# Towards Landau-Ginzburg models for cominuscule Spaces via the exceptional
cominuscule family

Peter Spacek  Technische Universität Chemnitz, Germany<EMAIL_ADDRESS>chemnitz.de and  Charles Wang  Harvard University, USA
<EMAIL_ADDRESS>
###### Abstract.

We present projective Landau-Ginzburg models for the exceptional cominuscule
homogeneous spaces $\mathbb{OP}^{2}=E_{6}^{\mathrm{sc}}/P_{6}$ and
$E_{7}^{\mathrm{sc}}/P_{7}$, known respectively as the Cayley plane and the
Freudenthal variety. These models are defined on the complement
$X^{\vee}_{\mathrm{can}}$ of an anti-canonical divisor of the “Langlands dual
homogeneous spaces” $\mathbb{X}^{\vee}=P^{\vee}\backslash G^{\vee}$ in terms
of generalized Plücker coordinates, analogous to the canonical models defined
for Grassmannians, quadrics and Lagrangian Grassmannians in [MR20, PRW16,
PR13]. We prove that these models for the exceptional family are isomorphic to
the Lie-theoretic mirror models defined in [Rie08] using a restriction to an
algebraic torus, also known as the _Lusztig torus_ , as proven in [Spa21]. We
also give a cluster structure on $\mathbb{C}[X^{\vee}]$, prove that the
Plücker coordinates form a Khovanskii basis for a valuation defined using the
Lusztig torus, and compute the Newton-Okounkov body associated to this
valuation. Although we present our methods for the exceptional types, they
generalize immediately to the members of other cominuscule families.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Conventions and notation
    2. 2.2 Rietsch’s Lie-theoretic mirror model and a local Laurent polynomial expression
    3. 2.3 Generalized Plücker coordinates and the coordinate ring of $(U_{-}^{P})^{T}$
    4. 2.4 The exceptional cominuscule family
  3. 3 The canonical models for cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$
    1. 3.1 The canonical model for the Cayley plane.
    2. 3.2 The canonical model for the Freudenthal variety.
    3. 3.3 Overview of the isomorphisms
    4. 3.4 The coordinate ring of $U_{-}^{P}$ for the Cayley plane
    5. 3.5 The coordinate ring of $U_{-}^{P}$ for the Freudenthal variety
    6. 3.6 Proof of the isomorphism of LG-models on cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$
  4. 4 Cluster structures on $\mathbb{X}^{\vee}$
    1. 4.1 Cluster algebras and varieties
    2. 4.2 Cluster structures on homogeneous spaces
    3. 4.3 Cluster algebras and mirror symmetry
    4. 4.4 The Cayley plane
    5. 4.5 The Freudenthal variety
  5. 5 Valuations, Khovanskii bases, and Newton-Okounkov bodies
  6. A Torus expansion example
  7. B Plücker coordinate valuations

##  1\. Introduction

Let $X$ be a projective homogeneous space $X=G/P$, for a simple and simply-
connected $G$ and parabolic subgroup $P$. The (small) quantum cohomology of
such $X$ has a long history of research, starting initially with the
consideration of Grassmannians by Witten in [Wit95]—see [Ful04] for a survey.
In this article, we will be considering mirror symmetry statements regarding
the (small) quantum cohomology ring of $X$ via a _Landau-Ginzburg model_. In
this context, a Landau-Ginzburg model consists of a mirror space $X^{\vee}$
and a superpotential $\mathcal{W}_{q}$ which is a regular function on
$X^{\vee}$ such that the _Jacobi ring_ of $\mathcal{W}_{q}$ is isomorphic to
the (small) quantum cohomology ring of $X$ with localized quantum parameters,
i.e.

$qH^{*}(X)_{\mathrm{loc}}\cong\mathbb{C}[X^{\vee}]/\langle\partial\mathcal{W}_{q}\rangle.$

In [Rie08], a Lie-theoretic construction of a Landau-Ginzburg model is given
and shown to be isomorphic to Peterson’s presentation of the small quantum
cohomology of general projective homogeneous spaces as given in [Pet97]. The
superpotential of this model is defined completely intrinsically, using
natural maps on the Langlands dual Lie group $G^{\vee}$, namely the dual maps
associated to a set of Chevalley generators on the Lie algebra
$\mathfrak{g}^{\vee}$ of $G^{\vee}$.

In practice it is more convenient to have a Landau-Ginzburg model whose
superpotential is expressed in terms of coordinates on the mirror space. This
is especially the case for homogeneous spaces that have natural coordinates in
the form of (generalized) Plücker coordinates. Hence, the results in [Rie08]
were considered for a number of families of homogeneous spaces that were
particularly accessible, namely Grassmannians [MR20], quadrics [PRW16] and
Lagrangian Grassmannians [PR13]. These families are all _cominuscule_
homogeneous spaces, which implies that the Plücker coordinates on the
Langlands dual homogeneous space $\mathbb{X}^{\vee}=P^{\vee}\backslash
G^{\vee}$ can be defined in terms of projections on a _minuscule_
representation. The structure of these representations is particularly well-
understood—see e.g. [Gre08] or [Gec17]—and for the families mentioned above
the representations are classical matrix representations.

In this paper, we consider the exceptional cominuscule family: the Cayley
plane, $E_{6}^{\mathrm{sc}}/P_{6}$ (often written as $\mathbb{OP}^{2}$), and
the Freudenthal variety, $E_{7}^{\mathrm{sc}}/P_{7}$, where
$E_{n}^{\mathrm{sc}}$ denotes the simply-connected Lie group of type $E$ and
rank $n$. Although less studied than the classical homogeneous spaces, the
geometry of the exceptional cominuscule family is still well-known—see e.g.
[IM05] for an overview and calculation of the Chow ring of the Cayley plane.
Moreover, the ring structure of the quantum cohomologies of the exceptional
family has been described in detail in [CMP08]; there the quantum cohomologies
of all cominuscule homogeneous spaces are presented in a uniform way. However,
to our knowledge, the only known Landau-Ginzburg models available until now
are Rietsch’s Lie-theoretic model and the first author’s Laurent polynomial
expression for the model on an algebraic torus, sometimes called the _Lusztig
torus_ , as given in [Spa21], neither of which give expression in terms of
coordinates on the mirror space.

We present Plücker coordinate Landau-Ginzburg models
$(X^{\vee}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})$ for the exceptional
cominuscule family. The superpotentials are defined as

$\displaystyle\mathcal{W}_{\mathrm{can}}$
$\displaystyle=\frac{p_{1}}{p_{0}}+\frac{p_{9}^{\prime}}{p_{8}}+\frac{q_{13}}{q_{12}}+\frac{q_{17}}{q_{16}}+\frac{q_{21}}{q_{20}}+\frac{q_{25}}{q_{24}}+q\frac{p_{5}^{\prime\prime}}{p_{16}},$
$\displaystyle\mathcal{W}_{\mathrm{can}}$
$\displaystyle=\frac{p_{1}}{p_{0}}+\frac{q_{19}}{q_{18}}+\frac{q_{28}}{q_{27}}+\frac{q_{37}}{q_{36}}+\frac{q_{37}^{\prime}}{q_{36}^{\prime}}+\frac{q_{46}}{q_{45}}+\frac{q_{55}}{q_{54}}+q\frac{p_{10}}{p_{27}}$

for $E_{6}^{\mathrm{sc}}/P_{6}$ and $E_{7}^{\mathrm{sc}}/P_{7}$ respectively,
where the $p_{i}$ are Plücker coordinates and the $q_{i}$ are certain
polynomials in the Plücker coordinates given in Sections 3.1 and 3.2. The
superpotentials are regular functions on $\mathbb{X}^{\vee}$ outside of an
anticanonical divisor $D$ which defines the mirror space as
$X^{\vee}_{\mathrm{can}}=\mathbb{X}^{\vee}\setminus D$; we follow the
convention of [PRW16] in calling such models _canonical Landau-Ginzburg
models_. Our formulas give an interesting quantum Schubert calculus
interpretation of our superpotentials above (see Remark 3.3 for more details):
for a given term of a superpotential, its numerator is equal to the quantum
product of $\sigma_{1}$ with its denominator, multiplied by the degree of its
denominator as a polynomial in the Plücker coordinates. Because the sum of the
degrees of the denominators is equal to the index $I$ in each case — $12$ for
the Cayley plane and $18$ for the Freudenthal variety — we see that
$\mathcal{W}_{\mathrm{can}}$ is equal in some sense to $I$ copies of the
hyperplane class, and thus represents the anticanonical class, e.g. in the
Jacobi ring. Similar observations were made in type $A$ in [MR20] and in type
$C$ in [PR13]. We are working on a generalization of this observation that we
believe holds for all cominuscule spaces, and this result will appear in a
future article.

Our main tool for producing these formulas is the expression of generalized
minors in terms of the coordinates of the Lusztig torus, which we summarize in
Algorithms 3.11 and 3.12. Two of our main results are the following:

######  Theorem 1.1 (Theorem 3.1).

For cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$, the variety
$X^{\vee}_{\mathrm{can}}$ is isomorphic to the open Richardson variety
$X^{\vee}_{\mathrm{Lie}}=\mathcal{R}^{\vee}_{w_{P},w_{0}}$.

######  Theorem 1.2 (Theorem 3.2).

For cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$, the pull-back of
$\mathcal{W}_{\mathrm{can}}$ under the isomorphism above equals
$\mathcal{W}_{\mathrm{Lie}}$. In particular, the Landau-Ginzburg model
$(X^{\vee}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})$ is isomorphic to
Rietsch’s Lie-theoretic Landau-Ginzburg model
$(X^{\vee}_{\mathrm{Lie}},\mathcal{W}_{\mathrm{Lie}})$.

Including our results, canonical models are now available for four out of the
five cominuscule families: Grassmannians, Lagrangian Grassmannians, quadrics,
and the exceptional family. The remaining family, consisting of the orthogonal
Grassmannians $\mathrm{OG}(n,2n)$ of maximal isotropic subspaces of
$\mathbb{C}^{2n}$ with respect to the standard inner product, is expected to
have a model that is mostly similar to that of the Lagrangian Grassmannians.
All canonical models, including for the exceptional case, have superpotentials
consisting of the sum of quotients of polynomials in the Plücker coordinates.
For Grassmannians these polynomials were linear, whereas for quadrics,
Lagrangian Grassmannians, and (conjecturally) orthogonal Grassmannians these
polynomials are quadratic. In contrast, the canonical model for the Cayley
plane has a term with cubic polynomials, and the canonical model for the
Freudenthal variety has a term with quartic polynomials as well as two terms
with cubic polynomials.

In fact, our methods apply to all cominuscule homogeneous spaces, with
completely analogous proofs. The models obtained in this way will coincide
with the canonical models for the Grassmannians, Lagrangian Grassmannians, and
quadrics, and our methods will provide similar models for the orthogonal
family. In the present form, our approach only handles one member of a family
at the time; however, with general representation theoretic knowledge and the
associated combinatorial tools, it should be possible to perform our methods
either inductively or in generality.

In [PRW16], it is noted that the results of [GLS08] imply that the coordinate
rings of $X^{\vee}_{\mathrm{can}}$ and $\mathbb{X}^{\vee}$ carry a cluster
structure. In particular, we view $X^{\vee}_{\mathrm{can}}$ as an open cluster
variety and $\mathbb{X}^{\vee}$ as a compactification of
$X^{\vee}_{\mathrm{can}}$ given by frozen variables, whose divisors form an
anticanonical divisor. On the level of coordinate rings, this means that we
obtain $\mathbb{C}[X^{\vee}_{\mathrm{can}}]$ from
$\mathbb{C}[\mathbb{X}^{\vee}]$ by inverting frozen variables. In Section 4,
we give a Plücker coordinate description for the exceptional cominuscule
family of the cluster structures given in [GLS08]. The cluster variables are
given in Lemmas 4.3 and 4.5, and the quivers are given in Figures 4.1 and 4.2.
In order to compute the Plücker coordinate expressions for the cluster
variables, we formulated Algorithm 4.6 for expressing generalized minors in
terms of Plücker coordinates. This algorithm is reminiscent of the _subduction
algorithm_ for _Khovanskii bases_ , and we were able to use it to define a
_valuation_ $\nu$ on the coordinate ring $\mathbb{C}[\mathbb{X}^{\vee}]$. With
this, we proved the following theorems:

######  Theorem 1.3 (Lemma 5.2 and Proposition 5.6).

For minuscule $\mathbb{X}^{\vee}=(E_{n}^{\mathrm{sc}}/P_{n})^{\vee}$, the
valuation $\nu$ defined on $\mathbb{C}[\mathbb{X}^{\vee}]$ above has _one-
dimensional leaves_ , and the associated _Newton-Okounkov body_
$\Delta(\mathbb{C}[\mathbb{X}^{\vee}],\nu)$ is equal to the convex hull of the
valuations of the Plücker coordinates on $\mathbb{X}^{\vee}$.

######  Theorem 1.4 (Proposition 5.7).

For minuscule $\mathbb{X}^{\vee}=(E_{n}^{\mathrm{sc}}/P_{n})^{\vee}$, the
Plücker coordinates form a Khovanskii basis for
$\mathbb{C}[\mathbb{X}^{\vee}]$ with respect to the valuation $\nu$ defined on
$\mathbb{C}[\mathbb{X}^{\vee}]$ above.

Although we only gave the statements above for
$\mathbb{X}^{\vee}=E_{n}^{\mathrm{sc}}/P_{n}$, our proofs do not depend on the
exceptional types in a meaningful way, and we propose in Conjecture 5.9 that
the Plücker coordinates form a Khovanskii basis for a family of valuations
depending on a choice of reduced expression for a particular Weyl group
element. In particular, we obtain from this a family of Newton-Okounkov bodies
as the convex hulls of the valuations of Plücker coordinates. While our
conjecture makes sense for any homogeneous space, we are unsure what
additional conditions—if any—are required to obtain Khovanskii bases and
Newton-Okounkov bodies.

We expect that our main results—these Plücker coordinate canonical models, the
Plücker coordinate cluster structures, and the Khovanskii bases and Newton-
Okounkov bodies—can be formulated type-independently for cominuscule and
possibly more general homogeneous spaces. This stems from the fact that many
of the intermediate results are type-independent themselves: the Lie-theoretic
model of [Rie08] holds for general homogeneous spaces; the Laurent polynomial
expression for this model in [Spa21] holds for all cominuscule spaces and is
currently being extended to adjoint spaces by the first author; the Lie-
theoretic and canonical models are compared using results by [GLS11] which
hold on the coordinate rings of general unipotent cells; and finally the
cluster structures given by [GLS08] are also valid on general unipotent cells.
In fact, in Proposition 3.8 we give a criterion expressing the generators of
the coordinate rings given in [GLS11] in terms of Plücker coordinates that
holds for general cominuscule homogeneous spaces. Thus, the only obstruction
to presenting canonical models and cluster structures type-independently is
expressing the generalized minors appearing in [GLS08] and [GLS11] in terms of
Plücker coordinates.

In [RW19], the authors study a combinatorial version of mirror symmetry in the
form of an equality of polytopes defined using the auxiliary data of _cluster
structures_. There, cluster structures are used to construct a family of
Newton-Okounkov associated with $X$ and a family of polytopes associated with
the superpotential on $X^{\vee}_{\mathrm{can}}$. These families are indexed by
_seeds_ in the cluster structures, and it is shown that these two polytopes
coincide for any choice of seed. While we do not pursue the story here, we
provide many of the necessary ingredients, and it would be interesting to
explore the combinatorics and see whether there is any relation between our
Newton-Okounkov bodies and those coming from the cluster structure.

Apart from being of interest on their own, the canonical models for
homogeneous spaces have appeared in a number of other contexts as well.
Firstly, it was conjectured in [Rie08] that solutions to the quantum
differential equations of homogeneous spaces can be obtained from oscillatory
integrals associated to the Landau-Ginzburg models. In [LT17] this conjecture
was proven for _minuscule_ homogeneous spaces using the language of
$D$-modules and the Langlands correspondence. However, as noted there, the
isomorphism given in this proof is not constructed explicitly. In [PRW16], an
explicit injective morphism between the $D$-modules in question is given using
the canonical mirror model for quadrics; for odd-dimensional quadrics (that
are cominuscule but _not_ minuscule) an implicit isomorphism was known before
so that their morphism gave an isomorphism; for even-dimensional quadrics
(that are both cominuscule _and_ minuscule) the results of [LT17] implied that
their injective morphism is in fact an isomorphism. We believe that the
canonical model presented for the exceptional cominuscule family will lead to
an analogous explicit morphism of $D$-modules, which will then be an
isomorphism by [LT17], but we do not verify this here.

Finally, in [Cas20], the canonical model for Grassmannians was modified to
prove _homological mirror symmetry_ for Grassmannians $\mathrm{Gr}(k,n)$ with
$n$ prime. This is an isomorphism of triangulated categories between the
Fukaya category of a variety and the (bounded) derived category of coherent
sheaves on its mirror. With canonical models available for four out of the
five cominuscule families, these results form promising first steps in
homological mirror symmetry for homogeneous spaces.

The outline of this article is as follows. In Section $2$, we set up our
conventions and notation, and recall background on Rietsch’s Lie-theoretic
Langdau-Ginzburg models, the Cayley plane, and the Freudenthal variety. In
Section $3$, we define our canonical models, give their Plücker coordinate
expressions, and prove the isomorphism with Rietsch’s Lie-theoretic model. In
Section $4$, we review background on cluster structures and give Plücker
coordinate cluster structures for the Cayley plane and Freudenthal variety,
and in Section $5$, we define our Newton-Okounkov body and show that the
Plücker coordinates give Khovanskii bases in both cases. In the appendices, we
provide extra data and examples from our computations in Sections $3$ and $5$.

#### Acknowledgements

We implemented Algorithms 3.11 and 4.6 in Sage [The21] and Macaulay2 [GS], and
used Polymake [GJ00, AGH+17] for Newton-Okounkov body computations. Our code
is available at https://github.com/foxflo/ExceptionalCominusculeSpaces. The
authors are grateful to Clélia Pech and Lauren Williams for many helpful
comments and suggestions.

##  2\. Preliminaries

In the first two subsections (Subsections 2.1 and 2.2), we will consider
general complete homogeneous spaces $X$ for a simple and simply-connected
complex algebraic group $G$ of rank $n$. These spaces are also often called
_generalized flag varieties_. We will work with these spaces in their
(Plücker) embeddings as the closed orbits of the highest weight vectors in the
projectivizations of the corresponding irreducible representations.
Afterwards, beginning from Subsection 2.2.2, we will restrict to the cases
when $X$ is _cominuscule_.

###  2.1. Conventions and notation

We write $\mathfrak{g}$ for the Lie algebra of $G$ and choose a set
$(e_{1},f_{1},h_{1},\ldots,e_{n},f_{n},h_{n})$ of Chevalley generators, i.e.
triples $(e_{i},f_{i},h_{i}=[e_{i},f_{i}])$ that generate Lie subalgebras
isomorphic to $\mathfrak{sl}_{2}$. These generators give a Cartan (or root
space) decomposition of the Lie algebra as
$\mathfrak{g}=\mathfrak{u}_{+}\oplus\mathfrak{t}\oplus\mathfrak{u}_{-}$ with
$\mathfrak{u}_{+}$ and $\mathfrak{u}_{-}$ generated by $\\{e_{i}\\}$ and
$\\{f_{i}\\}$ respectively, and with $\mathfrak{t}$ spanned by $\\{h_{i}\\}$.

The Cartan decomposition of $\mathfrak{g}$ gives rise to subgroups of $G$: let
$U_{+}$ and $U_{-}$ be the (upper and lower) unipotent subgroups of $G$ with
$\mathfrak{u}_{+}$ and $\mathfrak{u}_{-}$ as Lie algebras, and let $T$ be the
maximal torus of $G$ with $\mathfrak{t}$ as Lie algebra. We obtain a Borel
subgroup $B_{+}=TU_{+}$ and its opposite $B_{-}=TU_{-}$; they have
$\mathfrak{t}\oplus\mathfrak{u}_{+}$ and $\mathfrak{t}\oplus\mathfrak{u}_{-}$
as Lie algebras. The Borel subgroup defines aa unique parabolic subgroup $P$
containing $B_{+}$ such that $X=G/P$. Its Lie algebra $\mathfrak{p}$ satisfies
$\mathfrak{u}_{+}\oplus\mathfrak{t}\subset\mathfrak{p}\subset\mathfrak{u}_{+}\oplus\mathfrak{t}\oplus\mathfrak{u}_{-}=\mathfrak{g}$;
we will denote by $I_{P}\subset\\{1,\ldots,n\\}$ the set of indices such that

$\mathfrak{p}=\langle e_{i},h_{i},f_{j}~{}|~{}i\in\\{1,\ldots,n\\},~{}j\in
I_{P}\rangle$ (2.1)

and its complement is denoted by $I^{P}=\\{1,\ldots,n\\}\setminus I_{P}$.

Denote by $\Phi$ the set of roots and let
$\Delta=\\{\alpha_{1},\ldots,\alpha_{n}\\}$ be the base of simple roots
determined by the Chevalley generators. The positive and negative determined
by the base are denoted by $\Pi$ and $\Pi_{-}$ respectively. Writing
$\Phi^{\vee}$ for the coroots of $G$, there is a unique pair
$(G^{\vee},T^{\vee})$ of a Lie group $G^{\vee}$ and maximal torus
$T^{\vee}\subset G^{\vee}$ that have the coroots $\Phi^{\vee}$ of $G$ as roots
and the cocharacter lattice of $G$ as character lattice; this is called the
_Langlands dual pair_ and we will refer to $G^{\vee}$ as the _Langlands dual
group_.

######  Remark 2.1.

As $G$ is assumed to be simply-connected, $G^{\vee}$ will be adjoint.

$\Delta\\!^{\vee}=\\{\alpha^{\vee}_{1},\ldots,\alpha^{\vee}_{n}\\}$ forms a
base of simple roots for $G^{\vee}$ and determines both a decomposition
$\mathfrak{g}^{\vee}=\mathfrak{u}^{\vee}_{-}\oplus\mathfrak{t}^{\vee}\oplus\mathfrak{u}^{\vee}_{+}$
of the Lie algebra of $G^{\vee}$ and positive $\Pi^{\vee}$ and negative roots
$\Pi^{\vee}_{-}$ for $G^{\vee}$. We obtain the Langlands dual groups
$U^{\vee}_{+}$, $U^{\vee}_{-}$, $B^{\vee}_{+}$, $B^{\vee}_{-}$ and $P^{\vee}$
analogously to above.

We will also need the universal enveloping algebras of
$\mathfrak{u}^{\vee}_{+}$ and $\mathfrak{u}^{\vee}_{-}$, which we will denote
by $\mathcal{U}^{\vee}_{+}$ and $\mathcal{U}^{\vee}_{-}$, and their
completions $\widehat{\mathcal{U}}^{\vee}_{+}$ and
$\widehat{\mathcal{U}}^{\vee}_{-}$. Note that we can consider
$U^{\vee}_{+}\subset\widehat{\mathcal{U}}^{\vee}_{+}$ and
$U^{\vee}_{-}\subset\widehat{\mathcal{U}}^{\vee}_{-}$ by identifying the one-
parameter subgroups

$x^{\vee}_{i}(a)=\exp(a\,e^{\vee}_{i})\quad\text{and}\quad
y^{\vee}_{i}(a)=\exp(a\,f^{\vee}_{i})$ (2.2)

for $i\in\\{1,\ldots,n\\}$ and $a\in\mathbb{C}$. Here
$(e^{\vee}_{1},f^{\vee}_{1},h^{\vee}_{1},\ldots,e^{\vee}_{n},f^{\vee}_{n},h^{\vee}_{n})$
are the Chevalley generators for $\mathfrak{g}^{\vee}$ and
$\exp(a\,e^{\vee}_{i})=1+a\,e^{\vee}_{i}+\frac{1}{2}a^{2}(e^{\vee}_{i})^{2}+\ldots\in\widehat{\mathcal{U}}^{\vee}_{+}$.
Note that the parabolic subgroup $P^{\vee}$ is associated to the same set
$I_{P}$ as $P$: that is, its Lie algebra is given by

$\mathfrak{p}^{\vee}=\langle
e^{\vee}_{i},h^{\vee}_{i},f^{\vee}_{j}~{}|~{}i\in\\{1,\ldots,n\\},~{}j\in
I_{P}\rangle.$ (2.3)

The complement of the set of indices is the same, and is denoted by
$I^{P}=\\{1,\ldots,n\\}\setminus I_{P}$ as well.

######  Remark 2.2.

Note that Langlands duality reverses arrows in the Dynkin diagram of the type
of $G$; hence, for $G=E_{n}^{\mathrm{sc}}$ we have
$G^{\vee}=E_{n}^{\mathrm{ad}}$ the adjoint Lie group of type $E_{n}$.

We will write
$(e^{\vee}_{i})^{*},(f^{\vee}_{i})^{*}\in(\mathfrak{g}^{\vee})^{*}$ for the
dual maps corresponding to the Chevalley generators
$(e^{\vee}_{1},f^{\vee}_{1},h^{\vee}_{1},\ldots,e^{\vee}_{n},f^{\vee}_{n},h^{\vee}_{n})$
for $\mathfrak{g}^{\vee}$; they are given by

$(e^{\vee}_{i})^{*}(e^{\vee}_{j})=\delta_{ij}=(f^{\vee}_{i})^{*}(f^{\vee}_{j})\quad\text{and}\quad(e^{\vee}_{i})^{*}(f^{\vee}_{j})=0=(f^{\vee}_{i})^{*}(e^{\vee}_{j})$
(2.4)

and are identically zero on the remaining root spaces and the Cartan algebra.
We extend these maps to $U^{\vee}_{+}$ and $U^{\vee}_{-}$ using the the
identification of the one-parameter subgroups $x^{\vee}_{i}(a)\in
U^{\vee}_{+}$ and $y^{\vee}_{i}(a)\in U^{\vee}_{-}$ with
$\exp(a\,e^{\vee}_{i})\in\widehat{\mathcal{U}}^{\vee}_{+}$ and
$\exp(a\,f^{\vee}_{i})\in\widehat{\mathcal{U}}^{\vee}_{-}$ in combination with
the inclusions of $\mathfrak{u}^{\vee}_{+}$ and $\mathfrak{u}^{\vee}_{-}$ into
their completed universal algebras $\widehat{\mathcal{U}}^{\vee}_{+}$ and
$\widehat{\mathcal{U}}^{\vee}_{-}$. Equivalently, $(e^{\vee}_{i})^{*}$ and
$(f^{\vee}_{i})^{*}$ are defined as the unique group homomorphisms
$U^{\vee}_{+}\to\mathbb{C}$ and $U^{\vee}_{-}\to\mathbb{C}$ such that

$(e^{\vee}_{i})^{*}(x^{\vee}_{j}(a))=a\delta_{ij}=(f^{\vee}_{i})^{*}(y^{\vee}_{j}(a)).$
(2.5)

As $G^{\vee}$ is in general not simply-connected, we will need to consider the
universal cover $\widetilde{G}^{\vee}$ of $G^{\vee}$. As before, we define the
universal covers $\widetilde{P}^{\vee}$, $\widetilde{T}^{\vee}$,
$\widetilde{B}^{\vee}_{+}$ and $\widetilde{B}^{\vee}_{-}$. Note that the cover
of $U^{\vee}_{+}$ is in fact isomorphic to $U^{\vee}_{+}$ and the same holds
for $U^{\vee}_{-}$, so we simply identify them.

######  Remark 2.3.

Note that for Lie groups with simply-laced Dynkin diagrams Remark 2.2 tells us
that $\widetilde{G}^{\vee}\cong G$ as they are both simply-connected and of
the same type.

Denote by $W$ the Weyl group associated to (the Coxeter diagram of)
$G^{\vee}$; $G$ and $G^{\vee}$ share the same Coxeter diagram and hence give
rise to the same Weyl group. We write $s_{i}=s_{\alpha^{\vee}_{i}}\in W$ for
the simple reflections, $\alpha^{\vee}_{i}\in\Delta\\!^{\vee}$, which form a
set of generators. A decomposition for $w\in W$ as $w=s_{i_{1}}\cdots
s_{i_{j}}$ is called a _reduced expression_ if $j$ is minimal and $\ell(w)=j$
is called the length of $w$. There is a unique element of greatest length
denoted by $w_{0}\in W$ and we abbreviate its length by
${\ell_{0}}=\ell(w_{0})$. There is also a Weyl group associated to $P^{\vee}$,
given by the subgroup $W_{P}\subset W$ generated by the simple reflections
$\\{s_{i}~{}|~{}i\in I_{P}\\}$, compare equation (2.3). Note that $W_{P}$ is a
Weyl group in its own right, associated to the Dynkin diagram of $G^{\vee}$
with the vertices marked by $I^{P}$ removed. We write $w_{P}$ for the longest
element of $W_{P}$ and ${\ell_{P}}=\ell(w_{P})$ for its length.

We are particularly interested in the quotient $W/W_{P}$. We fix a set
$W^{P}\subset W$ of minimal coset representatives of this quotient (i.e. for
every coset the representative of minimal length), and we write the minimal
representative of $w_{0}W_{P}$ as $w^{P}$. Thus $w_{0}=w^{P}w_{P}$, and we fix
reduced expressions

$w^{P}=s_{r_{1}}\cdots s_{r_{\ell}}\quad\text{and}\quad
w_{P}=s_{r_{\ell+1}}\cdots s_{r_{\ell+{\ell_{P}}}},$ (2.6)

so we obtain a reduced expression $w_{0}=s_{r_{1}}\cdots
s_{r_{\ell}}s_{r_{\ell+1}}\cdots s_{r_{{\ell_{0}}}}$ (observe that
${\ell_{0}}=\ell+{\ell_{P}}$). However, it will be more convenient to use the
reversed reduced expression for $w_{0}$

$w_{0}=s_{r_{{\ell_{0}}}}\cdots s_{r_{\ell+1}}s_{r_{\ell}}\cdots s_{r_{1}},$
(2.7)

obtained from $w_{0}=w_{0}^{-1}$.

Finally, we associate two elements in $G^{\vee}$ to each $s_{i}\in W$:

$\dot{s}_{i}=x^{\vee}_{i}(1)y^{\vee}_{i}(-1)x^{\vee}_{i}(1)\quad\text{and}\quad\bar{s}_{i}=x^{\vee}_{i}(-1)y^{\vee}_{i}(1)x^{\vee}_{i}(-1)=\dot{s}_{i}^{-1}.$
(2.8)

For an arbitrary $w\in W$ with reduced expression $w=s_{i_{1}}\cdots
s_{i_{d}}$ we find an associated element in $G^{\vee}$ by setting
$\dot{w}=\dot{s}_{i_{1}}\cdots\dot{s}_{i_{d}}$ and
$\bar{w}=\bar{s}_{i_{1}}\cdots\bar{s}_{i_{d}}$. Note that $\bar{w}$ and
$\dot{w}^{-1}$ have the reverse product of simple reflections, i.e.
$\dot{w}^{-1}=\bar{s}_{i_{d}}\cdots\bar{s}_{i_{1}}$. We write

$T^{P}=(T^{\vee})^{W_{P}}\subset T^{\vee}$ (2.9)

for the part of $T^{\vee}$ that is invariant under the action $W_{P}\times
T^{\vee}\to T^{\vee}$ given by $(w,t)\mapsto\dot{w}t\dot{w}^{-1}$. Clearly,
$T^{P}$ has dimension $\\#I^{P}$.

###  2.2. Rietsch’s Lie-theoretic mirror model and a local Laurent polynomial
expression

The mirror models we construct will be shown to be isomorphic to the Landau-
Ginzburg models that have been constructed for general homogeneous spaces in
[Rie08]. In this subsection we will briefly discuss the construction given in
[Rie08] and the local Laurent polynomial expression for its superpotential
given by the first author in [Spa21].

####  2.2.1. The Lie-theoretic mirror model

In this subsection $X=G/P$ denotes a general homogeneous spaces with $P$ an
arbitrary parabolic subgroup. The mirror model consists of a subvariety of the
open Richardson variety associated to $(w_{P},w_{0})$. This open Richardson
variety is given by:

$X^{\vee}_{\mathrm{Lie}}=\mathcal{R}^{\vee}_{w_{P},w_{0}}=\bigl{(}B^{\vee}_{+}w_{P}B^{\vee}_{-}\cap
B^{\vee}_{-}w_{0}B^{\vee}_{-}\bigr{)}/B^{\vee}_{-}~{}\subset~{}G^{\vee}/B^{\vee}_{-}.$
(2.10)

There exists an isomorphism

$\Psi:X^{\vee}_{\mathrm{Lie}}\times
T^{P}\overset{\sim}{\longrightarrow}\mathcal{Z}^{\vee}_{P}$ (2.11)

to the variety

$\mathcal{Z}^{\vee}_{P}=B^{\vee}_{-}\bar{w}_{0}^{-1}\cap
U^{\vee}_{+}T^{P}\bar{w}_{P}U^{\vee}_{-}~{}\subset~{}G^{\vee}.$ (2.12)

(See [Rie08], Section 4.)

######  Remark 2.4.

Compared to [Rie08], we have translated the definition of
$\mathcal{Z}^{\vee}_{P}$ by right multiplication by $\bar{w}_{0}^{-1}$. For a
detailed description of $\Psi$, see Remark 3.1 of [Spa21].

In [Pet97], the quantum cohomology of a generalized flag variety $G/P$ is
presented as the coordinate ring of what is now known as the _Peterson
variety_ (see e.g. [Rie08], paragraph 3.2). Inside this (non-reduced) variety
there is an open stratum whose coordinate ring is isomorphic to the quantum
cohomology localised at the quantum parameters (see e.g. [Rie08], equation
(3.2)). In [Rie08], a function on $\mathcal{Z}^{\vee}_{P}$ is defined whose
critical locus is isomorphic to this open stratum. The isomorphism $\Psi$ of
equation (2.11) then yields a subvariety of $X^{\vee}_{\mathrm{Lie}}\times
T^{P}$ whose coordinate ring is isomorphic to the localized quantum cohomology
of $X$.

These results have been reformulated in [MR20], Theorem 6.5; see also [PR13],
Section 4. There, a Jacobi ring of $X^{\vee}_{\mathrm{Lie}}\times T^{P}$ with
respect to a superpotential replaces the critical locus in the original
formulation. That is, the localised quantum cohomology of $X=G/P$ is
isomorphic to the coordinate ring of $X^{\vee}_{\mathrm{Lie}}\times T^{P}$
modulo the ideal generated by derivatives of the superpotential:

######  Theorem 2.5 ([Rie08], Theorem 4.1).

Let $X=G/P$ be a complete homogeneous space with $G$ a simple, simply-
connected algebraic group over $\mathbb{C}$ and with $P$ a parabolic subgroup.
There exists a superpotential
$\mathcal{W}_{\mathrm{Lie}}:X^{\vee}_{\mathrm{Lie}}\times T^{P}\to\mathbb{C}$
(given in Definition 2.8) such that

$qH^{*}(X)_{\mathrm{loc}}\cong\mathbb{C}[X^{\vee}_{\mathrm{Lie}}\times
T^{P}]/\langle\partial\mathcal{W}_{\mathrm{Lie}}\rangle,$

where $qH^{*}(X)_{\mathrm{loc}}$ is the (small) quantum cohomology of $X$ with
all quantum parameters inverted and where
$\langle\partial\mathcal{W}_{\mathrm{Lie}}\rangle$ is the ideal generated by
the derivatives of $\mathcal{W}_{\mathrm{Lie}}$ along
$X^{\vee}_{\mathrm{Lie}}$.

The superpotential $\mathcal{W}_{\mathrm{Lie}}$ is presented in [PR13] as the
pull-back of a superpotential defined on $\mathcal{Z}^{\vee}_{P}$ along the
isomorphism $\Psi:X^{\vee}_{\mathrm{Lie}}\times
T^{P}\overset{\sim}{\longrightarrow}\mathcal{Z}^{\vee}_{P}$ from equation
(2.11). To state this superpotential, we introduce to the following subset of
$U^{\vee}_{-}$:

$U_{-}^{P}=U^{\vee}_{-}\cap
B^{\vee}_{+}\bar{w}_{P}\bar{w}_{0}B^{\vee}_{+}~{}\subset~{}U^{\vee}_{-}.$
(2.13)

This set has the following property:

######  Lemma 2.6 ([PR13], Section 4).

Every $z\in\mathcal{Z}^{\vee}_{P}$ has a _unique_ decomposition
$z=u_{+}t\bar{w}_{P}u_{-}$ with $u_{+}\in U^{\vee}_{+}$, $t\in T^{P}$ and
$u_{-}\in U_{-}^{P}$.

######  Remark 2.7.

The proof of this result in [PR13] can be carried over to the general case
without any modification, so it will be omitted here.

The superpotential on $\mathcal{Z}^{\vee}_{P}$ is now defined as follows:

######  Definition 2.8.

Define the superpotential
$\mathcal{W}_{\mathcal{Z}^{\vee}_{P}}:\mathcal{Z}^{\vee}_{P}\to\mathbb{C}$ as
the map:

$\mathcal{W}_{\mathcal{Z}^{\vee}_{P}}:\quad
z=u_{+}t\bar{w}_{P}u_{-}\longmapsto\mathcal{E}^{*}(u_{+}^{-1})+\mathcal{F}^{*}(u_{-}),$

where $\mathcal{E}^{*}=\sum_{i=1}^{n}(e^{\vee}_{i})^{*}$ and
$\mathcal{F}^{*}=\sum_{i=1}^{n}(f^{\vee}_{i})^{*}$, and where the
decomposition of $z=u_{+}t\bar{w}_{P}u_{-}$ is the unique decomposition with
$u_{-}\in U_{-}^{P}$ as stated in Lemma 2.6. Moreover, the superpotential
$\mathcal{W}_{\mathrm{Lie}}:X^{\vee}_{\mathrm{Lie}}\times T^{P}\to\mathbb{C}$
mentioned in Theorem 2.5 is given by
$\mathcal{W}_{\mathrm{Lie}}=\mathcal{W}_{\mathcal{Z}^{\vee}_{P}}\circ\Psi$
with $\Psi$ given in equation (2.11).

####  2.2.2. A local Laurent polynomial expression for the superpotential

From this point forward, we will restrict our attention to _cominuscule_
homogeneous spaces. Recall that parabolic subgroups containing a given Borel
subgroup can be associated to a subsets of vertices of the Dynkin diagram,
with the Borel subgroup corresponding to the full diagram and a maximal
parabolic corresponding to a single vertex. We will write $P=P_{k}$ if $P$ is
maximal parabolic and associated to the $k$th vertex. The homogeneous space
$X=G/P_{k}$ is called cominuscule if the fundamental coweight
${\varpi^{\vee}_{k}}$ is _minuscule_ , i.e. it satisfies

$\langle\\!\langle{\varpi^{\vee}_{k}},\alpha\rangle\\!\rangle\in\\{-1,0,+1\\},\quad\forall\alpha\in\Phi,$
(2.14)

where $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle$ denotes the dual pairing
between the character and the cocharacter lattice.

In [Spa21] a Laurent polynomial expression is given for
$\mathcal{W}_{\mathcal{Z}^{\vee}_{P}}$ restricted to the following open, dense
subvariety:

$\mathcal{Z}^{\circ}_{P}=B^{\vee}_{-}\bar{w}_{0}^{-1}\cap
U^{\vee}_{+}T^{P}\bar{w}_{P}U^{\circ}_{-}~{}\subset~{}\mathcal{Z}^{\vee}_{P},$

where $U^{\circ}_{-}\subset U_{-}^{P}$ is the open, dense algebraic torus
defined by

$U^{\circ}_{-}=\\{y^{\vee}_{r_{\ell}}(a_{\ell})\cdots
y^{\vee}_{r_{1}}(a_{1})~{}|~{}a_{i}\in\mathbb{C}^{*}\\}.$ (2.15)

Recall that $(r_{1},\ldots,r_{\ell})$ is the sequence of indices of our fixed
reduced expression for $w^{P}$ (see equation 2.6). The fact that
$U^{\circ}_{-}\subset U_{-}^{P}$ is open and dense is well-known, see for
example Lemma 4.8(ii) of [PR13] or Lemma 5.2 of [Spa21]. The superpotential is
parametrized by a certain Weyl group element: let $w^{\prime\prime}$ be the
minimal representative of the coset $w_{P}s_{k}W_{P}$ and set
$w^{\prime}=w^{P}(w^{\prime\prime})^{-1}$, writing its length as
$\ell(w^{\prime})={\ell^{\prime}}$, then define $\mathcal{I}$ as the set of
reduced subexpressions of $w^{\prime}$ in the fixed reduced expression of
$w^{P}$.

The Laurent polynomial superpotential is now given as follows:

######  Proposition 2.9 ([Spa21], Theorem 5.7).

Let $X=G/P$ be a cominuscule complete homogeneous space with $G$ a simply-
connected, simple, complex algebraic group and $P=P_{k}$ a (maximal) parabolic
subgroup. The restriction $\mathcal{W}_{\mathcal{Z}^{\circ}_{P}}$ of
$\mathcal{W}_{\mathcal{Z}^{\vee}_{P}}$ to $\mathcal{Z}^{\circ}_{P}$ has the
following Laurent polynomial expression:

$\mathcal{W}_{\mathcal{Z}^{\circ}_{P}}(z)=\sum_{i=1}^{\ell}a_{i}+q\frac{\sum_{(i_{j})\in\mathcal{I}}\prod_{j=1}^{{\ell^{\prime}}}a_{i_{j}}}{\prod_{i=1}^{\ell}a_{i}}.$

Here $z\in\mathcal{Z}^{\circ}_{P}$ is uniquely decomposed as
$z=u_{+}t\bar{w}_{P}u_{-}$ with $u_{-}=y^{\vee}_{r_{\ell}}(a_{\ell})\cdots
y^{\vee}_{r_{1}}(a_{1})\in U^{\circ}_{-}$. Also, $q\in\mathbb{C}^{*}$ is given
by $q=\alpha^{\vee}_{k}(t)$ with $t\in T^{P}$.

In Corollary 8.12 of [Spa21], it is shown that the numerator of the quantum
term can be enumerated using subsets of a quiver defined in [CMP08] and that
these subsets can be obtained from a given one using two elementary moves,
which can be represented graphically in the same way: by replacing a vertex in
the subset with the one below (resp. above) it in the same column under the
condition that all outgoing (resp. incoming) arrows at the original vertex do
not point to elements in the subset. (See Lemma 8.9 and Proposition 8.11 of
[Spa21].) This quiver will appear again later, see Remark 4.4. We would like
to note that these combinatorial operations coincide with the “excited moves”
defined on $d$-complete posets in [NO19], where equivariant $K$-theory is used
to derive a skew hook formula.

###  2.3. Generalized Plücker coordinates and the coordinate ring of
$(U_{-}^{P})^{T}$

For cominuscule homogeneous spaces $X=G/P_{k}$, a consequence of the geometric
Satake correspondence is that the Schubert classes that form a basis of the
cohomology are in one-to-one correspondence with a set of projective
coordinates called _(generalized) Plücker coordinates_ on the _Langlands dual
homogeneous space_

$\mathbb{X}^{\vee}=P^{\vee}_{k}\backslash
G^{\vee}=\widetilde{P}^{\vee}_{k}\backslash\widetilde{G}^{\vee}.$ (2.16)

(Both expressions give the same homogeneous space, and we mostly use the
second.) See [MR20], p. 3, for more references regarding the geometric Satake
correspondence. Thus, if we reformulate the superpotential in Definition 2.8
in terms of Plücker coordinates, we obtain a description of the localized
quantum cohomology in terms of Schubert classes and relations between them.
Plücker coordinate expressions for Rietsch’s superpotential have been given
for Grassmannians [MR20], quadrics [PRW16], and Lagrangian Grassmannians
[PR13]. In [PRW16], this was obtained using a description of the coordinate
ring of $U_{-}^{P}$ by [GLS11].

####  2.3.1. Minuscule representations and generalized Plücker coordinates.

With the assumption that $X=G/P$ is cominuscule, we will be able to define the
(generalized) Plücker coordinates on
$\mathbb{X}^{\vee}=\widetilde{P}^{\vee}\backslash\widetilde{G}^{\vee}$ using a
minuscule representation. To do this, we first recall a few properties of such
representations. Since we consider minuscule representations of
$\widetilde{G}^{\vee}$ and $\mathfrak{g}^{\vee}$ with corresponding roots
$\Phi^{\vee}$, we adapt our notation to this situation.

First, we will need a general statement concerning the structure of minuscule
representations and the action of Chevalley generators; this result can be
found in [Gre08] or [Gec17]. Fix a non-degenerate, symmetric, bilinear form
$(\cdot,\cdot)$ on the weight lattice of $\mathfrak{g}^{\vee}$ (e.g. the
Killing form) and let
$c_{i}(\mu^{\vee})=2(\mu^{\vee},\alpha^{\vee}_{i})/(\alpha^{\vee}_{i},\alpha^{\vee}_{i})$
for $\mu^{\vee}$ an element of the weight lattice.

######  Theorem 2.10.

Let $V(\lambda^{\vee})$ be a minuscule representation, i.e. a highest weight
representation with as highest weight $\lambda^{\vee}$ satisfying (2.14).
Denote its set of weights by $M(\lambda^{\vee})$. Then the following holds:

  * (i)

For all $\mu^{\vee}\in M(\lambda^{\vee})$ and $i\in\\{1,\ldots,n\\}$,
$c_{i}(\mu^{\vee})\in\\{-1,0,+1\\}$.

  * (ii)

Each weight space is one-dimensional.

  * (iii)

Given a highest weight vector $v_{\lambda^{\vee}}^{+}$, there exists a basis
of weight vectors $\\{v_{\mu^{\vee}}\\}$ such that
$v_{\lambda^{\vee}}=v_{\lambda^{\vee}}^{+}$ and such that

$\displaystyle e^{\vee}_{i}\cdot v_{\mu^{\vee}}$
$\displaystyle=\left\\{\begin{array}[]{ll}v_{\mu^{\vee}+\alpha^{\vee}_{i}},&\text{if
$c_{i}(\mu^{\vee})=-1$,}\\\ 0,&\text{otherwise,}\end{array}\right.$
$\displaystyle f^{\vee}_{i}\cdot v_{\mu^{\vee}}$
$\displaystyle=\left\\{\begin{array}[]{ll}v_{\mu^{\vee}-\alpha^{\vee}_{i}},&\text{if
$c_{i}(\mu^{\vee})=+1$,}\\\ 0,&\text{otherwise,}\end{array}\right.$
$\displaystyle h^{\vee}_{i}\cdot v_{\mu^{\vee}}$
$\displaystyle=c_{i}(\mu^{\vee})\,v_{\mu^{\vee}}.$

  * (iv)

For any vector $v\in V(\lambda^{\vee})$ and $i,j\in\\{1,\ldots,n\\}$,

$\displaystyle(e^{\vee}_{i})^{2}\cdot v$
$\displaystyle=0=(f^{\vee}_{i})^{2}\cdot v,$ $\displaystyle
e^{\vee}_{i}e^{\vee}_{j}e^{\vee}_{i}\cdot v$
$\displaystyle=0=f^{\vee}_{i}f^{\vee}_{j}f^{\vee}_{i}\qquad\text{if
$a_{ij}=c_{i}(\alpha^{\vee}_{j})=-1$,}$ $\displaystyle
e^{\vee}_{i}f^{\vee}_{j}\cdot v$ $\displaystyle=0=f^{\vee}_{j}e_{i}\cdot
v~{}\qquad\text{if $a_{ij}<0$.}$

Combining this result with the definition of $\dot{s}_{i}$ and $\bar{s}_{i}$
in (2.8), we find

######  Corollary 2.11.

With the notation of Theorem 2.10,
$s_{i}(\mu^{\vee})=\mu^{\vee}-c_{i}(\mu^{\vee})\,\alpha^{\vee}_{i}$ and

$\displaystyle\dot{s}_{i}\cdot v_{\mu^{\vee}}$
$\displaystyle=\left\\{\begin{array}[]{ll}v_{\mu^{\vee}+\alpha^{\vee}_{i}}=e^{\vee}_{i}\cdot
v_{\mu^{\vee}},&\text{if $c_{i}(\mu^{\vee})=-1$,}\\\ v_{\mu^{\vee}},&\text{if
$c_{i}(\mu^{\vee})=\hphantom{+}0$},\\\
-v_{\mu^{\vee}-\alpha^{\vee}_{i}}=-f^{\vee}_{i}\cdot v_{\mu^{\vee}},&\text{if
$c_{i}(\mu^{\vee})=+1$,}\end{array}\right.$ $\displaystyle\bar{s}_{i}\cdot
v_{\mu^{\vee}}$
$\displaystyle=\left\\{\begin{array}[]{ll}-v_{\mu^{\vee}+\alpha^{\vee}_{i}}=-e^{\vee}_{i}\cdot
v_{\mu^{\vee}},&\text{if $c_{i}(\mu^{\vee})=-1$,}\\\ v_{\mu^{\vee}},&\text{if
$c_{i}(\mu^{\vee})=\hphantom{+}0$,}\\\
v_{\mu^{\vee}-\alpha^{\vee}_{i}}=f^{\vee}_{i}\cdot v_{\mu^{\vee}},&\text{if
$c_{i}(\mu^{\vee})=+1$.}\end{array}\right.$ Conversely, we have $\displaystyle
e^{\vee}_{i}\cdot v_{\mu^{\vee}}$
$\displaystyle=\left\\{\begin{array}[]{ll}v_{\mu^{\vee}+\alpha^{\vee}_{i}}=\dot{s}_{i}\cdot
v_{\mu^{\vee}}=-\bar{s}_{i}\cdot v_{\mu^{\vee}},&\text{if
$c_{i}(\mu^{\vee})=-1$,}\\\ 0,&\text{otherwise,}\end{array}\right.$
$\displaystyle f^{\vee}_{i}\cdot v_{\mu^{\vee}}$
$\displaystyle=\left\\{\begin{array}[]{ll}v_{\mu^{\vee}-\alpha^{\vee}_{i}}=\bar{s}_{i}\cdot
v_{\mu^{\vee}}=-\dot{s}_{i}\cdot v_{\mu^{\vee}},&\text{if
$c_{i}(\mu^{\vee})=+1$,}\\\ 0,&\text{otherwise.}\end{array}\right.$

With these general facts about the structure of minuscule representations, we
return to defining the Plücker coordinates on $\mathbb{X}^{\vee}$. Recall that
by assumption ${\varpi^{\vee}_{k}}$ is minuscule; this implies that the
fundamental weight $-w_{0}\cdot{\varpi^{\vee}_{k}}$ is minuscule as well:

$\langle\\!\langle-
w_{0}\cdot{\varpi^{\vee}_{k}},\alpha\rangle\\!\rangle=\langle\\!\langle-
w_{0}\cdot{\varpi^{\vee}_{k}},-w_{0}\cdot\alpha^{\prime}\rangle\\!\rangle=\langle\\!\langle{\varpi^{\vee}_{k}},\alpha^{\prime}\rangle\\!\rangle\in\\{-1,0,+1\\}$

(where $\alpha\in\Phi$ and $\alpha^{\prime}=-w_{0}\cdot\alpha$), compare
(2.14). Note that $-w_{0}\cdot{\varpi^{\vee}_{k}}$ equals either
${\varpi^{\vee}_{k}}$ (in the cases of type $B_{n}$, $C_{n}$, $D_{2n}$ and
$E_{7}$) or ${\varpi^{\vee}_{\sigma(k)}}$ for
$\sigma:\\{1,\ldots,n\\}\to\\{1,\ldots,n\\}$ the symmetry involution of the
Dynkin diagram of $\widetilde{G}^{\vee}$ (in the cases of $A_{n}$, $D_{2n+1}$
and $E_{6}$). We will use the minuscule representation
$V(-w_{0}\cdot{\varpi^{\vee}_{k}})$ to define the coordinates, so to simplify
notation we will use $\sigma:\\{1,\ldots,n\\}\to\\{1,\ldots,n\\}$ to denote
_either_ the Dynkin diagram symmetry involution _or_ the identity map
depending on the case as described above; with this notation, we can simply
write $V(-w_{0}\cdot{\varpi^{\vee}_{k}})=V({\varpi^{\vee}_{\sigma(k)}})$.

By Theorem 2.10, there is a natural weight basis for
$V({\varpi^{\vee}_{\sigma(k)}})$, which we denote by $\\{v_{i}\\}$. Writing
$v_{0}$ for the basis vector of _lowest_ weight (i.e. of weight
$-{\varpi^{\vee}_{k}}$), the dual representation
$V({\varpi^{\vee}_{\sigma(k)}})^{*}$ has highest weight vector $v_{0}^{*}$ (of
weight ${\varpi^{\vee}_{k}}$), where $\\{v_{i}^{*}\\}$ is the dual basis. Note
that $V({\varpi^{\vee}_{\sigma(k)}})^{*}$ is a _right_ -representation with
action $v_{i}^{*}\cdot g=\bigl{(}v\mapsto v_{i}^{*}(g\cdot v)\bigr{)}$, and
that
$[v_{0}^{*}]\cdot\widetilde{P}^{\vee}=[v_{0}^{*}]\in\mathbb{P}\bigl{(}V({\varpi^{\vee}_{\sigma(k)}})^{*}\bigr{)}$.
Hence, we obtain an embedding
$\mathbb{X}^{\vee}\hookrightarrow\mathbb{P}\bigl{(}V({\varpi^{\vee}_{\sigma(k)}})^{*}\bigr{)}$
given by $\widetilde{P}^{\vee}g\mapsto[v_{0}^{*}\cdot g]$, using the fact that
$\widetilde{P}^{\vee}$ and $\\{y^{\vee}_{k}(a)~{}|~{}a\in\mathbb{C}\\}$
generate $\widetilde{G}^{\vee}$. This induces the coordinates:

######  Definition 2.12.

For any weight basis vector $v_{i}$ as above and $g\in\widetilde{G}^{\vee}$,
define $p_{i}:\widetilde{G}^{\vee}\to\mathbb{C}$ by

$p_{i}(g)=(v_{0}^{*}\cdot g)(v_{i})=v_{0}^{*}(g\cdot v_{i}).$ (2.17)

These maps induce projective coordinates on $\mathbb{X}^{\vee}$ called
_(generalized) Plücker coordinates_.

We will abuse notation and call the maps
$p_{i}:\widetilde{G}^{\vee}\to\mathbb{C}$ Plücker coordinates as well.

####  2.3.2. The coordinate ring of
$U_{+}^{P}=(U_{-}^{P})^{T}\subset\widetilde{G}^{\vee}$.

Recall from equation (2.13) that we defined $U_{-}^{P}=U^{\vee}_{-}\cap
B^{\vee}_{-}(\dot{w}^{P})^{-1}B^{\vee}_{-}$. In Proposition 8.5 of [GLS11],
the coordinate rings of these _unipotent cells_ have been described using a
dual PBW basis compatible with a reduced expression. However, the result is
formulated using cells of $U^{\vee}_{+}$ instead of $U^{\vee}_{-}$: we will
have to use the _transposition anti-automorphism_ $\cdot^{T}$ to translate
their results to cells of $U^{\vee}_{-}$. Recall that this anti-automorphism
is given by mapping $x^{\vee}_{i}(a)$ to $y^{\vee}_{i}(a)$ and vice versa,
while acting trivially on $\widetilde{T}^{\vee}$. Note that
$\bar{s}_{i}^{T}=\dot{s}_{i}=\bar{s}_{i}^{-1}$.

We would like to emphasize that the results in [GLS11] hold for general
unipotent cells, but we will only consider the case of $U_{-}^{P}$ for
$P=P_{k}$ with ${\varpi_{k}}$ cominuscule here.

Writing $U_{+}^{P}$ for the transpose of $U_{-}^{P}$, we clearly have

$U_{+}^{P}=(U_{-}^{P})^{T}=U^{\vee}_{+}\cap
B^{\vee}_{+}\dot{w}^{P}B^{\vee}_{+}.$ (2.18)

The coordinates are defined on $U_{+}^{P}$ using the embedding of
$U^{\vee}_{+}$ in the completed universal enveloping algebra
$\widehat{\mathcal{U}}^{\vee}_{+}$ of $\mathfrak{u}^{\vee}_{+}$. Namely, they
form a subset of a basis for the _graded dual_ of
$\widehat{\mathcal{U}}^{\vee}_{+}$. As the universal enveloping algebra is in
particular a infinite-dimensional vector space, we need to be more careful in
defining the dual. We will make use of the Chevalley generators
$(e^{\vee}_{1},f^{\vee}_{1},\ldots,e^{\vee}_{n},f^{\vee}_{n})$.

Consider the Cartan (or root space) decomposition

$\mathfrak{g}^{\vee}=\mathfrak{t}^{\vee}\oplus\bigoplus_{\alpha^{\vee}\in\Phi^{\vee}}\mathfrak{u}^{\vee}_{\alpha^{\vee}},$

where for $\alpha^{\vee}\in\Pi^{\vee}$ a positive root we have that
$\mathfrak{u}^{\vee}_{+\alpha^{\vee}}\subset\mathfrak{u}^{\vee}_{+}$ and
$\mathfrak{u}^{\vee}_{-\alpha^{\vee}}\subset\mathfrak{u}^{\vee}_{-}$ are one-
dimensional subspaces (see for example [Hum78], Section 8.5). Clearly,
$\mathfrak{u}^{\vee}_{+\alpha^{\vee}_{i}}=\mathbb{C}e^{\vee}_{i}$ is spanned
by the corresponding Chevalley generator,
$\mathfrak{u}^{\vee}_{+s_{i}(\alpha^{\vee}_{j})}=\mathbb{C}[e^{\vee}_{i},e^{\vee}_{j}]$
is spanned by the commutator of the generators if $(s_{i}s_{j})^{2}\neq 1$,
and so on.

Now, consider the _positive root lattice_
$\mathbb{N}_{0}\Delta\\!^{\vee}\cong\mathbb{N}_{0}^{n}$, i.e. all non-negative
integral linear combinations of the simple roots $\alpha^{\vee}_{i}$. For
$\alpha^{\vee}=\sum_{i=1}^{n}d_{i}\alpha^{\vee}_{i}\in\mathbb{N}_{0}\Delta\\!^{\vee}$,
denote by $\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee})$ the subspace
spanned by those pure tensors $e^{\vee}_{i_{1}}\cdots e^{\vee}_{i_{j}}$ such
that exactly $d_{i}$ of the factors have index $i$. For example,
$\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee}_{i})=\mathbb{C}e^{\vee}_{i}=\mathfrak{u}^{\vee}_{+\alpha^{\vee}_{i}}$
for $\alpha^{\vee}_{i}\in\Delta\\!^{\vee}$, and
$\widehat{\mathcal{U}}^{\vee}_{+}(2\alpha^{\vee}_{1}+\alpha^{\vee}_{2})=\mathbb{C}e^{\vee}_{1}e^{\vee}_{1}e^{\vee}_{2}\oplus\mathbb{C}e^{\vee}_{1}e^{\vee}_{2}e^{\vee}_{1}\oplus\mathbb{C}e^{\vee}_{2}e^{\vee}_{1}e^{\vee}_{1}$.
(If $2\alpha^{\vee}_{1}+\alpha^{\vee}_{2}\in\Pi^{\vee}$, note that
$\mathfrak{u}^{\vee}_{2\alpha^{\vee}_{1}+\alpha^{\vee}_{2}}=\mathbb{C}[e^{\vee}_{1},[e^{\vee}_{1},e^{\vee}_{2}]]\subset\widehat{\mathcal{U}}^{\vee}_{+}(2\alpha^{\vee}_{1}+\alpha^{\vee}_{2})$
.) This endows $\widehat{\mathcal{U}}^{\vee}_{+}$ with an
$\mathbb{N}_{0}\Delta\\!^{\vee}$-grading that naturally extends the root space
decomposition of $\mathfrak{u}^{\vee}_{+}$ (i.e.
$\mathfrak{u}^{\vee}_{+\alpha^{\vee}}\subset\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee})$
for $\alpha^{\vee}\in\Pi^{\vee}\subset\mathbb{N}_{0}\Delta\\!^{\vee}$). The
graded dual is defined as the direct sum of the duals of these finite-
dimensional subspaces,

$\bigl{(}\widehat{\mathcal{U}}^{\vee}_{+}\bigr{)}^{*}_{\mathrm{gr}}=\bigoplus_{\alpha^{\vee}\in\mathbb{N}_{0}\Delta\\!^{\vee}}\bigl{(}\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee})\bigr{)}^{*}.$

To define a basis on this dual, we start by recalling that we fixed a reduced
expression for the minimal representative $w^{P}\in W^{P}$ of the coset
$w_{0}W_{P}$ and denoted its sequence of indices by $(r_{i})_{i=1}^{\ell}$
where $\ell=\ell(w^{P})$, see equation (2.6). As a Weyl group element, $w^{P}$
maps a set
$\Pi^{\vee}_{w^{P}}=\\{{\beta^{\vee}_{P}}(j)~{}|~{}j=1,\ldots,\ell\\}\subset\Pi^{\vee}$
of positive roots to negative roots. It is well known that these are given by

${\beta^{\vee}_{P}}(1)=\alpha^{\vee}_{r_{\ell}}\quad\text{and}\quad{\beta^{\vee}_{P}}(j)=s_{r_{\ell}}\cdots
s_{r_{\ell-j+2}}(\alpha^{\vee}_{r_{\ell-j+1}})\quad\text{for
$j\in\\{2,\ldots,\ell\\}$,}$ (2.19)

(for example, this follows from [Hum78], Section 10.2). Note that
$s_{r_{\ell}}\cdots s_{r_{\ell-j+2}}$ are the first $j-1$ factors of the
reduced expression for $(w^{P})^{-1}$ obtained by reversing the simple
reflections in (2.6). Since $w_{0}=w^{P}w_{P}$, it is easy to see that
$\Pi^{\vee}=\Pi^{\vee}_{w^{P}}\sqcup\Pi^{\vee}_{P}$, where $\Pi^{\vee}_{P}$ is
the set of positive (co-) roots mapped to negative ones by $w_{P}$. We also
remark that both $\Pi^{\vee}_{w^{P}}$ and $\Pi^{\vee}_{P}$ are _bracket
closed_ subsets: if $\alpha^{\vee},\beta^{\vee}\in\Pi^{\vee}_{w^{P}}$ and
$\alpha^{\vee}+\beta^{\vee}\in\Pi^{\vee}$ then
$\alpha^{\vee}+\beta^{\vee}\in\Pi^{\vee}_{w^{P}}$, and the same holds for
roots in $\Pi^{\vee}_{P}$. (This follows from the linearity of action of the
Weyl group elements on roots.) See also [GLS11], Section 4.3, and the
references therein.

For each of the positive roots ${\beta^{\vee}_{P}}(m)$,
$m\in\\{1,\ldots,\ell\\}$, we choose a basis element $e^{\vee}_{w^{P}}(m)$ for
$\mathfrak{u}^{\vee}_{+{\beta^{\vee}_{P}}(m)}$. In particular, we take
$e^{\vee}_{w^{P}}(1)=e^{\vee}_{k}$, as
${\beta^{\vee}_{P}}(1)=\alpha^{\vee}_{k}$.

######  Remark 2.13.

In [GLS11], the basis elements for the one-dimensional subspaces
$\mathfrak{u}^{\vee}_{+{\beta^{\vee}_{P}}(m)}$ are not defined more
explicitly, nor will we need an explicit choice of basis vector in the
following. However, it is easy to define a set of basis elements
$e^{\vee}_{w^{P}}(m)\in\mathfrak{u}^{\vee}_{+{\beta^{\vee}_{P}}(m)}$ for
$m\in\\{1,\ldots,\ell\\}$ using the following algorithm: consider the defining
expression of ${\beta^{\vee}_{P}}(m)$ in (2.19); remove all simple reflections
acting trivially; replace each remaining simple reflection $s_{i}$ by the
corresponding $\mathop{\mathrm{ad}}\nolimits(e^{\vee}_{i})$; and finally
replace $\alpha^{\vee}_{i}$ with the corresponding $e^{\vee}_{i}$.

Now, we complete the elements
$\\{e^{\vee}_{w^{P}}(m)~{}|~{}m=1,\ldots,\ell\\}$ to a basis of
$\mathfrak{u}^{\vee}_{+}$ by choosing generating elements in
$\mathfrak{u}^{\vee}_{+\alpha^{\vee}}$ for
$\alpha^{\vee}\in\Pi^{\vee}\setminus\Pi^{\vee}_{w^{P}}=\Pi^{\vee}_{P}$. To
simplify notation, we will denote these additional elements by
$e^{\vee}_{w^{P}}(m)$ for $m\in\\{\ell+1,\ldots,\ell+{\ell_{P}}\\}$, where
$\ell+{\ell_{P}}={\ell_{0}}=\ell(w_{0})$ is the number of positive roots. Here
we will also assume that the Chevalley generators $e^{\vee}_{i}$ are part of
this basis. With a basis for $\mathfrak{u}^{\vee}_{+}$, we can in turn define
a PBW basis for $\widehat{\mathcal{U}}^{\vee}_{+}$, the elements of which we
denote by

$e^{\vee}_{w^{P}}(\mathbf{m})=\bigr{(}e^{\vee}_{w^{P}}(1)\bigr{)}^{m_{1}}\bigl{(}e^{\vee}_{w^{P}}(2)\bigr{)}^{m_{2}}\cdots\bigl{(}e^{\vee}_{w^{P}}({\ell_{0}})\bigr{)}^{m_{\ell_{0}}},$
(2.20)

for $\mathbf{m}=(m_{1},\ldots,m_{{\ell_{0}}})\in\mathbb{N}_{0}^{{\ell_{0}}}$.
As each $e^{\vee}_{w^{P}}(m)$ is homogeneous with respect to the grading on
$\widehat{\mathcal{U}}^{\vee}_{+}$ (this follows from Remark 2.13), the PBW
basis elements are homogeneous as well.

We can now define the dual PBW basis for
$\bigl{(}\widehat{\mathcal{U}}^{\vee}_{+}\bigr{)}^{*}_{\mathrm{gr}}$ by taking
the finite-dimensional vector space dual bases and extending the functionals
trivially on the complement. We denote these dual basis elements
$p^{\text{GLS}}_{\mathbf{m}}$, so that
$p^{\text{GLS}}_{\mathbf{m}}\bigl{(}e^{\vee}_{w^{P}}(\mathbf{n})\bigr{)}$
equals $1$ if $\mathbf{m}=\mathbf{n}\in\mathbb{N}_{0}^{\ell_{0}}$ and $0$
otherwise. Whenever $\mathbf{m}=(0,\ldots,0,1,0,\ldots,0)$ with a $1$ at the
$m$th position, we simply write
$p^{\text{GLS}}_{m}=p^{\text{GLS}}_{\mathbf{m}}$; indeed, in this case we have
$e^{\vee}_{w^{P}}(\mathbf{m})=e^{\vee}_{w^{P}}(m)$. This basis is called the
_dual PBW basis compatible with the reduced expression for $w^{P}$_. Using the
natural embedding of $U^{\vee}_{+}$ into $\widehat{\mathcal{U}}^{\vee}_{+}$,
we can consider the maps $p^{\text{GLS}}_{m}$ on $U^{\vee}_{+}$ as well.

From the results in [GLS11], it follows that the coordinate ring of
$U_{+}^{P}$ is generated by the dual basis elements $p^{\text{GLS}}_{m}$ for
$m\in\\{1,\ldots,\ell\\}$ where $\ell=\ell(w^{P})$. However, to fully express
the coordinate ring, we also need to introduce a number of _generalized
minors_ in the sense of Fomin and Zelevinsky [FZ99]. Recall that the highest
weight space of any highest weight representation is one-dimensional. Thus,
taking a highest weight vector $v_{{\varpi^{\vee}_{i}}}^{+}$, we can define
the projection
$V({\varpi^{\vee}_{i}})\to\mathbb{C}v_{{\varpi^{\vee}_{i}}}^{+}$ (parallel to
a basis of weight vectors). We denote the coefficient $c$ such that a vector
$v\in V({\varpi^{\vee}_{i}})$ projects to $cv_{{\varpi^{\vee}_{i}}}^{+}$ by

$\bigl{\langle}v,v_{{\varpi^{\vee}_{i}}}^{+}\bigr{\rangle}=c.$ (2.21)

Note that the highest weight representation $V({\varpi^{\vee}_{i}})$ is
implicit in the notation. Using the action of $\widetilde{G}^{\vee}$ on these
representations, we obtain maps $\widetilde{G}^{\vee}\to\mathbb{C}$ given by
$g\mapsto\langle g\cdot
v_{{\varpi^{\vee}_{i}}}^{+},v_{{\varpi^{\vee}_{i}}}^{+}\rangle$. For
$w_{1},w_{2}\in W$, the generalized minors are given in Definition 1.4 of
[FZ99] as

$\Delta_{w_{1}({\varpi^{\vee}_{i}}),w_{2}({\varpi^{\vee}_{i}})}(g)=\Bigl{\langle}(\bar{w}_{1})^{-1}g\bar{w}_{2}\cdot
v_{{\varpi^{\vee}_{i}}}^{+},v_{{\varpi^{\vee}_{i}}}^{+}\Bigr{\rangle}.$ (2.22)

([FZ99] gives a different formula, but their formula can easily be seen to
agree with this expression. See for example Proposition 7.2 of [GLS11].) In
particular, we will be concerned with the following minors:

$\Delta_{{\varpi^{\vee}_{i}},(w^{P})^{-1}({\varpi^{\vee}_{i}})}(g)=\Bigl{\langle}g(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{i}}}^{+},v_{{\varpi^{\vee}_{i}}}^{+}\Bigr{\rangle}.$ (2.23)

Note that we wrote $(\dot{w}^{P})^{-1}=\overline{(w^{P})^{-1}}$ to simplify
the expression; it has the reverse reduced expression compared to $w^{P}$. We
can now state the result:

######  Proposition 2.14 ([GLS11, Proposition 8.5]).

The cell $U_{+}^{P}=(U_{-}^{P})^{T}\subset\widetilde{G}^{\vee}$, see equation
(2.18), is an affine variety with coordinate ring isomorphic to

$\mathbb{C}[p^{\text{GLS}}_{1},\ldots,p^{\text{GLS}}_{\ell}][\Delta_{{\varpi^{\vee}_{1}},(w^{P})^{-1}({\varpi^{\vee}_{1}})}^{-1},\ldots,\Delta_{{\varpi^{\vee}_{n}},(w^{P})^{-1}({\varpi^{\vee}_{n}})}^{-1}].$
(2.24)

###  2.4. The exceptional cominuscule family

Of the exceptional types, only $E_{6}$ and $E_{7}$ allow cominuscule weights.
There are two cominuscule weights, ${\varpi_{1}}$ and ${\varpi_{6}}$, in type
$E_{6}$, although the homogeneous spaces $E_{6}^{\mathrm{sc}}/P_{1}$ and
$E_{6}^{\mathrm{sc}}/P_{6}$ are isomorphic. There is only one cominuscule
weight in type $E_{7}$, namely ${\varpi_{7}}$. Thus, in practice, we only need
to consider two exceptional cominuscule homogeneous spaces: the Cayley plane
$\mathbb{OP}^{2}=E_{6}^{\mathrm{sc}}/P_{6}$ and the Freudenthal variety
$E_{7}^{\mathrm{sc}}/P_{7}$. Whenever we consider both varieties at the same
time, we will call them “cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$” (we are
adding the adjective “cominuscule” to exclude the _quasi-cominuscule_
$E_{8}^{\mathrm{sc}}/P_{8}$).

We follow the labeling on the Dynkin diagrams used by [Bou68]:

$E_{6}:\quad\leavevmode\hbox to77.45pt{\vbox
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{}{}{}\pgfsys@moveto{0.0pt}{102.43097pt}\pgfsys@lineto{68.28731pt}{102.43097pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{34.14365pt}{102.43097pt}\pgfsys@lineto{34.14365pt}{119.50279pt}\pgfsys@stroke\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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E_{7}:\quad\leavevmode\hbox to94.53pt{\vbox
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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The Cayley plane and the Freudenthal variety are similar enough to allow for
the same approach, but they do have some notable differences. For example, for
type $E_{7}$ the longest Weyl group element $w_{0}$ acts on the fundamental
weights as minus the identity, whereas the longest Weyl group element $w_{0}$
in type $E_{6}$ acts as minus the obvious Dynkin diagram symmetry. Note that
this implies that ${\varpi^{\vee}_{\sigma(6)}}={\varpi^{\vee}_{1}}$ for
$E_{6}$ and ${\varpi^{\vee}_{\sigma(7)}}={\varpi^{\vee}_{7}}$ for $E_{7}$,
recalling that we defined $\sigma$ as acting by the Dynkin diagram symmetry
involution if applicable and by the identity otherwise.

In fact, we will need very few geometric properties of the exceptional
cominuscule homogeneous spaces other than the structure of their cohomologies.
The cohomology rings, and indeed the (small) quantum cohomology rings, of both
varieties are already known; see [CMP08] for presentations of both as well as
for further references.

####  2.4.1. The Cayley plane.

The exceptional cominuscule variety $E_{6}^{\mathrm{sc}}/P_{6}$ is often
denoted by $\mathbb{OP}^{2}$ as it can be interpreted as the variety of
(complex) octonion lines in three-dimensional (complex) octonion space.
However, we will not use this description here and simply consider the Cayley
plane as a homogeneous space of type $E_{6}$. The geometry and the Chow ring
(and thus the cohomology) of $\mathbb{OP}^{2}$ is discussed in [IM05]111Note
that a different labeling of the Dynkin diagram is used in [IM05]..

For the Cayley plane $X=E_{6}^{\mathrm{sc}}/P_{6}$, the embedding of the
Langlands dual homogeneous space is
$\mathbb{X}^{\vee}\hookrightarrow\mathbb{P}\bigl{(}V({\varpi^{\vee}_{1}})^{*}\bigr{)}$.
Since the fundamental weight representation $V({\varpi^{\vee}_{1}})$ is
minuscule, the Weyl group acts transitively on the weights. Thus, we can
define a graph that the weights of $V({\varpi^{\vee}_{1}})$ as vertices and
has an edge labeled $i$ between two weights if $s_{i}$ maps one weight to the
other (and vice versa). The graph we obtain is the following:

654233214125146153163536446252625431$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}^{\prime}$$v_{4}^{\prime\prime}$$v_{5}^{\prime}$$v_{5}^{\prime\prime}$$v_{6}^{\prime}$$v_{6}^{\prime\prime}$$v_{7}^{\prime}$$v_{7}^{\prime\prime}$$v_{8}$$v_{8}^{\prime}$$v_{8}^{\prime\prime}$$v_{9}^{\prime}$$v_{9}^{\prime\prime}$$v_{10}^{\prime}$$v_{10}^{\prime\prime}$$v_{11}^{\prime}$$v_{11}^{\prime\prime}$$v_{12}^{\prime}$$v_{12}^{\prime\prime}$$v_{13}$$v_{14}$$v_{15}$$v_{16}$
(2.25)

Here, we have set the highest weight space at the left and denoted a choice of
highest weight vector by $v_{16}$. Applying equation (2.17) to each of the
basis vectors $v_{i}^{(j)}$ gives the full set of Plücker coordinates.

Using Corollary 2.11, we see that the action of the representatives
$\dot{s}_{i},\bar{s}_{i}\in\widetilde{G}^{\vee}$ can be directly related to
the action of $e^{\vee}_{i},f^{\vee}_{i}\in\mathfrak{g}^{\vee}$. Here, applied
to $V({\varpi^{\vee}_{1}})$, this translates as follows: an edge marked $i$
between two vectors $v$ (on the left) and $w$ (on the right) means that
$\bar{s}_{i}\cdot v=f^{\vee}_{i}\cdot v=w$ and $\dot{s}_{i}\cdot
w=e^{\vee}_{i}\cdot w=v$.

The diagram for $V({\varpi^{\vee}_{6}})$ is in fact the same, however, now the
highest weight space is on the right, and the actions of $\bar{s}_{i}$ and
$\dot{s}_{i}$ go in the opposite direction. (In other words, exchange the
position of $v$ and $w$ in the above.)

More importantly, this graph is identical to the _Hasse diagram_ for the
Cayley plane: the Hasse diagram has as vertices the Schubert classes, and
edges mark inclusions. The identification between the graph above and the
Hasse diagram is obtained by replacing the weight vector $v_{i}^{(j)}$ with
the Schubert class $\sigma_{i}^{(j)}$, where we use the labeling of Schubert
classes given in [CMP08], Section 2.3.222Note that [CMP08] consider
$\mathbb{OP}^{2}$ as $E_{6}^{\mathrm{sc}}/P_{1}$, so the diagram would have
the reverse labeling of edges. This correspondence is implied by the geometric
Satake correspondence, which we mentioned at the start of this section.
Indeed, we have chosen the labeling for the basis of the representation in
such a way that these match up with the labeling for the Schubert classes of
[CMP08]. Note that the fundamental class $[X]=\sigma_{0}=1\in H^{0}(X)$ of the
Cayley plane is on the right, and $\sigma_{1}$ is the class of a hyperplane.
Considering the diagram in this way, an edge $i$ from $\sigma_{w_{1}}$ (on the
left) to $\sigma_{w_{2}}$ (on the right) denotes that $w_{2}=s_{i}w_{1}$.
(Here $\sigma_{w}$ is the class of the Schubert variety
$\overline{B_{+}wP/P}$.)

Now, the interpretation of (2.25) as the Hasse diagram indicates a final
interpretation: we can interpret the diagram as the partial ordering of the
elements of $W^{P}$ under the Bruhat order given by $v\leq w$ if there is a
$u\in W$ such that $w=uv$ and $\ell(w)=\ell(u)+\ell(v)$. Thus, any path from
the left to the right in the diagram in (2.25) gives a reduced expression for
the minimal coset representative $w^{P}$ of $w_{0}W_{P}$; we will fix the
following:

$w^{P}=s_{1}s_{3}s_{4}s_{2}s_{5}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{3}s_{2}s_{4}s_{5}s_{6}.$
(2.26)

Whenever we consider the Cayley plane, the sequence $(r_{i})_{i=1}^{16}$ will
denote the sequence of indices of this reduced expression by, i.e. $r_{1}=1$,
$r_{2}=3$ and so on.

####  2.4.2. The Freudenthal variety.

The Freudenthal variety is also related to (complex) octonions, namely it is
the closed orbit of $E_{7}$ on the _Zorn algebra_ , see Section 2.3 of
[CMP08]. The most extensive analysis of the Freudenthal variety can be found
in [Fre64].

For the Freudenthal variety, we embed the Langlands dual homogeneous space
$\mathbb{X}^{\vee}$ is naturally embedded as
$\mathbb{X}^{\vee}\hookrightarrow\mathbb{P}\bigl{(}V({\varpi^{\vee}_{7}})^{*}\bigr{)}$.
The minuscule fundamental weight representation $V({\varpi^{\vee}_{7}})$ has
the following weight structure:

7654233214125146153716351736437462465642647346371536173516415214123324567$v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}^{\prime}$$v_{5}^{\prime\prime}$$v_{6}^{\prime}$$v_{6}^{\prime\prime}$$v_{7}^{\prime}$$v_{7}^{\prime\prime}$$v_{8}^{\prime}$$v_{8}^{\prime\prime}$$v_{9}$$v_{9}^{\prime}$$v_{9}^{\prime\prime}$$v_{10}$$v_{10}^{\prime}$$v_{10}^{\prime\prime}$$v_{11}$$v_{11}^{\prime}$$v_{11}^{\prime\prime}$$v_{12}$$v_{12}^{\prime}$$v_{12}^{\prime\prime}$$v_{13}$$v_{13}^{\prime}$$v_{13}^{\prime\prime}$$v_{14}^{\prime\prime}$$v_{14}^{\prime}$$v_{14}$$v_{15}^{\prime\prime}$$v_{15}^{\prime}$$v_{15}$$v_{16}^{\prime\prime}$$v_{16}^{\prime}$$v_{16}$$v_{17}^{\prime\prime}$$v_{17}^{\prime}$$v_{17}$$v_{18}^{\prime\prime}$$v_{18}^{\prime}$$v_{18}$$v_{19}^{\prime\prime}$$v_{19}^{\prime}$$v_{20}^{\prime\prime}$$v_{20}^{\prime}$$v_{21}^{\prime\prime}$$v_{21}^{\prime}$$v_{22}^{\prime\prime}$$v_{22}^{\prime}$$v_{23}$$v_{24}$$v_{25}$$v_{26}$$v_{27}$
(2.27)

Here we used the same conventions as for the diagram in (2.25). Note that the
center consists of the following commutative cube:

752527725257$v_{12}^{\prime}$$v_{13}$$v_{13}^{\prime}$$v_{13}^{\prime\prime}$$v_{14}^{\prime\prime}$$v_{14}^{\prime}$$v_{14}$$v_{15}^{\prime}$

with all southwest-northeast edges marked $2$, all west-east edges marked $5$
and all northwest-southeast edges marked $7$. We have again used the same
labeling for the basis vectors as used for the Schubert classes of the
cohomology of $E_{7}^{\mathrm{sc}}/P_{7}$ in [CMP08] in accordance with the
geometric Satake correspondence.

We have the same interpretations of diagram (2.27) as the actions of
$e^{\vee}_{i}$ and $f^{\vee}_{i}$ on $V({\varpi^{\vee}_{1}})$, as the Hasse
diagram for the cohomology, and as the partial order on the elements of
$W^{P}$ under the Bruhat order. Although any path from left to right in (2.27)
gives a reduced expression for $w^{P}$, we fix the following:

$w^{P}=s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{4}s_{5}s_{6}s_{1}s_{3}s_{4}s_{2}s_{5}s_{7}s_{4}s_{3}s_{1}s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}.$
(2.28)

Note that we have $(w^{P})^{-1}=w^{P}$, which did not hold in the case of the
Cayley plane.

##  3\. The canonical models for cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$

In this section we will define _canonical mirror models_
$(X^{\vee}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})$ for the Cayley plane
and the Freudenthal variety in terms of Plücker coordinates. (The terminology
“canonical” follows that of [PRW16].) In addition to the Plücker coordinate
expressions, our main results are:

######  Theorem 3.1.

For cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$, the variety
$X^{\vee}_{\mathrm{can}}$ is isomorphic to the open Richardson variety
$X^{\vee}_{\mathrm{Lie}}=\mathcal{R}^{\vee}_{w_{P},w_{0}}$.

######  Theorem 3.2.

For cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$, the pull-back of
$\mathcal{W}_{\mathrm{can}}$ under the isomorphism of Theorem 3.1 equals
$\mathcal{W}_{\mathrm{Lie}}$. In other words, the Landau-Ginzburg model
$(X^{\vee}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})$ is isomorphic to
Rietsch’s Lie-theoretic Landau-Ginzburg model
$(X^{\vee}_{\mathrm{Lie}},\mathcal{W}_{\mathrm{Lie}})$.

We follow the general approach as used in [PR13, PRW16]. In particular, we
will use a presentation of the coordinate ring of $U_{-}^{P}$ derived from the
results in [GLS11] (which we introduced in Section 2.3) and express it in
terms of Plücker coordinates using representation theory. However, the
varieties considered in these references are homogeneous under
$\mathrm{Sp}_{2n}$ and $\mathrm{Spin}_{2n}$, and the representation theory of
these groups is well-established (although by no means trivial). For the
groups of type $E_{6}$ and $E_{7}$, the representations have a more
complicated structure. Moreover, in contrast with [PR13, PRW16], we will
perform most calculations restricted to the open, dense torus $U^{\circ}_{-}$
(or more precisely: its transpose) to simplify the computations.

We will denote the homogeneous polynomials appearing in the numerators and
denominators of our superpotentials $\mathcal{W}_{\mathrm{can}}$ by $q_{i}$;
this should not cause any confusion with quantum parameters as cominuscule
homogeneous spaces always have
$\mathop{\mathrm{rk}}\nolimits_{\mathbb{Z}}H_{2}(X,\mathbb{Z})=1$ and thus
also only a single quantum parameter, which we denote by $q$.

###  3.1. The canonical model for the Cayley plane.

The homogeneous polynomials in Plücker coordinates appearing in the
denominators are the following:

$\displaystyle q_{12}$
$\displaystyle=p_{1}p_{11}^{\prime\prime}-p_{0}p_{12}^{\prime\prime},\qquad
q_{16}=p_{7}^{\prime}p_{9}^{\prime}-p_{8}p_{8}^{\prime},\qquad
q_{20}=p_{5}^{\prime\prime}p_{15}-p_{4}^{\prime\prime}p_{16},$ (3.1)
$\displaystyle q_{24}$
$\displaystyle=p_{0}(p_{11}^{\prime}p_{13}-p_{12}^{\prime}p_{12}^{\prime\prime})-p_{1}(p_{10}^{\prime\prime}p_{13}-p_{11}^{\prime\prime}p_{12}^{\prime})+p_{2}(p_{10}^{\prime\prime}p_{12}^{\prime\prime}-p_{11}^{\prime}p_{11}^{\prime\prime});$

and the following will be numerators:

$\displaystyle q_{13}$
$\displaystyle=p_{2}p_{11}^{\prime\prime}-p_{0}p_{13},\qquad
q_{17}=p_{7}^{\prime}p_{10}^{\prime}-p_{8}p_{9}^{\prime\prime},\qquad
q_{21}=p_{6}^{\prime\prime}p_{15}-p_{5}^{\prime}p_{16},$ (3.2) $\displaystyle
q_{25}$
$\displaystyle=p_{0}p_{11}^{\prime}p_{14}-p_{1}p_{10}^{\prime\prime}p_{14}+p_{3}(p_{10}^{\prime\prime}p_{12}^{\prime\prime}-p_{11}^{\prime}p_{11}^{\prime\prime}).$

Using these coordinates, define the following rational map
$\mathbb{X}^{\vee}=\widetilde{P}^{\vee}\backslash\widetilde{G}^{\vee}\to\mathbb{C}$,
called the _canonical superpotential_ :

$\displaystyle\mathcal{W}_{\mathrm{can}}={}$
$\displaystyle\frac{p_{1}}{p_{0}}+\frac{p_{9}^{\prime}}{p_{8}}+\frac{q_{13}}{q_{12}}+\frac{q_{17}}{q_{16}}+\frac{q_{21}}{q_{20}}+\frac{q_{25}}{q_{24}}+q\frac{p_{5}^{\prime\prime}}{p_{16}}.$
(3.3)

The denominators of the rational map of equation (3.3) define principal
divisors on $\mathbb{X}^{\vee}$ which we will denote as follows:

$\displaystyle D_{0}^{(1)}$ $\displaystyle=\\{p_{0}=0\\},\quad
D_{8}^{(1)}=\\{p_{8}=0\\},\quad D_{16}^{(1)}=\\{p_{16}=0\\},$ $\displaystyle
D_{12}^{(2)}$ $\displaystyle=\\{q_{12}=0\\},~{}\quad
D_{16}^{(2)}=\\{q_{16}=0\\},\quad D_{20}^{(2)}=\\{q_{20}=0\\},$ $\displaystyle
D_{24}^{(3)}$ $\displaystyle=\\{q_{24}=0\\}.$

Take $D=\sum D_{i}^{(j)}$ and let
$X^{\vee}_{\mathrm{can}}=\mathbb{X}^{\vee}\setminus D$ be the complement of
this anticanonical divisor. The restriction of $\mathcal{W}_{\mathrm{can}}$ to
$X^{\vee}_{\mathrm{can}}$ is regular and will also be denoted by
$\mathcal{W}_{\mathrm{can}}$. The pair
$(X^{\vee}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})$ forms the canonical LG
model for the Cayley plane.

###  3.2. The canonical model for the Freudenthal variety.

The canonical superpotential for the Freudenthal variety will have the
following denominators:

$\displaystyle q_{18}$ $\displaystyle=p_{1}p_{17}-p_{0}p_{18},\qquad
q_{36}=p_{10}p_{26}-p_{9}p_{27},$ (3.4) $\displaystyle q_{27}$
$\displaystyle=p_{0}p_{27}-p_{1}p_{26}+p_{2}p_{25}-p_{3}p_{24}+p_{4}p_{23}-p_{5}^{\prime\prime}p_{22}^{\prime\prime}+p_{6}^{\prime\prime}p_{21}^{\prime\prime},$
$\displaystyle q_{45}$
$\displaystyle=(p_{9}p_{11}-p_{10}p_{10}^{\prime})p_{25}-(p_{8}^{\prime}p_{11}-p_{9}^{\prime}p_{10})p_{26}+(p_{8}^{\prime}p_{10}^{\prime}-p_{9}p_{9}^{\prime})p_{27},$
$\displaystyle q_{36}^{\prime}$
$\displaystyle=(p_{0}p_{14}^{\prime}-p_{1}p_{13}^{\prime\prime}+p_{2}p_{12}^{\prime\prime})p_{22}^{\prime}+(-p_{0}p_{13}+p_{1}p_{12}^{\prime}-p_{2}p_{11}^{\prime\prime})p_{23}$
$\displaystyle\quad+(p_{0}p_{12}-p_{1}p_{11}^{\prime}+p_{2}p_{10}^{\prime\prime})p_{24}-p_{2}p_{9}^{\prime\prime}p_{25}+p_{1}p_{9}^{\prime\prime}p_{26}-p_{0}p_{9}^{\prime\prime}p_{27},$
$\displaystyle q_{54}$
$\displaystyle=(p_{5}^{\prime\prime}p_{7}^{\prime\prime}-p_{6}^{\prime}p_{6}^{\prime\prime})(p_{20}^{\prime\prime}p_{22}^{\prime\prime}-p_{21}^{\prime}p_{21}^{\prime\prime})-(p_{4}p_{7}^{\prime\prime}-p_{5}^{\prime}p_{6}^{\prime\prime})(p_{20}^{\prime\prime}p_{23}-p_{21}^{\prime\prime}p_{22}^{\prime})$
$\displaystyle\quad+(p_{4}p_{6}^{\prime}-p_{5}^{\prime}p_{5}^{\prime\prime})(p_{21}^{\prime}p_{23}-p_{22}^{\prime}p_{22}^{\prime\prime})+q_{27}^{\prime}q_{27}^{\prime\prime},$

where

$\displaystyle q_{27}^{\prime}$
$\displaystyle=-p_{0}p_{27}+p_{1}p_{26}-p_{2}p_{25}+p_{3}p_{24},$
$\displaystyle q_{27}^{\prime\prime}$
$\displaystyle=-p_{6}^{\prime\prime}p_{21}^{\prime\prime}-p_{7}^{\prime}p_{20}^{\prime}+p_{7}^{\prime\prime}p_{20}^{\prime\prime}+p_{8}^{\prime}p_{19}^{\prime}-p_{9}p_{18}+p_{10}p_{17};$

and the following numerators:

$\displaystyle q_{19}$ $\displaystyle=p_{2}p_{17}-p_{0}p_{19}^{\prime},\qquad
q_{28}=p_{5}^{\prime}p_{23}-p_{6}^{\prime}p_{22}^{\prime\prime}+p_{7}^{\prime\prime}p_{21}^{\prime\prime},\qquad
q_{37}=p_{11}p_{26}-p_{10}^{\prime}p_{27},$ (3.5) $\displaystyle q_{46}$
$\displaystyle=(p_{9}p_{12}-p_{10}p_{11}^{\prime})p_{25}+(-p_{8}^{\prime}p_{12}+p_{10}p_{10}^{\prime\prime})p_{26}+(p_{8}^{\prime}p_{11}^{\prime}-p_{9}p_{10}^{\prime\prime})p_{27},$
$\displaystyle q_{37}^{\prime}$
$\displaystyle=(p_{0}p_{15}^{\prime}-p_{1}p_{14}+p_{3}p_{12}^{\prime\prime})p_{22}^{\prime}+(-p_{0}p_{14}^{\prime\prime}+p_{1}p_{13}^{\prime}-p_{3}p_{11}^{\prime\prime})p_{23}$
$\displaystyle\quad+p_{3}p_{10}^{\prime\prime}p_{24}+(p_{0}p_{12}-p_{1}p_{11}^{\prime}-p_{3}p_{9}^{\prime\prime})p_{25}+p_{1}p_{10}^{\prime\prime}p_{26}-p_{0}p_{10}^{\prime\prime}p_{27},$
$\displaystyle q_{55}$
$\displaystyle=(p_{5}^{\prime\prime}p_{8}^{\prime\prime}-p_{6}^{\prime\prime}p_{7}^{\prime})(p_{20}^{\prime\prime}p_{22}^{\prime\prime}-p_{21}^{\prime}p_{21}^{\prime\prime})-p_{4}p_{8}^{\prime\prime}(p_{20}^{\prime\prime}p_{23}-p_{21}^{\prime\prime}p_{22}^{\prime})+p_{4}p_{7}^{\prime}(p_{21}^{\prime}p_{23}-p_{22}^{\prime}p_{22}^{\prime\prime})$
$\displaystyle\quad+p_{5}^{\prime}p_{24}(-p_{5}^{\prime\prime}p_{21}^{\prime}+p_{6}^{\prime\prime}p_{20}^{\prime\prime})+p_{4}p_{24}(-p_{4}p_{23}+p_{5}^{\prime}p_{22}^{\prime}+p_{5}^{\prime\prime}p_{22}^{\prime\prime}-p_{6}^{\prime\prime}p_{21}^{\prime\prime})$
$\displaystyle\quad+(p_{4}p_{24}-p_{7}^{\prime}p_{21}^{\prime}+p_{8}^{\prime\prime}p_{20}^{\prime\prime})(-p_{0}p_{27}+p_{1}p_{26}-p_{2}p_{25}+p_{3}p_{24}).$

The _canonical superpotential_ is the following map
$\mathbb{X}^{\vee}\to\mathbb{C}$:

$\displaystyle\mathcal{W}_{\mathrm{can}}=\frac{p_{1}}{p_{0}}+\frac{q_{19}}{q_{18}}+\frac{q_{28}}{q_{27}}+\frac{q_{37}}{q_{36}}+\frac{q_{37}^{\prime}}{q_{36}^{\prime}}+\frac{q_{46}}{q_{45}}+\frac{q_{55}}{q_{54}}+q\frac{p_{10}}{p_{27}}$
(3.6)

The sum of the loci where the denominators of this superpotential are zero
forms an anticanonical divisor $D$; we will write
$X^{\vee}_{\mathrm{can}}=\mathbb{X}^{\vee}\setminus D$ for its complement. We
will also write $\mathcal{W}_{\mathrm{can}}$ for the regular restriction of
the superpotential to $X^{\vee}_{\mathrm{can}}$. The pair
$(X^{\vee}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})$ is the canonical LG
model for the Freudenthal variety.

######  Remark 3.3.

We observed that the numerators and denominators in our superpotentials are
related in the quantum cohomology ring of $X$ as follows. After identifying
Plücker coordinates with Schubert classes via the geometric Satake
correspondence, we may interpret the numerator and denominator of each term in
our superpotentials as elements of the quantum cohomology ring by interpreting
all products as quantum products. We observe after these identifications that
for each term in our superpotentials, its numerator equals the quantum product
of its denominator with the hyperplane class $\sigma_{1}$, scaled by the
degree of the denominator (or numerator) as a polynomial in Plücker
coordinates. For example, for our Cayley plane superpotential in equation
(3.3), we compute that $q_{25}=3\sigma_{1}*q_{24}$ where all products
(including those in the Plücker coordinate expressions of $q_{24}$ and
$q_{25}$) are interpreted as quantum products.

In order to verify this relationship between the numerators and denominators
of the superpotential, we used the description of the quantum cohomology ring
for the Cayley plane presented in Proposition 3 and Theorem 31 of [CMP08] as
well as the description of the quantum cohomology ring of the Freudenthal
variety using Theorems 6 and 34 of [CMP08].

Similar statements have appeared in type $A$ in e.g. [MR20, Kal20], and in
type $B$ in [PR18, Section 9]. This scaling by degree was not observed in the
type-$A$ Grassmannians $\mathrm{Gr}(k,n)$ because there each term was a ratio
of degree-$1$ expressions in the Plücker coordinates, while the type $B$
superpotential had one term ($W_{m-1}$ in the notation of [PR18]) which
contributed only $p_{1}$, despite being a ratio of degree-$2$ polynomials.
Furthermore, in type $C$, [PR13] observed that each term of their
superpotential contributes only one hyperplane class $p_{1}$. These
observations suggest the existence of a general construction of canonical
Plücker coordinate superpotentials for certain homogeneous spaces, and we are
investigating this topic in an upcoming article.

###  3.3. Overview of the isomorphisms

The proof of Theorem 3.1 requires an auxiliary variety, $U_{-}^{P}$, which we
defined in Subsection 2.2. Namely, we will construct the isomorphism as the
following composition of maps:

$X^{\vee}_{\mathrm{Lie}}\overset{\varphi}{\longrightarrow}U_{-}^{P}\overset{\pi}{\longrightarrow}\mathbb{X}^{\vee}\supset
X^{\vee}_{\mathrm{can}}.$ (3.7)

Recall from equation (2.13) that

$U_{-}^{P}=U^{\vee}_{-}\cap B^{\vee}_{+}\bar{w}_{P}\bar{w}_{0}B^{\vee}_{+}.$

The map $\pi:U_{-}^{P}\to\mathbb{X}^{\vee}$ is the restriction of the quotient
map
$\widetilde{G}^{\vee}\to\widetilde{P}^{\vee}\backslash\widetilde{G}^{\vee}=\mathbb{X}^{\vee}$
to $U_{-}^{P}$. (Note that we identify $U^{\vee}_{-}\subset G^{\vee}$ and its
universal cover in $\widetilde{G}^{\vee}$.)

The map
$\varphi:X^{\vee}_{\mathrm{Lie}}\overset{\sim}{\longrightarrow}U_{-}^{P}$ is
an isomorphism that cannot be expressed so explicitly. It is given by mapping
$u_{+}\bar{w}_{P}B^{\vee}_{-}\in X^{\vee}_{\mathrm{Lie}}$ (with $u_{+}\in
U^{\vee}_{+}$) to the unique $u_{-}\in U_{-}^{P}$ such that
$u_{+}\bar{w}_{P}u_{-}\in\mathcal{Z}^{\vee}_{P}$. Here, the variety
$\mathcal{Z}^{\vee}_{P}$ was defined in equation (2.12) as

$\mathcal{Z}^{\vee}_{P}=B^{\vee}_{-}\bar{w}_{0}^{-1}\cap
U^{\vee}_{+}T^{P}\bar{w}_{P}U^{\vee}_{-}~{}\subset~{}G^{\vee},$

and we noted in Lemma 2.6 that each $z\in\mathcal{Z}^{\vee}_{P}$ has a unique
decomposition as $z=u_{+}t\bar{w}_{P}u_{-}$ with $u_{+}\in U^{\vee}_{+}$,
$t\in T^{P}$ and $u_{-}\in U_{-}^{P}$. The map can also be obtained by
composing the isomorphism $\Psi:X^{\vee}_{\mathrm{Lie}}\times
T^{P}\to\mathcal{Z}^{\vee}_{P}$ of equation (2.11) mentioned in 2.4 with the
isomorphism $\mathcal{Z}^{\vee}_{P}\to U_{-}^{P}\times T^{P}$ sending $z$ to
the unique $(u_{-},t)$ such that $z=u_{+}t\bar{w}_{P}u_{-}$ for some $u_{+}\in
U^{\vee}_{+}$, and then restricting the resulting map
$X^{\vee}_{\mathrm{Lie}}\times T^{P}\to U_{-}^{P}\times T^{P}$ to the unit
element in $T^{P}$.

To show that $\pi$ is an isomorphism between $U_{-}^{P}$ and
$X^{\vee}_{\mathrm{can}}\subset\mathbb{X}^{\vee}$ we use the description of
the coordinate ring of $U_{+}^{P}=(U_{-}^{P})^{T}$ given by [GLS11], which we
discussed in Subsection 2.3. The first step for this is to express the
coordinate ring in terms of the Plücker coordinates.

It turns out that there is an easy criterion for a generalized Plücker
coordinate $p_{i}$ defined in equation (2.17) to coincide with a coordinate
$p^{\text{GLS}}_{m}$ used by [GLS11] under transposition. Recall from
Subsection 2.3 that $p^{\text{GLS}}_{m}$ was defined as the dual map
associated to a basis element $e^{\vee}_{w^{P}}(m)$ for
$\mathfrak{u}^{\vee}_{-{\beta^{\vee}_{P}}(m)}$, where ${\beta^{\vee}_{P}}(m)$
is the $m$th positive root mapped to a negative root by $w^{P}$, see equation
(2.19).

######  Remark 3.4.

This criterion holds for general cominuscule homogeneous spaces, and we will
prove it in this context. Thus, until the end of this subsection, $X=G/P_{k}$
refers to a general cominuscule homogeneous space with the $k$th vertex of the
Dynkin diagram corresponding to a cominuscule fundamental weight.

We first need the following characterization of the set
$\Pi^{\vee}_{w^{P}}=\\{{\beta^{\vee}_{P}}(m)~{}|~{}m=1,\ldots,\ell\\}$, and of
its complement $\Pi^{\vee}_{P}$ which coincides with the positive roots mapped
to negative roots by $w_{P}$:

######  Lemma 3.5.

Every positive root
$\alpha^{\vee}=\sum_{i=1}^{n}d_{i}\alpha^{\vee}_{i}\in\Pi^{\vee}=\Pi^{\vee}_{w^{P}}\sqcup\Pi^{\vee}_{P}$
has $d_{k}\in\\{0,1\\}$. Moreover, $\alpha^{\vee}\in\Pi^{\vee}_{P}$ if and
only if $d_{k}=0$; and $\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}$ if and only if
$d_{k}=1$.

###### Proof.

The fact that ${\varpi^{\vee}_{k}}$ is minuscule is well-known to be
equivalent to the fact that the longest root $\alpha^{\vee}_{0}\in\Phi^{\vee}$
has coefficient $1$ in front of $\alpha^{\vee}_{k}$ in the decomposition of
$\alpha^{\vee}_{0}$ in terms of simple roots. (See e.g. exercise VI.24(c) of
[Bou68].) Thus, we know that $d_{k}\in\\{0,1\\}$.

Now, it is well known that there is a sequence of (not necessarily distinct)
simple roots $(\alpha^{\vee}_{i_{j}})_{j=1}^{h}$, where
$h=0pt(\alpha^{\vee})=\sum_{i=1}^{n}d_{i}$ is the height of $\alpha^{\vee}$,
such that we have $\alpha^{\vee}=\sum_{j=1}^{h}\alpha^{\vee}_{i_{j}}$ and such
that for every $h^{\prime}\leq h$ the partial sum
$\sum_{j=1}^{h^{\prime}}\alpha^{\vee}_{i_{j}}\in\Phi^{\vee}$ is a root as
well. (See Corollary 10.2 of [Hum78].)

Suppose $d_{k}=0$, then we clearly have that $i_{j}\neq k$ for all $j$. Since
$\\{\alpha^{\vee}_{i}~{}|~{}i\neq k\\}\subset\Pi^{\vee}_{P}$ and
$\Pi^{\vee}_{P}$ is bracket closed, every partial sum lies in $\Pi^{\vee}_{P}$
and thus $\alpha^{\vee}$ as well. Conversely, since $\Pi^{\vee}_{P}$ can be
considered as the positive roots of the root system with as Dynkin diagram the
diagram of $G^{\vee}$ with the $k$th vertex removed, we have that
$\alpha^{\vee}\in\Pi^{\vee}_{P}$ implies that $d_{k}=0$.

As $\Pi^{\vee}_{w^{P}}$ is the complement of $\Pi^{\vee}_{P}$ in $\Pi^{\vee}$,
the statement follows. ∎

Recall that we defined a $\mathbb{N}_{0}\Delta\\!^{\vee}$-grading on the
completed universal enveloping algebra $\widehat{\mathcal{U}}^{\vee}_{+}$ of
$\mathfrak{u}^{\vee}_{+}$ in Section 2.3. Here,
$\mathbb{N}_{0}\Delta\\!^{\vee}\cong\mathbb{N}_{0}^{n}$ is the positive root
lattice of all non-negative integral linear combinations of the simple roots
$\alpha^{\vee}_{i}$. We defined
$\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee})\subset\widehat{\mathcal{U}}^{\vee}_{+}$
as the subspace spanned by pure tensors $e^{\vee}_{i_{1}}\cdots
e^{\vee}_{i_{j}}$ such that exactly $d_{i}$ of the factors have index $i$,
where
$\alpha^{\vee}=\sum_{i=1}^{n}d_{i}\alpha^{\vee}_{i}\in\mathbb{N}_{0}\Delta\\!^{\vee}$.
We noted there that
$\mathfrak{u}^{\vee}_{+\alpha^{\vee}}\subset\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee})$
for $\alpha^{\vee}\in\Pi^{\vee}$, so we directly conclude that:

######  Corollary 3.6.

For $j\in\\{1,\ldots,{\ell_{0}}\\}$,
$e^{\vee}_{w^{P}}(j)\in\widehat{\mathcal{U}}^{\vee}_{+}({\beta^{\vee}_{P}}(j))$.
Let $d_{i}\in\mathbb{N}_{0}$ such that
${\beta^{\vee}_{P}}(j)=\sum_{i=1}^{n}d_{i}\alpha^{\vee}_{i}$. If
$j\in\\{1,\ldots,\ell\\}$, then $d_{k}=1$. Else, i.e.
$j\in\\{\ell+1,\ldots,{\ell_{0}}\\}$, we have $d_{k}=0$.

We also need the following simple observation regarding the action of the
elements of the completed universal enveloping algebra on
$V({\varpi^{\vee}_{\sigma(k)}})$:

######  Lemma 3.7.

If $u\in\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee})$ for
$\alpha^{\vee}\in\mathbb{N}_{0}\Delta\\!^{\vee}$ and $v\in
V({\varpi^{\vee}_{\sigma(k)}})$ a vector of weight $\mu$, then $u\cdot v$ is
either a vector of weight $\mu+\alpha^{\vee}$ or the zero vector.

###### Proof.

By Theorem 2.10, the statement holds for the Chevalley generators
$e^{\vee}_{i}\in\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee}_{i})$. By
definition $u\in\widehat{\mathcal{U}}^{\vee}_{+}(\alpha^{\vee})$ for
$\alpha^{\vee}=\sum_{i=1}^{n}d_{i}\alpha^{\vee}_{i}$ can be decomposed into a
linear combination of pure tensors in which each $e^{\vee}_{i}$ appears
$d_{i}$ times. The statement now follows by induction. ∎

Now, recall that the Plücker coordinates $p_{i}:g\mapsto v_{0}^{*}(g\cdot
v_{i})$ are defined using the minuscule representation
$V({\varpi^{\vee}_{\sigma(k)}})$ and its dual, where
$-w_{0}\cdot{\varpi^{\vee}_{k}}=-w^{P}\cdot{\varpi^{\vee}_{k}}$ and this
either equals ${\varpi^{\vee}_{k}}$ or ${\varpi^{\vee}_{\sigma(k)}}$ for
$\sigma:\\{1,\ldots,n\\}\to\\{1,\ldots,n\\}$ the Dynkin diagram symmetry; see
the discussion of Plücker coordinates of Section 2.3. The criterion is given
as follows:

######  Proposition 3.8.

If $v_{i}$ is of weight $-{\varpi^{\vee}_{k}}+\alpha^{\vee}(v_{i})$ and
$\alpha^{\vee}(v_{i})={\beta^{\vee}_{P}}(m)\in\Pi^{\vee}_{w^{P}}$, then
$u\mapsto p_{i}(u^{T})$ coincides up to a constant with $p^{\text{GLS}}_{m}$
on $U_{+}^{P}$.

###### Proof.

Let $u\in U_{+}^{P}\hookrightarrow\widehat{\mathcal{U}}^{\vee}_{+}$ and
consider the decomposition of $u$ with respect to the PBW basis. If the
coefficient in front of $e^{\vee}_{w^{P}}(m)$ is $c$, then we have by
definition that $p^{\text{GLS}}_{m}(u)=c$.

On the other hand, $p_{i}(u^{T})$ is the coefficient in front of $v_{0}$ of
the projection of $u^{T}\cdot v_{i}$ to $\mathbb{C}v_{0}\subset
V({\varpi^{\vee}_{\sigma(k)}})$. It suffices to show that
$e^{\vee}_{w^{P}}(m)^{T}$ is the only transposed PBW basis element mapping
$v_{i}$ to a non-zero multiple of $v_{0}$. Equivalently, we need to show that
$e^{\vee}_{w^{P}}(m)$ is the only PBW basis element mapping $v_{0}$ to a non-
zero multiple of $v_{i}$. One can see this equivalence as follows: For
Chevalley generators, Theorem 2.10 implies that $e^{\vee}_{j}$ maps $v$ to
$v^{\prime}$ if and only if $(e^{\vee}_{i})^{T}=f^{\vee}_{j}$ maps
$v^{\prime}$ to $v$; by induction, the same holds for all pure tensors;
finally, each PBW basis element is a linear combination of these pure tensors,
so that the equivalence follows.

By Lemma 3.7, only the elements of
$\widehat{\mathcal{U}}^{\vee}_{+}\bigl{(}{\beta^{\vee}_{P}}(m)\bigr{)}$ map
$v_{0}$ to $\mathbb{C}v_{i}$. Thus, we can restrict ourselves to PBW basis
elements $e^{\vee}_{w^{P}}(\mathbf{m})$ lying in
$\widehat{\mathcal{U}}^{\vee}_{+}\bigl{(}{\beta^{\vee}_{P}}(m)\bigr{)}$. Now,
$e^{\vee}_{w^{P}}(m)\in\mathfrak{u}^{\vee}_{+{\beta^{\vee}_{P}}(m)}\subset\widehat{\mathcal{U}}^{\vee}_{+}\bigl{(}{\beta^{\vee}_{P}}(m)\bigr{)}$.
Moreover, Lemma 3.5 implies that $d_{k}=1$ in the decomposition
${\beta^{\vee}_{P}}(m)=\sum_{i=1}^{n}d_{i}\alpha^{\vee}_{i}$.

First, let us consider
$e^{\vee}_{w^{P}}(\mathbf{m})\in\widehat{\mathcal{U}}^{\vee}_{+}\bigl{(}{\beta^{\vee}_{P}}(m)\bigr{)}$
not equal to $e^{\vee}_{w^{P}}(m)$. Since for $j\neq m$ we have
$e^{\vee}_{w^{P}}(j)\notin\widehat{\mathcal{U}}^{\vee}_{+}\bigl{(}{\beta^{\vee}_{P}}(m)\bigr{)}$,
this assumption means that we are considering those
$e^{\vee}_{w^{P}}(\mathbf{m})$ with
$\mathbf{m}=(m_{1},m_{2},\ldots,m_{\ell_{0}})$ satisfying
$\sum_{j=1}^{{\ell_{0}}}m_{j}\geq 2$. Corollary 3.6 implies that of the
coefficients $m_{1},\ldots,m_{\ell}$ (where $\ell=\ell(w^{P})$) exactly one is
non-zero, and that coefficient equals $1$, as these are the only coefficients
contributing to $d_{k}=1$. Thus, at least one of
$m_{\ell+1},\ldots,m_{{\ell_{0}}}$ is non-zero; let
$j\in\\{\ell+1,\ldots,{\ell_{0}}\\}$ be the largest index such that $m_{j}\neq
0$. Recalling the definition of the PBW basis elements from equation (2.20),
we find that $e^{\vee}_{w^{P}}(\mathbf{m})\cdot
v_{0}=\bigl{(}e^{\vee}_{w^{P}}(1)\bigr{)}^{m_{1}}\cdots\bigl{(}e^{\vee}_{w^{P}}(j)\bigr{)}^{m_{j}}\cdot
v_{0}$. However, $e^{\vee}_{w^{P}}(j)\cdot v_{0}=0$: we have by Corollary 3.6
that
$e^{\vee}_{w^{P}}(j)\in\widehat{\mathcal{U}}^{\vee}_{+}\bigl{(}{\beta^{\vee}_{P}}(m)\bigr{)}$
where
${\beta^{\vee}_{P}}=\sum_{j^{\prime}=1}^{n}d_{j^{\prime}}^{\prime}\alpha^{\vee}_{i}$
with $d_{k}^{\prime}=0$, which implies that $e^{\vee}_{w^{P}}(j)$ can be
written as a linear combination of pure tensors involving the Chevalley
generators $e^{\vee}_{i^{\prime}}$ for $i^{\prime}\neq k$, but by Theorem 2.10
we know that $e^{\vee}_{i^{\prime}}\cdot v_{0}=0$ for $i^{\prime}\neq k$.

Thus, it remains to show that $e^{\vee}_{w^{P}}(m)\cdot v_{0}$ is a non-zero
multiple of $v_{i}$. Suppose it acts trivially, then we find that
$\mathbb{C}v_{i}\cap\widehat{\mathcal{U}}^{\vee}_{+}v_{0}=\\{0\\}$, which
contradicts the irreducibility of $V({\varpi^{\vee}_{\sigma(k)}})$. ∎

Using this criterion, we will be able to identify all of the generators of the
coordinate ring of $U_{-}^{P}$ given by [GLS11] with generalized Plücker
coordinates.

Moreover, we can identify the generalized minors defined in (2.23) with the
denominators of the superpotentials for the cominuscule exceptional
homogeneous spaces as given in (3.1) and (3.4). We will identify these
functions by checking that their expressions in terms of the coordinates
$\\{a_{i}\\}_{i=1}^{\ell}$ on the algebraic torus $U^{\circ}_{-}$ of (2.15)
agree.

###  3.4. The coordinate ring of $U_{-}^{P}$ for the Cayley plane

Recall the conventions for the Cayley plane $E_{6}^{\mathrm{sc}}/P_{6}$ from
Section 2.4. In particular, we have $\ell=\ell(w^{P})=16$. Applying this to
the coordinate ring of $U_{+}^{P}$ described in Proposition 2.14 (see
Proposition 8.5 of [GLS11]), we find that it is generated by the dual PBW
elements $p^{\text{GLS}}_{m}$, $m\in\\{1,\ldots,16\\}$, and localized at the
generalized minors
$\Delta_{{\varpi^{\vee}_{j}},(w^{P})^{-1}({\varpi^{\vee}_{j}})}$,
$j\in\\{1,\ldots,6\\}$. Using the criterion in Proposition 3.8, we find that
the generators $p^{\text{GLS}}_{m}$ can be expressed in terms of Plücker
coordinates as follows:

######  Lemma 3.9.

On $U_{+}^{P}\subset\widetilde{G}^{\vee}$, we have the following
identifications:

$p_{0}$$p_{1}$$p_{2}$$p_{3}$$p_{4}^{\prime}$$p_{4}^{\prime\prime}$$p_{5}^{\prime}$$p_{5}^{\prime\prime}$$p_{6}^{\prime}$$p_{6}^{\prime\prime}$$p_{7}^{\prime}$$p_{7}^{\prime\prime}$$p_{8}^{\prime}$$p_{8}^{\prime\prime}$$p_{9}^{\prime\prime}$$p_{10}^{\prime\prime}$$p_{11}^{\prime\prime}$$1$$p^{\text{GLS}}_{1}$$p^{\text{GLS}}_{2}$$p^{\text{GLS}}_{3}$$p^{\text{GLS}}_{4}$$p^{\text{GLS}}_{5}$$p^{\text{GLS}}_{6}$$p^{\text{GLS}}_{9}$$p^{\text{GLS}}_{7}$$p^{\text{GLS}}_{10}$$p^{\text{GLS}}_{8}$$p^{\text{GLS}}_{11}$$p^{\text{GLS}}_{12}$$p^{\text{GLS}}_{13}$$p^{\text{GLS}}_{14}$$p^{\text{GLS}}_{15}$$p^{\text{GLS}}_{16}$
(3.8)

Here, a vertex marked with $p_{i}^{(j)}$ below and $p^{\text{GLS}}_{m}$ above
denotes that $p_{i}^{(j)}(u_{+}^{T})$ equals $p^{\text{GLS}}_{m}(u_{+})$ up to
a constant for $u_{+}\in U_{-}^{P}$.

For example, we have that $p_{7}^{\prime\prime}(u_{+}^{T})$ equals
$p^{\text{GLS}}_{11}(u_{+})$ up to a constant for $u_{+}\in U_{+}^{P}$. In
particular, the rightmost vertex tells us that $p_{0}(u_{+}^{T})$ is constant,
and it is easy to see that this constant is $1$. Note that the diagram of
equation (3.8) is a subdiagram of the one in equation (2.25). Upon removing
the rightmost vertex, this subdiagram is in fact isomorphic to the Hasse
diagram of
$\mathrm{OG}(5,10)=\mathrm{PSO}_{10}/P_{5}=D_{5}^{\mathrm{sc}}/P_{5}$. This
variety is also known as a _spinor variety_. See [IM05, Section 4], for its
role in the geometry of the Cayley plane. The appearance of this variety is
not too much of a surprise, as we obtain the Dynkin diagram of type $D_{5}$
upon removing the sixth vertex from the diagram of type $E_{6}$; in other
words $W_{P}$ is of type $D_{5}$.

###### Proof.

We can compute the weights of the minuscule fundamental weight representation
$V({\varpi^{\vee}_{6}})$ directly using diagram (2.25) and Theorem 2.10. On
the other hand, the positive roots ${\beta^{\vee}_{P}}(m)$,
$m\in\\{1,\ldots,16\\}$ are straightforward to compute using the expression in
equation (2.19). A comparison of the results gives the following equalities:

$\begin{array}[]{llll}\alpha^{\vee}(v_{1})={\beta^{\vee}_{P}}(1)&\alpha^{\vee}(v_{2})={\beta^{\vee}_{P}}(2)&\alpha^{\vee}(v_{3})={\beta^{\vee}_{P}}(3)&\alpha^{\vee}(v_{4}^{\prime})={\beta^{\vee}_{P}}(4)\\\
\alpha^{\vee}(v_{4}^{\prime\prime})={\beta^{\vee}_{P}}(5)&\alpha^{\vee}(v_{5}^{\prime})={\beta^{\vee}_{P}}(6)&\alpha^{\vee}(v_{5}^{\prime\prime})={\beta^{\vee}_{P}}(9)&\alpha^{\vee}(v_{6}^{\prime})={\beta^{\vee}_{P}}(7)\\\
\alpha^{\vee}(v_{6}^{\prime\prime})={\beta^{\vee}_{P}}(10)&\alpha^{\vee}(v_{7}^{\prime})={\beta^{\vee}_{P}}(8)&\alpha^{\vee}(v_{7}^{\prime\prime})={\beta^{\vee}_{P}}(11)&\alpha^{\vee}(v_{8}^{\prime})={\beta^{\vee}_{P}}(12)\\\
\alpha^{\vee}(v_{8}^{\prime\prime})={\beta^{\vee}_{P}}(13)&\alpha^{\vee}(v_{9}^{\prime\prime})={\beta^{\vee}_{P}}(14)&\alpha^{\vee}(v_{10}^{\prime\prime})={\beta^{\vee}_{P}}(15)&\alpha^{\vee}(v_{11}^{\prime\prime})={\beta^{\vee}_{P}}(16)\end{array}$

Using Proposition 3.8, we find that $p_{i}^{(j)}(u_{+}^{T})$ equals
$p^{\text{GLS}}_{m}(u_{+})$ up to a constant as indicated in (3.8) for
$u_{+}\in U_{+}^{P}$. The only exception is the identification
$p_{0}(u_{-})=1$, but this identity is obvious from the fact that
$u_{+}^{T}\in U^{\vee}_{-}$ is a lower unipotent element and thus acts
trivially on the lowest weight vector $v_{0}$. ∎

To simplify expressions in the following, we will refer to the Plücker
coordinates appearing in the above as

$p_{\alpha^{\vee}}=p_{i}^{(j)}\text{ when
$\alpha^{\vee}=\alpha^{\vee}(v_{i}^{(j)})\in\Pi^{\vee}_{w^{P}}$.}$ (3.9)

Note that the set
$\\{p_{\alpha^{\vee}}~{}|~{}\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}\\}$ consists of
the Plücker coordinates appearing in Lemma 3.9 (it does not include $p_{0}$).
Moreover, we will write $p_{\alpha^{\vee}}^{T}:U_{+}^{P}\to\mathbb{C}$ for the
map $u_{+}\mapsto p_{\alpha^{\vee}}(u_{+}^{T})$.

######  Lemma 3.10.

The generalized minors defined in equation (2.23) can be expressed on
$U_{+}^{P}$ in terms of Plücker coordinates as

$\begin{array}[]{lll}\Delta_{{\varpi^{\vee}_{1}},(w^{P})^{-1}({\varpi^{\vee}_{1}})}(u_{+})=p_{16}(u_{+}^{T}),&\Delta_{{\varpi^{\vee}_{2}},(w^{P})^{-1}({\varpi^{\vee}_{2}})}(u_{+})=q_{12}(u_{+}^{T}),\\\
\Delta_{{\varpi^{\vee}_{3}},(w^{P})^{-1}({\varpi^{\vee}_{3}})}(u_{+})=q_{20}(u_{+}^{T}),&\Delta_{{\varpi^{\vee}_{4}},(w^{P})^{-1}({\varpi^{\vee}_{4}})}(u_{+})=q_{24}(u_{+}^{T}),\\\
\Delta_{{\varpi^{\vee}_{5}},(w^{P})^{-1}({\varpi^{\vee}_{5}})}(u_{+})=q_{16}(u_{+}^{T}),&\Delta_{{\varpi^{\vee}_{6}},(w^{P})^{-1}({\varpi^{\vee}_{6}})}(u_{+})=p_{8}(u_{+}^{T}),\end{array}$
(3.10)

where the expressions $q_{i}$ in Plücker coordinates were defined in (3.1).

###### Proof.

We will begin by considering the generalized minors. The action of
$(\dot{w}^{P})^{-1}$ on the highest weight vector
$v_{{\varpi^{\vee}_{j}}}^{+}$ of $V({\varpi^{\vee}_{j}})$ is straightforward
to calculate using the action of $(w^{P})^{-1}$ on the weight
${\varpi^{\vee}_{j}}$ and the following equality:

$\bar{s}_{i}\cdot v_{\mu^{\vee}}=\tfrac{1}{m!}(f^{\vee}_{i})^{m}\cdot
v_{\mu^{\vee}}\quad\text{when
$s_{i}(\mu^{\vee})=\mu^{\vee}-m\alpha^{\vee}_{i}$,}$ (3.11)

where $v_{\mu^{\vee}}\in V({\varpi^{\vee}_{j}})$ is assumed to be of weight
$\mu^{\vee}$. As the expression can get quite large, we will use the following
abbreviation:

$(f^{\vee}_{i_{1}})^{m_{1}}(f^{\vee}_{i_{2}})^{m_{2}}\cdots=f^{\vee}_{i_{1}^{m_{1}}i_{2}^{m_{2}}\cdots}$
(3.12)

These calculations yield (cf. equation (2.26)):

$\begin{array}[]{ll}(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{1}}}^{+}=f^{\vee}_{6542345613452431}\cdot
v_{{\varpi^{\vee}_{1}}}^{+},&(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}=\tfrac{1}{2}f^{\vee}_{6^{2}5423451342}\cdot
v_{{\varpi^{\vee}_{2}}}^{+},\\\ (\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{3}}}^{+}=\tfrac{1}{2^{5}}f^{\vee}_{6^{2}5^{2}4^{2}23^{2}4561^{2}345243}\cdot
v_{{\varpi^{\vee}_{3}}}^{+},&(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{4}}}^{+}=\tfrac{1}{2^{8}}\tfrac{1}{3^{2}}f^{\vee}_{6^{3}5^{3}4^{2}2^{2}3^{2}4^{2}561^{2}3^{2}4524}\cdot
v_{{\varpi^{\vee}_{4}}}^{+},\\\ (\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{5}}}^{+}=\tfrac{1}{2^{4}}f^{\vee}_{6^{2}5^{2}4^{2}2^{2}34561345}\cdot
v_{{\varpi^{\vee}_{5}}}^{+},&(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{6}}}^{+}=f^{\vee}_{65423456}\cdot
v_{{\varpi^{\vee}_{6}}}^{+}.\end{array}$

We now need to find the coefficient of $v_{{\varpi^{\vee}_{j}}}^{+}$ in the
product $u_{+}(\dot{w}^{P})^{-1}\cdot v_{{\varpi^{\vee}_{j}}}^{+}$ to
calculate the minors.

Instead of calculating these coefficients for general $u_{+}\in U_{+}^{P}$, we
will restrict to the open, dense algebraic torus $U^{\circ}_{+}\subset
U_{+}^{P}$ that is the image of $U^{\circ}_{-}$ under the transposition
isomorphism $\cdot^{T}:U_{-}^{P}\to U_{+}^{P}$. Recall from (2.15) that the
elements of $U^{\circ}_{-}$ have a specific decomposition; after transposing
these elements, we find that every $u_{+}\in U^{\circ}_{+}$ has a
decomposition of the form

$u_{+}=x^{\vee}_{r_{1}}(a_{1})x^{\vee}_{r_{2}}(a_{2})\cdots
x^{\vee}_{r_{16}}(a_{16}),\quad\text{where $a_{i}\in\mathbb{C}^{*}$.}$ (3.13)

Note that the sequence
$(r_{1},r_{2},\ldots,r_{16})=(1,3,4,2,5,4,3,1,6,5,4,3,2,4,5,6)$ is the same as
that of the fixed reduced expression for $w^{P}$ (see (2.26)), and thus the
inverse sequence compared to that of $(w^{P})^{-1}$.

Since $x^{\vee}_{i}(a)$ acts on $v\in V({\varpi^{\vee}_{j}})$ as

$x^{\vee}_{i}(a)\cdot
v=\bigl{(}1+a\,e^{\vee}_{i}+\tfrac{1}{2}a^{2}(e^{\vee}_{i})^{2}+\tfrac{1}{3!}a^{3}(e^{\vee}_{i})^{3}+\ldots\bigr{)}\cdot
v,$ (3.14)

we need to find all terms of $u_{+}$ that cancel each the factors
$f^{\vee}_{i}$ of $(\dot{w}^{P})^{-1}\cdot v_{{\varpi^{\vee}_{j}}}^{+}$.
However, Theorem 1.10 of [FZ99] tells us that the toric coordinates are
related to the generalized minors by an invertible monomial transformation; in
other words, there is exactly one term in the action of $u_{+}$ that cancels
out the $f^{\vee}_{i}$. In the following, we will argue that this term is the
one whose $e^{\vee}_{i}$ have the opposite sequence of indices (counting
doubles and triples).

First, consider $u_{+}(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{1}}}^{+}=u_{+}f^{\vee}_{6542345613452431}\cdot
v_{{\varpi^{\vee}_{1}}}^{+}$. As $(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{1}}}^{+}$ acts with 16 factors $f^{\vee}_{i}$, lowering the
weight by subtracting 16 simple roots, we need to act with 16 factors
$e^{\vee}_{i}$ to get back to a vector of the correct weight. However, since
the representation in question is minuscule, we know that $(e^{\vee}_{i})^{2}$
acts trivially on the representations for all $i\in\\{1,\ldots,6\\}$. (See
Theorem 2.10.) Thus, the decomposition of $u_{+}$ in equation (3.13) only has
one term with 16 factors acting non-trivially:

$\left(\textstyle\prod_{j=1}^{16}\displaystyle
a_{j}\right)\,e^{\vee}_{1342543165432456}.$

For $u_{+}(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{6}}}^{+}=u_{+}f^{\vee}_{65423456}\cdot
v_{{\varpi^{\vee}_{6}}}^{+}$, the computation is simplified by the Hasse
diagram in (2.25): we find $(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{6}}}^{+}=v_{8}$. (Note that in interpreting diagram (2.25)
as the weight spaces of the minuscule representation $V({\varpi^{\vee}_{6}})$,
we have put the highest weight vector $v_{{\varpi^{\vee}_{6}}}^{+}$ on the
right and denoted it $v_{0}$, which is the opposite of the way we interpreted
it for $V({\varpi^{\vee}_{1}})$.) The diagram shows that both
$e^{\vee}_{65423456}$ and $e^{\vee}_{65432456}$ map $v_{8}$ back to
$v_{0}=v_{{\varpi^{\vee}_{6}}}^{+}$. However, the decomposition of $u_{+}$
show that only the second occurs, so the only term in $u_{+}$ acting non-
trivially is

$\left(\textstyle\prod_{j=9}^{16}\displaystyle
a_{j}\right)e^{\vee}_{65432456}.$

The arguments for $u_{+}(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}=u_{+}\,\tfrac{1}{2}f^{\vee}_{6^{2}5423451342}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}$ are more involved. First, note that
$s_{i}({\varpi^{\vee}_{2}})={\varpi^{\vee}_{2}}-\delta_{i2}\alpha^{\vee}_{2}$
implies that of ${\varpi^{\vee}_{2}}-\alpha^{\vee}_{i}$ only $i=2$ is a weight
occurring in the representation $V({\varpi^{\vee}_{2}})$. Moreover, it tells
us that ${\varpi^{\vee}_{2}}-2\alpha^{\vee}_{2}$ is _not_ a weight of the
representation. Thus, the last factor in any term of $u_{+}$ acting non-
trivially must be $e^{\vee}_{2}$. There are only two indices $j$ such that
$r_{j}=2$ in the decomposition in (3.13), namely $j=4$ and $j=13$. However,
only the $j=4$ factor $x^{\vee}_{2}(a_{i})$ can contribute a factor
$e^{\vee}_{2}$ non-trivially: we will need the second factor to cancel out the
second factor $f^{\vee}_{2}$ in $(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}$. Thus, we reduced $u_{+}(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}$ to

$a_{4}\,e^{\vee}_{2}\,x^{\vee}_{r_{5}}(a_{5})x^{\vee}_{r_{6}}(a_{6})\cdots
x^{\vee}_{r_{16}}(a_{16})\,(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}.$

For the second step, we have
$s_{i}({\varpi^{\vee}_{2}}-\alpha^{\vee}_{2})={\varpi^{\vee}_{2}}-\alpha^{\vee}_{2}-\delta_{i4}\alpha^{\vee}_{4}$
for $i\neq 2$ ($s_{2}$ will raise the weight back to ${\varpi^{\vee}_{2}}$),
so that the next factor has to be $e^{\vee}_{4}$. Now, $r_{j}=4$ for
$j=3,6,11,14$, but we need $j>5$ by the above. Moreover, if we take $j=11$ or
$j=14$ we will not have sufficient factors to reach
$v_{{\varpi^{\vee}_{2}}}^{+}$ from $(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}$ (we need ten more factors). Thus, we reduced
$u_{+}(\dot{w}^{P})^{-1}\cdot v_{{\varpi^{\vee}_{2}}}^{+}$ to

$a_{4}a_{6}\,e^{\vee}_{24}\,x^{\vee}_{r_{7}}(a_{7})x^{\vee}_{r_{8}}(a_{8})\cdots
x^{\vee}_{r_{16}}(a_{16})\,(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}.$

In the third step, we have
$s_{i}({\varpi^{\vee}_{2}}-\alpha^{\vee}_{2}-\alpha^{\vee}_{4})={\varpi^{\vee}_{2}}-\alpha^{\vee}_{2}-\delta_{i3}\alpha^{\vee}_{3}-\alpha^{\vee}_{4}-\delta_{i5}\alpha^{\vee}_{5}$
for $i\neq 4$, so there are two “paths”: using an $e^{\vee}_{3}$ or an
$e^{\vee}_{5}$. For the latter path, we are looking for a $j$ with $r_{j}=5$
and $j>6$. This means that $j=10$ or $j=15$, but this would give a problem: as
$(\dot{w}^{P})^{-1}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}=\tfrac{1}{2}f^{\vee}_{6^{2}5423451342}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}$, we need at least one factor $e^{\vee}_{1}$, but
$r_{j}\neq 1$ for $j>10$. Thus, we need to take the path with $e^{\vee}_{3}$.
We find that $j=7$ gives the only possible factor $e^{\vee}_{3}$ using an
argument similar to the second step. Continuing to apply similar arguments, we
conclude that only the term

$\tfrac{1}{2}a_{4}a_{6}a_{7}a_{8}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}(a_{16})^{2}\,e^{\vee}_{24315432456^{2}}$

of $u_{+}$ gives a non-trivial contribution. One can easily see that

$\tfrac{1}{2}\,e^{\vee}_{24315432456^{2}}\cdot\tfrac{1}{2}f^{\vee}_{6^{2}5423451342}\cdot
v_{{\varpi^{\vee}_{2}}}^{+}=v_{{\varpi^{\vee}_{2}}}^{+},$

so that

$\Delta_{{\varpi^{\vee}_{2}},(w^{P})^{-1}({\varpi^{\vee}_{2}})}(u_{+})=a_{4}a_{6}a_{7}a_{8}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}^{2}.$

Analogous arguments to those used in the previous three calculations imply
that $u_{+}(\dot{w}^{P})^{-1}\cdot v_{{\varpi^{\vee}_{i}}}^{+}$ for $i=3,4,5$
only have the claimed contribution as well. Altogether, we find for $u_{+}\in
U^{\circ}_{+}$

$\displaystyle\Delta_{{\varpi^{\vee}_{1}},(w^{P})^{-1}({\varpi^{\vee}_{1}})}(u_{+})$
$\displaystyle=a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16},$
$\displaystyle\Delta_{{\varpi^{\vee}_{2}},(w^{P})^{-1}({\varpi^{\vee}_{2}})}(u_{+})$
$\displaystyle=a_{4}a_{6}a_{7}a_{8}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}^{2},$
$\displaystyle\Delta_{{\varpi^{\vee}_{3}},(w^{P})^{-1}({\varpi^{\vee}_{3}})}(u_{+})$
$\displaystyle=a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}^{2}a_{9}a_{10}a_{11}a_{12}^{2}a_{13}a_{14}^{2}a_{15}^{2}a_{16}^{2},$
$\displaystyle\Delta_{{\varpi^{\vee}_{4}},(w^{P})^{-1}({\varpi^{\vee}_{4}})}(u_{+})$
$\displaystyle=a_{3}a_{4}a_{5}a_{6}a_{7}^{2}a_{8}^{2}a_{9}a_{10}a_{11}^{2}a_{12}^{2}a_{13}^{2}a_{14}^{2}a_{15}^{3}a_{16}^{3},$
$\displaystyle\Delta_{{\varpi^{\vee}_{5}},(w^{P})^{-1}({\varpi^{\vee}_{5}})}(u_{+})$
$\displaystyle=a_{5}a_{6}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}^{2}a_{14}^{2}a_{15}^{2}a_{16}^{2},$
$\displaystyle\Delta_{{\varpi^{\vee}_{6}},(w^{P})^{-1}({\varpi^{\vee}_{6}})}(u_{+})$
$\displaystyle=a_{9}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}.$

To prove the equalities in (3.10), the next step is to evaluate the Plücker
coordinate expressions (defined in equation (3.1)) on the transpose of
$u_{+}\in U^{\circ}_{+}$. We will find that they have the same monomial
expressions in the toric coordinates $\\{a_{i}\\}$ of $U^{\circ}_{+}$. Recall
from Section 2.3 that the generalized Plücker coordinates are defined on the
fundamental weight representation $V({\varpi^{\vee}_{1}})$, which is
minuscule. As we saw in the above, this simplifies calculations considerably.

Recall from equation (2.17) that $p_{i}(g)=v_{0}^{*}(g\cdot v_{i})$, where the
$v_{i}$ have been defined using diagram (2.25). Thus, we need to find the
terms of $u_{+}^{T}\cdot v_{i}$ that are multiples of $v_{0}$. As
$V({\varpi^{\vee}_{1}})$ is minuscule, we know that each $y^{\vee}_{i}(a_{j})$
in the decomposition of $u_{+}^{T}\in U^{\circ}_{-}$ (see (2.15)) acts as
$1+a_{j}f^{\vee}_{i}$ on the representation. Now, every path from $v_{i}$ to
$v_{0}$ in diagram (2.25) gives a sequence $(b_{1},\ldots,b_{i})$ of indices
such that $\bar{s}_{b_{1}}\cdots\bar{s}_{b_{i}}\cdot v_{i}=v_{0}$. By
Corollary 2.11, we find that each $\bar{s}_{b_{j}}$ acts as
$f^{\vee}_{b_{j}}$, so that $f^{\vee}_{b_{1}}\cdots f^{\vee}_{b_{i}}\cdot
v_{i}=v_{0}$. Thus, for each sequence of indices $(b_{1},\ldots,b_{i})$
obtained from a path from $v_{i}$ to $v_{0}$, the term $f^{\vee}_{b_{1}}\cdots
f^{\vee}_{b_{i}}$ in $u_{+}^{T}$ will contribute its coefficient to
$p_{i}^{(j)}(u_{+}^{T})$.

To find these coefficients, apply the following algorithm: determine all the
paths $(b_{1},\ldots,b_{i})$ from $v_{i}$ to $v_{0}$ in diagram (2.25); find
all the subexpressions of the form $s_{b_{1}}\cdots s_{b_{i}}$ in
$(w^{P})^{-1}$; for each of the subexpressions, take the respective
coefficients in the decomposition of $u_{+}^{T}\in U^{\circ}_{-}$. Note that
this algorithm is reminiscent of the methods in [Spa21]; we can use the result
in Proposition 8.11 (applied to $(w^{P})^{-1}$; in other words flipping all
the arrows) there to obtain these subexpressions combinatorially.

Let us illustrate with $p_{7}^{\prime}(u_{+}^{T})$ for $u_{+}^{T}\in
U^{\circ}_{-}$. Using diagram (2.25) we find that there are two paths:
$(6,5,4,2,3,4,5)$ and $(6,5,4,3,2,4,5)$. The first gives the following
subexpressions:

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{\boldsymbol{s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}}}}s_{6}s_{1}s_{3}s_{4}s_{5}s_{2}s_{4}s_{3}s_{1},\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{\boldsymbol{s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}}}}s_{5}s_{6}s_{1}s_{3}s_{4}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{\boldsymbol{s_{5}}}}s_{2}s_{4}s_{3}s_{1},$
$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{\boldsymbol{s_{6}s_{5}s_{4}s_{2}s_{3}}}}s_{4}s_{5}s_{6}s_{1}s_{3}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{\boldsymbol{s_{4}s_{5}}}}s_{2}s_{4}s_{3}s_{1},\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{\boldsymbol{s_{6}s_{5}s_{4}s_{2}}}}s_{3}s_{4}s_{5}s_{6}s_{1}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\underline{\boldsymbol{s_{3}s_{4}s_{5}}}}s_{2}s_{4}s_{3}s_{1},$

The second does not give any subexpressions in $(w^{P})^{-1}$. We now compare
it with the decomposition of $u_{+}^{T}$ as

$\displaystyle u_{+}^{T}$
$\displaystyle=y^{\vee}_{6}(a_{16})y^{\vee}_{5}(a_{15})y^{\vee}_{4}(a_{14})y^{\vee}_{2}(a_{13})y^{\vee}_{3}(a_{12})y^{\vee}_{4}(a_{11})y^{\vee}_{5}(a_{10})y^{\vee}_{6}(a_{9})$
$\displaystyle\qquad\cdot
y^{\vee}_{1}(a_{8})y^{\vee}_{3}(a_{7})y^{\vee}_{4}(a_{6})y^{\vee}_{5}(a_{5})y^{\vee}_{2}(a_{4})y^{\vee}_{4}(a_{3})y^{\vee}_{3}(a_{2})y^{\vee}_{1}(a_{1}).$

to find the contributing terms (in order of the subexpressions):

$\displaystyle p_{7}^{\prime}(u_{+}^{T})$
$\displaystyle=a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}+a_{5}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}$
$\displaystyle\qquad+a_{5}a_{6}a_{12}a_{13}a_{14}a_{15}a_{16}+a_{5}a_{6}a_{7}a_{13}a_{14}a_{15}a_{16}$
$\displaystyle=\bigl{(}(a_{5}+a_{10})a_{11}a_{12}+a_{5}a_{6}(a_{7}+a_{12})\bigr{)}a_{13}a_{14}a_{15}a_{16}.$

Once we obtained all toric expressions for the Plücker coordinates,
calculating the expressions of equation (3.1) is straightforward. We find the
following for $u_{+}\in U^{\circ}_{+}$

$\displaystyle p_{8}(u_{+}^{T})$
$\displaystyle=a_{9}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16},$
$\displaystyle p_{16}(u_{+}^{T})$
$\displaystyle=a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16},$
$\displaystyle q_{12}(u_{+}^{T})$
$\displaystyle=a_{4}a_{6}a_{7}a_{8}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}^{2},$
$\displaystyle q_{16}(u_{+}^{T})$
$\displaystyle=a_{5}a_{6}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}^{2}a_{14}^{2}a_{15}^{2}a_{16}^{2},$
$\displaystyle q_{20}(u_{+}^{T})$
$\displaystyle=a_{2}a_{3}a_{4}a_{5}a_{6}a_{7}a_{8}^{2}a_{9}a_{10}a_{11}a_{12}^{2}a_{13}a_{14}^{2}a_{15}^{2}a_{16}^{2},$
$\displaystyle q_{24}(u_{+}^{T})$
$\displaystyle=a_{3}a_{4}a_{5}a_{6}a_{7}^{2}a_{8}^{2}a_{9}a_{10}a_{11}^{2}a_{12}^{2}a_{13}^{2}a_{14}^{2}a_{15}^{3}a_{16}^{3}.$

Comparing these with the expressions for the minors, we conclude that the
equalities in (3.10) hold, at least for $u_{+}\in U^{\circ}_{+}$. However,
$U^{\circ}_{+}\subset U_{+}^{P}$ is open and dense (this follows from
$U^{\circ}_{-}\subset U_{-}^{P}$ as discussed in Section 2.2) and $U_{+}^{P}$
is irreducible, so these equalities extend to all of $U_{-}^{P}$. ∎

We will need to calculate local expressions for generalized minors on the
torus $U^{\circ}_{+}$ again for the superpotential of the Freudenthal variety,
as well for the cluster structure of the mirrors of both varieties, so we will
summarize the algorithm performed in the proof above of Lemma 3.10, for ease
of reference. We will need only generalized minors of the form

$\Delta_{{\varpi^{\vee}_{i}},w({\varpi^{\vee}_{i}})}(u_{+})=\Bigl{\langle}u_{+}\bar{w}\cdot
v_{{\varpi^{\vee}_{i}}}^{+},v_{{\varpi^{\vee}_{i}}}^{+}\Bigr{\rangle},$

where $w\in W$, though the general form is entirely analogous.

######  Algorithm 3.11.

  1. (1)

In the weight space for the irreducible representation
$V({\varpi^{\vee}_{i}})$ with highest weight ${\varpi^{\vee}_{i}}$, find the
weight $w({\varpi^{\vee}_{i}})$ by acting on ${\varpi^{\vee}_{i}}$ with the
Weyl group element $w=s_{i_{1}}\dots s_{i_{k}}$. As shown in the proof above,
this corresponds to acting on $v_{{\varpi^{\vee}_{i}}}^{+}$ with the element
$\frac{1}{m_{1}!}\dots\frac{1}{m_{k}!}f^{\vee}_{i_{1}^{m_{1}}\dots
i_{k}^{m_{k}}}$.

  2. (2)

In order to find the coefficient of $v_{{\varpi^{\vee}_{i}}}^{+}$ in
$u_{+}\bar{w}\cdot v_{{\varpi^{\vee}_{i}}}^{+}$, we must find those terms in
$u_{+}$ which act on $\bar{w}\cdot v_{{\varpi^{\vee}_{i}}}^{+}$ by sending it
to a multiple of $v_{{\varpi^{\vee}_{i}}}^{+}$. To do this, first find all
sequences $(i_{1},\dots,i_{k})$ such that $e^{\vee}_{i_{1}\dots
i_{k}}(\bar{w}\cdot v_{{\varpi^{\vee}_{i}}}^{+})$ is a multiple of
$v_{{\varpi^{\vee}_{i}}}^{+}$. (We refer to these sequences as _paths_ in the
weight space of $V({\varpi^{\vee}_{i}})$ from $w({\varpi^{\vee}_{i}})$ to
${\varpi^{\vee}_{i}}$.)

  3. (3)

For each such $e^{\vee}_{i_{1},\dots,i_{k}}$, determine its coefficient in
$u_{+}$. One way to do this is to expand $u_{+}$ as a product of the
$x^{\vee}_{r_{i}}(a_{i})=(1+a_{i}e^{\vee}_{r_{i}}+\dots)$ and to collect terms
by the index sequence of $e^{\vee}_{i_{1}\dots i_{k}}$.

  4. (4)

The coefficient of each such term of $u_{+}$ above gives a polynomial in the
torus coordinates, and the sum of all these is the local expression of
$\Delta_{{\varpi^{\vee}_{i}},w({\varpi^{\vee}_{i}})}(u_{+})$ for $u_{+}\in
U^{\circ}_{+}$.

While calculating the superpotential of the Freudenthal variety and the
cluster structure for its mirror, we found that the algorithm is performed
much more efficiently by a computer algebra program when steps (2) and (3) are
performed simultaneously: to be more precise, while searching for paths from
$w({\varpi^{\vee}_{i}})$ to ${\varpi^{\vee}_{i}}$ as in step (2), we discard a
partial sequence that does not appear in the expansion of $u_{+}$.

In Appendix A, we give an example of performing this algorithm for the
Freudenthal variety.

Note that the calculation of Plücker coordinates in the proof of Lemma 3.10
above is actually a simplification (and translation in terms of combinatorial
objects) of the same algorithm. This simplification is mostly due to the fact
that Pücker coordinates are defined on _minuscule_ highest weight
representations. Since the algorithm works for the Plücker coordinates of any
minuscule homogeneous space, we will summarize it as well: