb{R}^{N}_{\boldsymbol{\zeta}}\
\big{|}\
\mathrm{Trop}\left(\mathcal{W}^{\mathbf{i}}_{t^{\lambda},\boldsymbol{z}}\right)(\boldsymbol{\alpha})\geq
0\right\\}$