######  Algorithm 3.12.

  1. (1)

For a given Plücker coordinate $p_{i}$ on a _minuscule_ homogeneous space
$P^{\vee}\backslash G^{\vee}$, find the corresponding vertex $v_{i}$ in the
weight diagram of the corresponding minuscule representation.

  2. (2)

Determine all the paths $(b_{1},\ldots,b_{i})$ in the weight diagram from
$v_{i}$ to $v_{0}$, the lowest weight vector.

  3. (3)

Find all the subexpressions of the form $s_{b_{1}}\cdots s_{b_{i}}$ in
$(w^{P})^{-1}$, using Proposition 8.11 of [Spa21] (replacing $w^{\prime}$ by
$s_{b_{1}}\cdots s_{b_{i}}$ and $w^{P}$ by $(w^{P})^{-1}$).

  4. (4)

For each of the subexpressions, take the respective coefficients in the
decomposition of $u_{+}^{T}\in U^{\circ}_{-}$, noting that at most linear
contributions occur.

Recalling that we have written $p_{i}^{T}:u_{+}\mapsto p_{i}(u_{+}^{T})$ for
$i=8,16$ and $q_{i}^{T}:u_{+}\mapsto q_{i}(u_{+}^{T})$ for $i=12,16,20,24$, we
conclude the following by combining Lemmas 3.9 and 3.10 with Proposition 2.14
giving the expression for the coordinate ring of $U_{+}^{P}$:

######  Corollary 3.13.

The coordinate ring of the affine variety $U_{+}^{P}$ is given by

$\mathbb{C}[U_{+}^{P}]=\mathbb{C}[p_{\alpha^{\vee}}^{T}\,|\,\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}][(p_{8}^{T})^{-1},(p_{16}^{T})^{-1},(q_{12}^{T})^{-1},(q_{16}^{T})^{-1},(q_{20}^{T})^{-1},(q_{24}^{T})^{-1}].$

This implies that the coordinate ring of the affine variety
$U_{-}^{P}=(U_{+}^{P})^{T}$ is given by

$\mathbb{C}[U_{-}^{P}]=\mathbb{C}[p_{\alpha^{\vee}}\,|\,\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}][p_{8}^{-1},p_{16}^{-1},q_{12}^{-1},q_{16}^{-1},q_{20}^{-1},q_{24}^{-1}].$

This in turn implies that $\pi(U_{-}^{P})\subseteq X^{\vee}_{\mathrm{can}}$.

Recall that the generating Plücker coordinates are the ones appearing in Lemma
3.9. and that the (generalized) Plücker coordinates are defined in equation
(2.17). The quadratic and cubic expressions $q_{i}$ are given in equation
(3.1).

######  Remark 3.14.

Note that $p_{8}$, $p_{16}$, $q_{12}$, $q_{16}$, $q_{20}$ and $q_{24}$ are not
a priori generated by the generators of the coordinate ring. Recall from
Corollary 3.13 that the coordinate ring is generated by the Plücker
coordinates $p_{\alpha^{\vee}}$ for $\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}$, i.e.

$p_{1},p_{2},p_{3},p_{4}^{\prime},p_{4}^{\prime\prime},p_{5}^{\prime},p_{5}^{\prime\prime},p_{6}^{\prime},p_{6}^{\prime\prime},p_{7}^{\prime},p_{7}^{\prime\prime},p_{8}^{\prime},p_{8}^{\prime\prime},p_{9}^{\prime\prime},p_{10}^{\prime\prime},p_{11}^{\prime\prime}.$

(Note that $p_{0}\equiv 1$ on $U^{\vee}_{-}$.) However, it turns out that
there are “Plücker relations” expressing the remaining Plücker coordinates in
terms of these. We have found these using the toric expressions for the
Plücker coordinates restricted to $U^{\circ}_{-}$; as $U^{\circ}_{-}\subset
U_{-}^{P}$ is dense and $U_{-}^{P}$ is irreducible, this suffices. The
relations are given as follows:

$\begin{array}[]{lcl}p_{8}=p_{1}p_{7}^{\prime}-p_{2}p_{6}^{\prime}+p_{3}p_{5}^{\prime}-p_{4}^{\prime}p_{4}^{\prime\prime},&&p_{9}^{\prime}=p_{1}p_{8}^{\prime}-p_{2}p_{7}^{\prime\prime}+p_{3}p_{6}^{\prime\prime}-p_{4}^{\prime}p_{5}^{\prime\prime},\\\
p_{10}^{\prime}=p_{1}p_{9}^{\prime\prime}-p_{2}p_{8}^{\prime\prime}+p_{4}^{\prime\prime}p_{6}^{\prime\prime}-p_{5}^{\prime}p_{5}^{\prime\prime},&&p_{11}^{\prime}=p_{1}p_{10}^{\prime\prime}-p_{3}p_{8}^{\prime\prime}+p_{4}^{\prime\prime}p_{7}^{\prime\prime}-p_{5}^{\prime\prime}p_{6}^{\prime},\\\
p_{12}^{\prime}=p_{2}p_{10}^{\prime\prime}-p_{3}p_{9}^{\prime\prime}+p_{4}^{\prime\prime}p_{8}^{\prime}-p_{5}^{\prime\prime}p_{7}^{\prime},&&p_{12}^{\prime\prime}=p_{1}p_{11}^{\prime\prime}-p_{4}^{\prime}p_{8}^{\prime\prime}+p_{5}^{\prime}p_{7}^{\prime\prime}-p_{6}^{\prime}p_{6}^{\prime\prime},\\\
p_{13}=p_{2}p_{11}^{\prime\prime}-p_{4}^{\prime}p_{9}^{\prime\prime}+p_{5}^{\prime}p_{8}^{\prime}-p_{6}^{\prime\prime}p_{7}^{\prime},&&p_{14}=p_{3}p_{11}^{\prime\prime}-p_{4}^{\prime}p_{10}^{\prime\prime}+p_{6}^{\prime}p_{8}^{\prime}-p_{7}^{\prime}p_{7}^{\prime\prime},\\\
p_{15}=p_{4}^{\prime\prime}p_{11}^{\prime\prime}-p_{5}^{\prime}p_{10}^{\prime\prime}+p_{6}^{\prime}p_{9}^{\prime\prime}-p_{7}^{\prime}p_{8}^{\prime\prime},&&p_{16}=p_{5}^{\prime\prime}p_{11}^{\prime\prime}-p_{6}^{\prime\prime}p_{10}^{\prime\prime}+p_{7}^{\prime\prime}p_{9}^{\prime\prime}-p_{8}^{\prime}p_{8}^{\prime\prime}.\end{array}$

Thus, the localized expressions are in fact expressible in terms of the
generators.

###  3.5. The coordinate ring of $U_{-}^{P}$ for the Freudenthal variety

The approach for the Freudenthal variety $E_{7}^{\mathrm{sc}}/P_{7}$ is
analogous to the Cayley plane. Recall the from Section 2.4 that
$\ell=\ell(w^{P})=27$. Thus, Proposition 2.14 (see Proposition 8.5 of [GLS11])
tells us that the coordinate ring of $U_{+}^{P}$ is generated by the dual PBW
elements $p^{\text{GLS}}_{m}$, $m\in\\{1,\ldots,27\\}$, and localized at the
generalized minors
$\Delta_{{\varpi^{\vee}_{j}},(w^{P})^{-1}({\varpi^{\vee}_{j}})}$,
$j\in\\{1,\ldots,7\\}$. The generators are expressed as follows:

######  Lemma 3.15.

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

$1$$p_{0}$$p^{\text{GLS}}_{1}$$p_{1}$$p^{\text{GLS}}_{2}$$p_{2}$$p^{\text{GLS}}_{3}$$p_{3}$$p^{\text{GLS}}_{4}$$p_{4}$$p^{\text{GLS}}_{6}$$p_{5}^{\prime}$$p^{\text{GLS}}_{5}$$p_{5}^{\prime\prime}$$p^{\text{GLS}}_{7}$$p_{6}^{\prime}$$p^{\text{GLS}}_{10}$$p_{6}^{\prime\prime}$$p^{\text{GLS}}_{8}$$p_{7}^{\prime}$$p^{\text{GLS}}_{11}$$p_{7}^{\prime\prime}$$p^{\text{GLS}}_{9}$$p_{8}^{\prime}$$p^{\text{GLS}}_{12}$$p_{8}^{\prime\prime}$$p^{\text{GLS}}_{13}$$p_{9}$$p^{\text{GLS}}_{14}$$p_{9}^{\prime}$$p^{\text{GLS}}_{15}$$p_{9}^{\prime\prime}$$p^{\text{GLS}}_{19}$$p_{10}^{\prime}$$p^{\text{GLS}}_{16}$$p_{10}^{\prime\prime}$$p^{\text{GLS}}_{20}$$p_{11}^{\prime}$$p^{\text{GLS}}_{17}$$p_{11}^{\prime\prime}$$p^{\text{GLS}}_{21}$$p_{12}^{\prime}$$p^{\text{GLS}}_{18}$$p_{12}^{\prime\prime}$$p^{\text{GLS}}_{22}$$p_{13}^{\prime}$$p^{\text{GLS}}_{23}$$p_{13}^{\prime\prime}$$p^{\text{GLS}}_{24}$$p_{14}$$p^{\text{GLS}}_{25}$$p_{15}$$p^{\text{GLS}}_{26}$$p_{16}$$p^{\text{GLS}}_{27}$$p_{17}$
(3.15)

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

Note that the diagram of equation (3.15) is a subdiagram of the one in
equation (2.27). Upon removing the rightmost vertex, this subdiagram is
isomorphic to the Hasse diagram of the Cayley plane $\mathbb{OP}^{2}$ (see
equation (2.25)).

###### Proof.

This is proven by computing the weight of each of the basis vectors
$v_{i}^{(j)}$ and the positive roots ${\beta^{\vee}_{P}}(m)$ that are mapped
to negative roots by $w^{P}$ ($m\in\\{1,\ldots,27\\}$), and using Proposition
3.8. Compare Lemma 3.9. ∎

######  Lemma 3.16.

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

$\begin{array}[]{lll}\Delta_{{\varpi^{\vee}_{1}},(w^{P})^{-1}({\varpi^{\vee}_{1}})}(u_{+})=q_{18}(u_{+}^{T}),&\Delta_{{\varpi^{\vee}_{2}},(w^{P})^{-1}({\varpi^{\vee}_{2}})}(u_{+})=q_{27}(u_{+}^{T}),\\\
\Delta_{{\varpi^{\vee}_{3}},(w^{P})^{-1}({\varpi^{\vee}_{3}})}(u_{+})=q_{36}^{\prime}(u_{+}^{T}),&\Delta_{{\varpi^{\vee}_{4}},(w^{P})^{-1}({\varpi^{\vee}_{4}})}(u_{+})=q_{54}(u_{+}^{T}),\\\
\Delta_{{\varpi^{\vee}_{5}},(w^{P})^{-1}({\varpi^{\vee}_{5}})}(u_{+})=q_{45}(u_{+}^{T}),&\Delta_{{\varpi^{\vee}_{6}},(w^{P})^{-1}({\varpi^{\vee}_{6}})}(u_{+})=q_{36}(u_{+}^{T}),\\\
\Delta_{{\varpi^{\vee}_{7}},(w^{P})^{-1}({\varpi^{\vee}_{7}})}(u_{+})=p_{27}(u_{+}^{T}),\end{array}$
(3.16)

where the expressions $q_{i}$ in Plücker coordinates were defined in (3.4).

###### Proof.

The arguments are analogous to the proof of Lemma 3.10. Restricting to
$u_{+}\in
U^{\circ}_{+}=\\{x^{\vee}_{r_{1}}(a_{1})x^{\vee}_{r_{2}}(a_{2})\cdots
x^{\vee}_{r_{27}}(a_{27})~{}|~{}a_{i}\in\mathbb{C}^{*}\\}$, where the sequence
of indices equals
$(r_{1},r_{2},\ldots,r_{27})=(7,6,5,4,3,2,4,5,6,1,3,4,2,5,7,4,3,1,6,5,4,2,3,4,5,6,7)$,
we find using Algorithm 3.11

$\displaystyle\Delta_{{\varpi^{\vee}_{1}},(w^{P})^{-1}({\varpi^{\vee}_{1}})}(u_{+})$
$\displaystyle=\frac{a_{27}}{a_{15}}\bigl{(}\textstyle\prod_{i=10}^{27}\displaystyle
a_{i}\bigr{)},$
$\displaystyle\Delta_{{\varpi^{\vee}_{2}},(w^{P})^{-1}({\varpi^{\vee}_{2}})}(u_{+})$
$\displaystyle=\frac{a_{18}a_{23}a_{24}a_{25}a_{26}a_{27}}{a_{10}}\bigl{(}\textstyle\prod_{i=6}^{27}\displaystyle
a_{i}\bigr{)},$
$\displaystyle\Delta_{{\varpi^{\vee}_{3}},(w^{P})^{-1}({\varpi^{\vee}_{3}})}(u_{+})$
$\displaystyle=\frac{a_{26}a_{27}}{a_{6}a_{14}a_{15}a_{19}}\bigl{(}\textstyle\prod_{i=13}^{27}\displaystyle
a_{i}\bigr{)}\bigl{(}\textstyle\prod_{j=5}^{27}\displaystyle a_{j}\bigr{)},$
$\displaystyle\Delta_{{\varpi^{\vee}_{4}},(w^{P})^{-1}({\varpi^{\vee}_{4}})}(u_{+})$
$\displaystyle=\frac{a_{25}a_{26}a_{27}}{a_{10}a_{11}a_{19}a_{20}}\bigl{(}\textstyle\prod_{i_{1}=17}^{27}\displaystyle
a_{i_{1}}\bigr{)}\bigl{(}\textstyle\prod_{i_{2}=8}^{27}\displaystyle
a_{i_{2}}\bigr{)}\bigl{(}\textstyle\prod_{i_{3}=4}^{27}\displaystyle
a_{i_{3}}\bigr{)},$
$\displaystyle\Delta_{{\varpi^{\vee}_{5}},(w^{P})^{-1}({\varpi^{\vee}_{5}})}(u_{+})$
$\displaystyle=a_{9}a_{22}a_{24}a_{25}a_{26}a_{27}\bigl{(}\textstyle\prod_{i_{1}=14}^{27}\displaystyle
a_{i_{1}}\bigr{)}\bigl{(}\textstyle\prod_{i_{2}=3}^{27}\displaystyle
a_{i_{2}}\bigr{)},$
$\displaystyle\Delta_{{\varpi^{\vee}_{6}},(w^{P})^{-1}({\varpi^{\vee}_{6}})}(u_{+})$
$\displaystyle=a_{15}\bigl{(}\textstyle\prod_{i_{1}=19}^{27}\displaystyle
a_{i_{1}}\bigr{)}\bigl{(}\textstyle\prod_{i_{2}=2}^{27}\displaystyle
a_{i_{2}}\bigr{)},$
$\displaystyle\Delta_{{\varpi^{\vee}_{7}},(w^{P})^{-1}({\varpi^{\vee}_{7}})}(u_{+})$
$\displaystyle=\textstyle\prod_{i=1}^{27}\displaystyle a_{i}.$

Using Algorithm 3.12 to determine expressions in toric coordinates for the
homogeneous polynomials in Plücker coordinates, we find that the expressions
coincide with those of the generalized minors above, so that we obtain the
equalities in (3.16) on $U^{\circ}_{+}$, and therefore on $U_{+}^{P}$ as well.
∎

Using the same notation as in (3.9), Lemmas 3.15, 3.16, and Proposition 2.14
imply:

######  Corollary 3.17.

The coordinate ring of the affine variety $U_{+}^{P}$ is given by:

$\mathbb{C}[U_{+}^{P}]=\mathbb{C}[p_{\alpha^{\vee}}^{T}\,|\,\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}][(p_{27}^{T})^{-1},(q_{18}^{T})^{-1},(q_{27}^{T})^{-1},(q_{36}^{T})^{-1},(q_{36}^{\prime
T})^{-1},(q_{45}^{T})^{-1},(q_{54}^{T})^{-1}].$

This implies that the coordinate ring of the affine variety
$U_{-}^{P}=(U_{+}^{P})^{T})$ is given by

$\mathbb{C}[U_{-}^{P}]=\mathbb{C}[p_{\alpha^{\vee}}\,|\,\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}][p_{27}^{-1},q_{18}^{-1},q_{27}^{-1},q_{36}^{-1},(q_{36}^{\prime})^{-1},q_{45}^{-1},q_{54}^{-1}].$

Consequently, $\pi(U_{-}^{P})\subseteq X^{\vee}_{\mathrm{can}}$.

Recall that the (generalized) Plücker coordinates $p_{i}^{(j)}$ are defined in
equation (2.17) and the quadratic, cubic and quartic expressions $q_{i}$ are
given in equation (3.4).

######  Remark 3.18.

Just as in the Remark 3.14, the localized expressions $p_{27}$, $q_{18}$,
$q_{36}$, $(q_{36}^{\prime})$, $q_{45}$ and $q_{54}$ are not a priori
generated by the generators of the coordinate ring. Recall that the generators
were $p_{\alpha^{\vee}}$ for $\alpha^{\vee}\in\Pi^{\vee}_{w^{P}}$, i.e.

$\displaystyle
p_{1},p_{2},p_{3},p_{4},p_{5}^{\prime},p_{5}^{\prime\prime},p_{6}^{\prime},p_{6}^{\prime\prime},p_{7}^{\prime},p_{7}^{\prime\prime},p_{8}^{\prime},p_{8}^{\prime\prime},p_{9},p_{9}^{\prime},p_{9}^{\prime\prime},p_{10}^{\prime},p_{10}^{\prime\prime},$
$\displaystyle
p_{11}^{\prime},p_{11}^{\prime\prime},p_{12}^{\prime},p_{12}^{\prime\prime},p_{13}^{\prime},p_{13}^{\prime\prime},p_{14},p_{15},p_{16},p_{17}.$

The remaining Plücker coordinates can be expressed using the following
“Plücker relations”:

$\displaystyle p_{10}$
$\displaystyle=p_{1}p_{9}-p_{2}p_{8}^{\prime}+p_{3}p_{7}^{\prime}-p_{4}p_{6}^{\prime}+p_{5}^{\prime}p_{5}^{\prime\prime},$
$\displaystyle p_{11}$
$\displaystyle=p_{1}p_{10}^{\prime}-p_{2}p_{9}^{\prime}+p_{3}p_{8}^{\prime\prime}-p_{4}p_{7}^{\prime\prime}+p_{5}^{\prime}p_{6}^{\prime\prime},$
$\displaystyle p_{12}$
$\displaystyle=p_{1}p_{11}^{\prime}-p_{2}p_{10}^{\prime\prime}+p_{3}p_{9}^{\prime\prime}-p_{5}^{\prime\prime}p_{7}^{\prime\prime}+p_{6}^{\prime}p_{6}^{\prime\prime},$
$\displaystyle p_{13}$
$\displaystyle=p_{1}p_{12}^{\prime}-p_{2}p_{11}^{\prime\prime}+p_{4}p_{9}^{\prime\prime}-p_{5}^{\prime\prime}p_{8}^{\prime\prime}+p_{6}^{\prime\prime}p_{7}^{\prime},$
$\displaystyle p_{14}^{\prime}$
$\displaystyle=p_{1}p_{13}^{\prime\prime}-p_{2}p_{12}^{\prime\prime}+p_{5}^{\prime}p_{9}^{\prime\prime}-p_{6}^{\prime}p_{8}^{\prime\prime}+p_{7}^{\prime}p_{7}^{\prime\prime},$
$\displaystyle p_{14}^{\prime\prime}$
$\displaystyle=p_{1}p_{13}^{\prime}-p_{3}p_{11}^{\prime\prime}+p_{4}p_{10}^{\prime\prime}-p_{5}^{\prime\prime}p_{9}^{\prime}+p_{6}^{\prime\prime}p_{8}^{\prime},$
$\displaystyle p_{15}^{\prime}$
$\displaystyle=p_{1}p_{14}-p_{3}p_{12}^{\prime\prime}+p_{5}^{\prime}p_{10}^{\prime\prime}-p_{6}^{\prime}p_{9}^{\prime}+p_{7}^{\prime\prime}p_{8}^{\prime},$
$\displaystyle p_{15}^{\prime\prime}$
$\displaystyle=p_{2}p_{13}^{\prime}-p_{3}p_{12}^{\prime}+p_{4}p_{11}^{\prime}-p_{5}^{\prime\prime}p_{10}^{\prime}+p_{6}^{\prime\prime}p_{9},$
$\displaystyle p_{16}^{\prime}$
$\displaystyle=p_{1}p_{15}-p_{4}p_{12}^{\prime\prime}+p_{5}^{\prime}p_{11}^{\prime\prime}-p_{7}^{\prime}p_{9}^{\prime}+p_{8}^{\prime}p_{8}^{\prime\prime},$
$\displaystyle p_{16}^{\prime\prime}$
$\displaystyle=p_{2}p_{14}-p_{3}p_{13}^{\prime\prime}+p_{5}^{\prime}p_{11}^{\prime}-p_{6}^{\prime}p_{10}^{\prime}+p_{7}^{\prime\prime}p_{9}$
$\displaystyle p_{17}^{\prime}$
$\displaystyle=p_{1}p_{16}-p_{5}^{\prime\prime}p_{12}^{\prime\prime}+p_{6}^{\prime}p_{11}^{\prime\prime}-p_{7}^{\prime}p_{10}^{\prime\prime}+p_{8}^{\prime}p_{9}^{\prime\prime},$
$\displaystyle p_{17}^{\prime\prime}$
$\displaystyle=p_{2}p_{15}-p_{4}p_{13}^{\prime\prime}+p_{5}^{\prime}p_{12}^{\prime}-p_{7}^{\prime}p_{10}^{\prime}+p_{8}^{\prime\prime}p_{9},$
$\displaystyle p_{18}$
$\displaystyle=p_{1}p_{17}-p_{6}^{\prime\prime}p_{12}^{\prime\prime}+p_{7}^{\prime\prime}p_{11}^{\prime\prime}-p_{8}^{\prime\prime}p_{10}^{\prime\prime}+p_{9}^{\prime}p_{9}^{\prime\prime},$
$\displaystyle p_{18}^{\prime}$
$\displaystyle=p_{2}p_{16}-p_{5}^{\prime\prime}p_{13}^{\prime\prime}+p_{6}^{\prime}p_{12}^{\prime}-p_{7}^{\prime}p_{11}^{\prime}+p_{9}p_{9}^{\prime\prime},$
$\displaystyle p_{18}^{\prime\prime}$
$\displaystyle=p_{3}p_{15}-p_{4}p_{14}+p_{5}^{\prime}p_{13}^{\prime}-p_{8}^{\prime}p_{10}^{\prime}+p_{9}p_{9}^{\prime},$
$\displaystyle p_{19}^{\prime}$
$\displaystyle=p_{2}p_{17}-p_{6}^{\prime\prime}p_{13}^{\prime\prime}+p_{7}^{\prime\prime}p_{12}^{\prime}-p_{8}^{\prime\prime}p_{11}^{\prime}+p_{9}^{\prime\prime}p_{10}^{\prime},$
$\displaystyle p_{19}^{\prime\prime}$
$\displaystyle=p_{3}p_{16}-p_{5}^{\prime\prime}p_{14}+p_{6}^{\prime}p_{13}^{\prime}-p_{8}^{\prime}p_{11}^{\prime}+p_{9}p_{10}^{\prime\prime},$
$\displaystyle p_{20}^{\prime}$
$\displaystyle=p_{3}p_{17}-p_{6}^{\prime\prime}p_{14}+p_{7}^{\prime\prime}p_{13}^{\prime}-p_{9}^{\prime}p_{11}^{\prime}+p_{10}^{\prime}p_{10}^{\prime\prime},$
$\displaystyle p_{20}^{\prime\prime}$
$\displaystyle=p_{4}p_{16}-p_{5}^{\prime\prime}p_{15}+p_{7}^{\prime}p_{13}^{\prime}-p_{8}^{\prime}p_{12}^{\prime}+p_{9}p_{11}^{\prime\prime},$
$\displaystyle p_{21}^{\prime}$
$\displaystyle=p_{4}p_{17}-p_{6}^{\prime\prime}p_{15}+p_{8}^{\prime\prime}p_{13}^{\prime}-p_{9}^{\prime}p_{12}^{\prime}+p_{10}^{\prime}p_{11}^{\prime\prime},$
$\displaystyle p_{21}^{\prime\prime}$
$\displaystyle=p_{5}^{\prime}p_{16}-p_{6}^{\prime}p_{15}+p_{7}^{\prime}p_{14}-p_{8}^{\prime}p_{13}^{\prime\prime}+p_{9}p_{12}^{\prime\prime},$
$\displaystyle p_{22}^{\prime}$
$\displaystyle=p_{5}^{\prime\prime}p_{17}-p_{6}^{\prime\prime}p_{16}+p_{9}^{\prime\prime}p_{13}^{\prime}-p_{10}^{\prime\prime}p_{12}^{\prime}+p_{11}^{\prime}p_{11}^{\prime\prime},$
$\displaystyle p_{22}^{\prime\prime}$
$\displaystyle=p_{5}^{\prime}p_{17}-p_{7}^{\prime\prime}p_{15}+p_{8}^{\prime\prime}p_{14}-p_{9}^{\prime}p_{13}^{\prime\prime}+p_{10}^{\prime}p_{12}^{\prime\prime},$
$\displaystyle p_{23}$
$\displaystyle=p_{6}^{\prime}p_{17}-p_{7}^{\prime\prime}p_{16}+p_{9}^{\prime\prime}p_{14}-p_{10}^{\prime\prime}p_{13}^{\prime\prime}+p_{11}^{\prime}p_{12}^{\prime\prime},$
$\displaystyle p_{24}$
$\displaystyle=p_{7}^{\prime}p_{17}-p_{8}^{\prime\prime}p_{16}+p_{9}^{\prime\prime}p_{15}-p_{11}^{\prime\prime}p_{13}^{\prime\prime}+p_{12}^{\prime}p_{12}^{\prime\prime},$
$\displaystyle p_{25}$
$\displaystyle=p_{8}^{\prime}p_{17}-p_{9}^{\prime}p_{16}+p_{10}^{\prime\prime}p_{15}-p_{11}^{\prime\prime}p_{14}+p_{12}^{\prime\prime}p_{13}^{\prime},$
$\displaystyle p_{26}$
$\displaystyle=p_{9}p_{17}-p_{10}^{\prime}p_{16}+p_{11}^{\prime}p_{15}-p_{12}^{\prime}p_{14}+p_{13}^{\prime}p_{13}^{\prime\prime}.$

We can express $p_{27}$ in terms of the generating Plücker coordinates as
well, but it is more conveniently expressed in terms of the other Plücker
coordinates as follows:

$p_{27}=p_{1}p_{26}-p_{2}p_{25}+p_{3}p_{24}-p_{4}p_{23}+p_{5}^{\prime}p_{22}^{\prime}+p_{5}^{\prime\prime}p_{22}^{\prime\prime}-p_{6}^{\prime}p_{21}^{\prime}+p_{7}^{\prime}p_{20}^{\prime}-p_{8}^{\prime}p_{19}^{\prime}+p_{9}p_{18}-p_{10}p_{17}.$

###  3.6. Proof of the isomorphism of LG-models on cominuscule
$E_{n}^{\mathrm{sc}}/P_{n}$

Recall that we want to show that $\pi:U_{-}^{P}\to\mathbb{X}^{\vee}$ is an
isomorphism to $X^{\vee}_{\mathrm{can}}\subset\mathbb{X}^{\vee}$

######  Remark 3.19.

The following proof follows the same structure as the proof of Theorem 3.9 in
Section 8 of [PR13].

###### Proof of Theorem 3.1.

Corollaries 3.13 and 3.17 imply $\pi(U_{-}^{P})\subseteq
X^{\vee}_{\mathrm{can}}$ for cominuscule $E_{n}^{\mathrm{sc}}/P_{n}$.

To show that $\pi:U_{-}^{P}\to X^{\vee}_{\mathrm{can}}$ is an isomorphism, it
suffices to show that
$\pi^{*}:\mathbb{C}[X^{\vee}_{\mathrm{can}}]\to\mathbb{C}[U_{-}^{P}]$ is an
isomorphism since both $U_{-}^{P}$ and $X^{\vee}_{\mathrm{can}}$ are affine
varieties (for $U_{-}^{P}$, this follows from Proposition 2.14; for
$X^{\vee}_{\mathrm{can}}$, this follows from the fact that it is the
complement of an ample divisor). The fact that $\pi^{*}$ is injective follows
from the fact that $\pi:U_{-}^{P}\to X^{\vee}_{\mathrm{can}}$ is dominant,
which follows from the fact that $U_{-}^{P}$ and $X^{\vee}_{\mathrm{can}}$
have the same dimension and that $\pi$ is an open map (as it is the quotient
map $u_{-}\mapsto\widetilde{P}^{\vee}u_{-}$).

For surjectivity of
$\pi^{*}:\mathbb{C}[X^{\vee}_{\mathrm{can}}]\to\mathbb{C}[U_{-}^{P}]$, it is
straightforward to check that each of the generators of
$\mathbb{C}[U_{-}^{P}]$ as given in Corollaries 3.13 and 3.17 is a well-
defined function on $X^{\vee}_{\mathrm{can}}$, and that each localized
expression is non-zero on $X^{\vee}_{\mathrm{can}}$. This is clear from the
fact that they are all generalized Plücker coordinates or expressions in
these, and from the definition of $X^{\vee}_{\mathrm{can}}$. ∎

######  Remark 3.20.

Note that this proof also shows that every coset in
$X^{\vee}_{\mathrm{can}}\subset\widetilde{P}^{\vee}\backslash\widetilde{G}^{\vee}$
can uniquely be represented by an element of $U_{-}^{P}=U^{\vee}_{-}\cap
B^{\vee}_{+}\bar{w}_{P}\bar{w}_{0}B^{\vee}_{+}$.

Now we know that $\pi:U_{-}^{P}\to X^{\vee}_{\mathrm{can}}$ is an isomorphism,
showing that $\pi\circ\varphi$ is an isomorphism of Landau-Ginzburg models
reduces to checking that
$(\pi\circ\varphi)^{*}(\mathcal{W}_{\mathrm{can}})=\mathcal{W}_{\mathrm{Lie}}$.

######  Remark 3.21.

We are abusing notation here: a more proper way to write this would be
$\bigl{(}(\pi\circ\varphi)\times\mathrm{Id}\bigr{)}^{*}(\mathcal{W}_{\mathrm{can}})=\mathcal{W}_{\mathrm{Lie}}$,
as $\mathcal{W}_{\mathrm{can}}:X^{\vee}_{\mathrm{can}}\times
T^{P}\to\mathbb{C}$ and
$\mathcal{W}_{\mathrm{Lie}}:X^{\vee}_{\mathrm{Lie}}\times T^{P}\to\mathbb{C}$.
We will continue to use this notation as it should not give rise to
ambiguities.

The proof of Theorem 3.2 is obtained by restricting to the torus
$U^{\circ}_{-}\subset U_{-}^{P}$ of (2.15) and showing that
$(\varphi^{-1})^{*}(\mathcal{W}_{\mathrm{Lie}})=\pi^{*}(\mathcal{W}_{\mathrm{can}})$
on $U^{\circ}_{-}\times T^{P}$. This is most conveniently done separately for
the Cayley plane and the Freudenthal variety:

###### Proof of Theorem 3.2 for the Cayley plane.

Applying the results of Proposition 2.9 to the Cayley plane, we see that
$(\varphi^{-1})^{*}(\mathcal{W}_{\mathrm{Lie}})$ can be written as follows on
$U^{\circ}_{-}\times T^{P}$:

$[(\varphi^{-1})^{*}(\mathcal{W}_{\mathrm{Lie}})](u_{-},t)=a_{1}+a_{2}+\ldots+a_{16}+q\frac{\sum_{(i_{j})\in\mathcal{I}}\prod_{j=1}^{{\ell^{\prime}}}a_{i_{j}}}{a_{1}\cdots
a_{16}},$

where $u_{-}=y^{\vee}_{r_{16}}(a_{16})y^{\vee}_{r_{15}}(a_{15})\cdots
y^{\vee}_{r_{1}}(a_{1})$ and $q=\alpha^{\vee}_{6}(t)$. (Recall that
$(r_{1},\ldots,r_{16})$ is the sequence of indices in the reduced expression
of $w^{P}$ in equation (2.26).) The set $\mathcal{I}$ consists of the reduced
subexpressions of the element $w^{\prime}$ in the fixed reduced expression of
$w^{P}$ as discussed above Propostion 2.9.

On the other hand we need to calculate the terms of
$\mathcal{W}_{\mathrm{can}}$ as given in equation (3.3) applied to an element
of $U^{\circ}_{-}$. We can use Algorithm 3.12 to calculate the toric
expressions for the Plücker coordinates, followed by elementary operations on
polynomials (addition, multiplication, and so on) that are somewhat cumbersome
to find toric expressions for the terms of the superpotential; we used a
computer algebra program and simply state the results here:

$\displaystyle\frac{p_{1}}{p_{0}}$
$\displaystyle=a_{9}+a_{16},\qquad\frac{p_{9}^{\prime}}{p_{8}}=a_{1}+a_{8},\qquad\frac{q_{13}}{q_{12}}=a_{5}+a_{10}+a_{15},$
$\displaystyle\frac{q_{17}}{q_{16}}$
$\displaystyle=a_{2}+a_{7}+a_{12},\qquad\frac{q_{21}}{q_{20}}=a_{4}+a_{13},\qquad\frac{q_{25}}{q_{24}}=a_{3}+a_{6}+a_{11}+a_{14},$
$\displaystyle q\frac{p_{5}^{\prime\prime}}{p_{16}}$
$\displaystyle=q\frac{\sum_{(i_{j})\in\mathcal{I}}\prod_{j=1}^{{\ell^{\prime}}}a_{i_{j}}}{a_{1}\cdots
a_{16}},$

where we suppressed the arguments of the Plücker coordinates (they are all
applied to $u_{-}$). Clearly, we have
$(\varphi^{-1})^{*}(\mathcal{W}_{\mathrm{Lie}})=\pi^{*}(\mathcal{W}_{\mathrm{can}})$
on $U^{\circ}_{-}\times T^{P}$, but since $U^{\circ}_{-}\subset U_{-}^{P}$ is
open and dense and $U_{-}^{P}$ is irreducible, we conclude that the equality
holds on all of $U_{-}^{P}\times T^{P}$. The theorem now follows in the case
of the Cayley plane. ∎

###### Proof of Theorem 3.2 for the Freudenthal variety.

In this case, the results of Propositon 2.9 yield the following expression for
$(\varphi^{-1})^{*}(\mathcal{W}_{\mathrm{Lie}})$ on $U^{\circ}_{-}\times
T^{P}$:

$[(\varphi^{-1})^{*}(\mathcal{W}_{\mathrm{Lie}})](u_{-},t)=a_{1}+a_{2}+\ldots+a_{27}+q\frac{\sum_{(i_{j})\in\mathcal{I}}\prod_{j=1}^{{\ell^{\prime}}}a_{i_{j}}}{a_{1}\cdots
a_{27}},$

where $u_{-}=y^{\vee}_{r_{27}}(a_{27})y^{\vee}_{r_{26}}(a_{26})\cdots
y^{\vee}_{r_{1}}(a_{1})$ and $q=\alpha^{\vee}_{7}(t)$. (Recall that
$(r_{1},\ldots,r_{27})$ is the sequence of indices in the reduced expression
of $w^{P}$ in equation (2.28).)

The terms of $\mathcal{W}_{\mathrm{can}}$ applied to an element of
$U^{\circ}_{-}$ can be computed using Algorithm 3.12 and a computer algebra
program to be the following:

$\displaystyle\frac{p_{1}}{p_{0}}$
$\displaystyle=a_{1}+a_{15}+a_{27},\qquad\frac{q_{19}}{q_{18}}=a_{2}+a_{9}+a_{19}+a_{26},\qquad\frac{q_{28}}{q_{27}}=a_{6}+a_{13}+a_{22}$
$\displaystyle\frac{q_{37}}{q_{36}}$
$\displaystyle=a_{10}+a_{18},\qquad\frac{q_{37}^{\prime}}{q_{36}^{\prime}}=a_{3}+a_{8}+a_{14}+a_{20}+a_{25},\qquad\frac{q_{46}}{q_{45}}=a_{5}+a_{11}+a_{17}+a_{23}$
$\displaystyle\frac{q_{55}}{q_{54}}$
$\displaystyle=a_{4}+a_{7}+a_{12}+a_{16}+a_{21}+a_{24},\qquad
q\frac{p_{10}}{p_{27}}=q\frac{\sum_{(i_{j})\in\mathcal{I}}\prod_{j=1}^{{\ell^{\prime}}}a_{i_{j}}}{a_{1}\cdots
a_{27}}$

Here all the Plücker coordinates are applied to $u_{-}\in U^{\circ}_{-}$, so
this is suppressed. This shows that
$(\varphi^{-1})^{*}(\mathcal{W}_{\mathrm{Lie}})=\pi^{*}(\mathcal{W}_{\mathrm{can}})$
on $U^{\circ}_{-}\times T^{P}$, implying that the equality holds on
$U_{-}^{P}\times T^{P}$, as $U^{\circ}_{-}\subset U_{-}^{P}$ is open and dense
and $U_{-}^{P}$ is irreducible. ∎

Thus, we have shown that the canonical models we defined in Section 3 for the
exceptional cominuscule homogeneous spaces are isomorphic to the Lie-theoretic
Landau-Ginzburg models given in [Rie08] (discussed in Section 2.2). From this
we conclude that the canonical models are indeed Landau-Ginzburg models,
satisfying

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

where $\langle\partial\mathcal{W}_{\mathrm{can}}\rangle$ is generated by the
derivatives of $\mathcal{W}_{\mathrm{can}}$ along $X_{\mathrm{can}}$. Due to
the geometric Satake correspondence, this isomorphism is given by associating
to $\sigma_{i}\in H^{*}(X)$ the Plücker coordinate
$p_{i}\in\mathbb{C}[X_{\mathrm{can}}]$. Furthermore, this isomorphism of
Landau-Ginzburg models implies in particular that our canonical models have
the expected number of critical points, namely $\dim H^{*}(X,\mathbb{C})$,
although the expressions of the superpotentials are too unwieldly to calculate
them directly.

##  4\. Cluster structures on $\mathbb{X}^{\vee}$

In this section we first review the cluster structures presented in [GLS08]
for homogeneous spaces of simply-laced type. We then give Plücker coordinate
descriptions for these cluster structures using an algorithm we formulated for
expressing generalized minors as polynomials in the Plücker coordinates.

###  4.1. Cluster algebras and varieties

A _cluster structure_ on a commutative algebra $A$ is a combinatorial
presentation of $A$ in terms of _seeds_ and _mutations_. Cluster algebras come
in two types, $\mathcal{A}$-cluster algebras and $\mathcal{X}$-cluster
algebras, differing in the definition of their mutations. We will only
consider $\mathcal{A}$-cluster algebras here. To define seeds, we need to fix
the field $\mathbb{C}(x_{1},\dots,x_{n})$ of rational functions in $n$
variables.

A seed $\Sigma$ of rank $l$ is a pair $\Sigma=(\mathbf{x},B)$ and an integer
$m\geq l$, where $\mathbf{x}\in\mathbb{C}(x_{1},\dots,x_{n})^{m}$ is a tuple
of algebraically independent elements, and $B$ is an $m\times l$ integer
matrix whose upper $l\times l$ submatrix $M$ is skew-symmetrizable, meaning
that there is a diagonal matrix $D$ with positive integer entries on the
diagonal such that $DM=-(DM)^{T}$. The elements of $\mathbf{x}$ are called
_cluster variables_ , and the first $l$ are called _mutable_ while the
remaining $m-l$ are called _frozen_. Furthermore, $B$ is called an _exchange
matrix_ , and in the case that $D=I$, then we may think of $B$ as the
adjacency matrix of a directed graph $Q$ known as a _quiver_. Mutation is an
operation on seeds that produces new seeds. For $1\leq k\leq l$, the mutation
of $\Sigma$ in direction $k$ is a new seed
$\Sigma^{\prime}=(\mu_{k}(\mathbf{x}),\mu_{k}(B))$. We will not work with
mutations here, so we omit their full definition.

Given an _initial seed_ $\Sigma$, let $S_{\\!\mathcal{A}}$ denote the set of
cluster variables appearing in any seed $\Sigma^{\prime}$ reachable from
$\Sigma$ by any (finite) sequence of $\mathcal{A}$-mutations. Then $A$ is an
$\mathcal{A}$-cluster algebra if $A=\mathbb{C}[S_{\mathcal{A}}]$. Finally, an
affine variety $X$ is an $\mathcal{A}$-cluster variety if $\mathbb{C}[X]$ is
an $\mathcal{A}$-cluster algebra. To each frozen variable $x$, we associate a
divisor $D_{x}=\\{p\in X~{}|~{}x(p)=0\\}$, and to each seed $\Sigma$ we
associate a torus $\mathbb{T}_{\\!\Sigma}\hookrightarrow X$. The union of all
tori $\mathbb{T}=\bigcup_{\Sigma}\mathbb{T}_{\\!\Sigma}$ is a dense, open
subset of $X$ often called the _open cluster variety_ , and
$\mathbb{T}\cup(\bigcup_{x}D_{x})$ is a _(partial) compactification_ of
$\mathbb{T}$, where the union is over the set of frozen variables. Roughly,
_cluster varieties_ can be thought of as generalizations of toric varieties
where the dense torus is replaced by a union of (infinitely many) _cluster
tori_ identified along birational _mutation_ maps, and the torus-invariant
divisors are replaced by divisors corresponding to _frozen variables_.

For our purposes, we will need a concrete description of the cluster structure
in terms of Plücker coordinates, and we will provide such descriptions for the
cominuscule exceptional types in Section 4. Cluster structures for homogeneous
spaces $P^{-}\backslash G$ were given in [GLS08] for $G$ of simply-laced
Dynkin type in terms of generalized minors. Their general construction depends
on a choice of Lie group $G$, parabolic subgroup $P$, and reduced expression
for the longest word $w_{0}$ of the Weyl group $W$ of $G$, having as prefix a
reduced expression for the longest word $w_{P}$ of the subgroup $W_{P}\subset
W$. We present a simplified version of their construction for our particular
cases of the Cayley plane and the Freudenthal variety.

###  4.2. Cluster structures on homogeneous spaces

This section follows Sections 9 and 10 of [GLS08]. Recall our fixed reduced
expression for $w_{0}=w_{0}^{-1}=w_{P}^{-1}(w^{P})^{-1}$ in equation (2.7):

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

which has the reverse numbering of indices compared to [GLS08].

To each simple reflection $s_{r_{i}}$ in the reduced expression for $w_{0}$,
we associate the generalized minor
$\phi(i,{\varpi^{\vee}_{r_{i}}})=\Delta_{u_{\geq
i}({\varpi^{\vee}_{r_{i}}}),w_{0}({\varpi^{\vee}_{r_{i}}})}$ where $u_{\geq
i}=s_{r_{{\ell_{0}}}}\dots s_{r_{i}}$ and
$\Delta_{u({\varpi^{\vee}_{i}}),v({\varpi^{\vee}_{i}})}$ is the generalized
minor defined in equation (2.22). There are $n$ indices $r_{i}$ such that
$\phi(i,{\varpi^{\vee}_{r_{i}}})|_{U_{+}^{P}}\equiv 1$: one for the last
occurrence of each simple transposition $s_{i}$ in our reduced expression for
$w_{0}$. To see this, observe that for these indices we have that $u_{\geq
i}({\varpi^{\vee}_{r_{i}}})=w_{0}({\varpi^{\vee}_{r_{i}}})$, which follows
from the fact that the remaining simple transpositions in $w_{0}$ act
trivially on the highest weight. Thus, we have for these last occurrences
$r_{i}$ that

$\phi(i,{\varpi^{\vee}_{r_{i}}})|_{U_{+}^{P}}=\Delta_{w_{0}({\varpi^{\vee}_{r_{i}}}),w_{0}({\varpi^{\vee}_{r_{i}}})}|_{U_{+}^{P}}={}\Bigr{\langle}(\overline{w_{0}})^{-1}g\,\overline{w_{0}}\cdot
v_{{\varpi^{\vee}_{r_{i}}}}^{+},v_{{\varpi^{\vee}_{r_{i}}}}^{+}\Bigr{\rangle}.$

In particular for any $g\in U^{\circ}_{+}=(U^{\circ}_{-})^{T}$ (see equation
(2.15)), we have that

$\Bigr{\langle}(\overline{w_{0}})^{-1}g\,\overline{w_{0}}\cdot
v_{{\varpi^{\vee}_{r_{i}}}}^{+},v_{{\varpi^{\vee}_{r_{i}}}}^{+}\Bigr{\rangle}=1,$

obtained from the constant term after expanding of the decomposition of
$g=x^{\vee}_{r_{1}}(a_{1})\cdots x^{\vee}_{r_{\ell}}(a_{\ell})$. This equality
extends to $U_{+}^{P}$ as $U^{\circ}_{+}\subset U_{+}^{P}$ dense, so that the
desired equality holds. The ${\ell_{0}}-n$ indices for which
$\phi(i,{\varpi^{\vee}_{r_{i}}})|_{U_{+}^{P}}\not\equiv 1$ are called
$w_{0}$-exchangeable, and we denote the set of $w_{0}$-exchangeable indices by
$e(w_{0})$.

We now describe the generalized minors appearing in our initial seed. We have
mutable variables given by $\phi(i,{\varpi^{\vee}_{r_{i}}})$ for
$i\in\\{1,\dots,\ell\\}\cap e(w_{0})$. In order to define the frozen
variables, we set $t_{i}=\min\\{t~{}|~{}r_{t}=i,t>l\\}$ unless $i\in I^{P}$,
in which case we set $t_{i}={\ell_{0}}+i$, and $u_{\geq t_{i}}=e$. Then we
have frozen variables given by

$\\{\phi(t_{i},{\varpi^{\vee}_{r_{t_{i}}}})~{}|~{}1\leq i\leq
n\\}\cup\\{\Delta_{{\varpi^{\vee}_{i}},{\varpi^{\vee}_{i}}}\mid i\in
I^{P}\\}.$

Note that in the case of the exceptional cominuscule family, $I^{P}=\\{n\\}$
is a singleton set.

######  Remark 4.1.

Note that $\Delta_{{\varpi^{\vee}_{i}},{\varpi^{\vee}_{i}}}|_{U_{+}^{P}}=1$ by
an analogous computation to the one above for
$\Delta_{w_{0}({\varpi^{\vee}_{r_{i}}}),w_{0}({\varpi^{\vee}_{r_{i}}})}|_{U_{+}^{P}}=1$.
In fact, the $\phi|_{U_{+}^{P}}$ are the cluster variables for a cluster
structure on $U_{+}^{P}$ (without any of the generalized minors
$\Delta_{{\varpi^{\vee}_{i}},{\varpi^{\vee}_{i}}}|_{U_{+}^{P}}$). However, we
have for each $i$ that $\Delta_{{\varpi^{\vee}_{i}},{\varpi^{\vee}_{i}}}\neq
1$ on $\mathbb{X}^{\vee}$, and we require these additional generalized minors
as frozen variables for the cluster structures on $\mathbb{X}^{\vee}$ given in
this section.

Finally, we define the quiver $\Gamma_{\\!w_{0}}$ with vertex set

$V(\Gamma_{\\!w_{0}})=\bigl{(}\\{1,2,\dots,\ell\\}\cap
e(w_{0})\bigr{)}\cup\bigl{(}\\{t_{i}~{}|~{}1\leq i\leq
n\\}\cup\\{\Delta_{{\varpi^{\vee}_{i}},{\varpi^{\vee}_{i}}}~{}|~{}i\in
I^{P}\\}\bigr{)},$

where the last two sets of vertices are frozen. To describe the arrows of the
quiver, define $m^{+}=\max\\{1\leq
l\leq{\ell_{0}}~{}|~{}l<m,~{}r_{l}=r_{m}\\}$. Then for $m<l\in
V(\Gamma_{\\!w_{0}})$ with at least one of $m,l\in e(w_{0})$, there is an edge
$m\rightarrow l$ if and only if $m^{+}=l$, and there is an edge $l\rightarrow
m$ if and only if both $l<m^{+}<l^{+}$ and $a_{r_{m},r_{l}}<0$, where $a_{ij}$
denotes the $(i,j)$-entry of the Cartan matrix of $G$. The edges involving the
remaining frozen vertices $\Delta_{{\varpi^{\vee}_{i}},{\varpi^{\vee}_{i}}}$
are determined through cluster category theory (see Theorem 10.2 of [GLS08]).
In our case, the extra vertex corresponding to
$\Delta_{{\varpi^{\vee}_{n}},{\varpi^{\vee}_{n}}}$ has an edge to the vertex
labeled by $p_{1}$ (which in each case corresponds to the generalized minor
$\phi(i,{\varpi^{\vee}_{r_{i}}})$ such that $r_{i}=\sigma(n)$ and $i$ is the
second smallest such index, $\sigma$ being the Dynkin diagram symmetry
involution if applicable and the identity else).

###  4.3. Cluster algebras and mirror symmetry

In the version of mirror symmetry we have been working with so far, the
superpotentials $\mathcal{W}_{\mathrm{can}}$ defined on the Langlands dual
spaces encode enumerative information about the original spaces via their
Jacobi rings. One of the main reuslts in [RW19] provides a combinatorial
interpretation of mirror symmetry using cluster structures on Grassmannians
through two families of polytopes, _Newton-Okounkov bodies_ and
_superpotential polytopes_. Although it has not yet been studied, we expect
that this statement holds for our homogeneous spaces as well, and we now give
$\mathcal{A}$-cluster structures in terms of Plücker coordinates for the
Langlands dual spaces (by giving an initial seed).

######  Remark 4.2.

While [GLS08] works with $P^{-}\backslash G$, where $P^{-}$ refers to an
opposite parabolic subgroup, we prefer to work with $P\backslash G$. Because
of this conventional difference, we worked with $u_{-}^{T}\in
U_{+}^{\circ}=(U_{-}^{\circ})^{T}$ in Lemma 3.10. In this section, we will
prefer to work directly on $U_{-}^{\circ}$ by using the “transposes” of the
generalized minors,
$\Delta^{T}_{w_{1}({\varpi^{\vee}_{i}}),w_{2}({\varpi^{\vee}_{i}})}(u_{-})$.
This corresponds in Algorithm 3.11 to looking for paths in
$V({\varpi^{\vee}_{i}})$ from $w_{1}({\varpi^{\vee}_{i}})$ to
$w_{2}({\varpi^{\vee}_{i}})$ (using $f^{\vee}_{i}$’s), as opposed to paths
from $w_{2}({\varpi^{\vee}_{i}})$ to $w_{1}({\varpi^{\vee}_{i}})$ (using
$e^{\vee}_{i}$’s), and replacing $u_{+}\in\\{x^{\vee}_{r_{1}}(a_{1})\dots
x^{\vee}_{r_{\ell}}(a_{\ell})\\}$ with its transpose
$u_{+}^{T}\in\\{y^{\vee}_{r_{\ell}}(a_{\ell})\dots
y^{\vee}_{r_{1}}(a_{1})\\}$. The rest of the computation is entirely
analogous. With respect to the notation, the transpose of a generalized minor
corresponds to switching the two subscripts and transposing the input. To be
more consistent with our previous choice, we could also have worked in the
representation with highest weight
${\varpi^{\vee}_{\sigma(i)}}=-w_{0}({\varpi^{\vee}_{i}})$ instead:

$\Delta^{T}_{w_{1}({\varpi^{\vee}_{i}}),w_{2}({\varpi^{\vee}_{i}})}(g)=\Delta^{T}_{w_{1}w_{0}\cdot
w_{0}({\varpi^{\vee}_{i}}),w_{2}w_{0}\cdot
w_{0}({\varpi^{\vee}_{i}})}(g)=\Delta_{w_{2}w_{0}({\varpi^{\vee}_{\sigma(i)}}),w_{1}w_{0}({\varpi^{\vee}_{\sigma(i)}})}(g^{T}).$

Note that the equality above is of polynomials in the torus coordinates. The
form of generalized minor that will appear in our cluster variables is
$\Delta^{T}_{u_{\geq i}({\varpi^{\vee}_{i}}),w_{0}({\varpi^{\vee}_{i}})}$. In
the notation of Lemma 3.10, these would be written as

$\Delta^{T}_{u_{\geq
i}({\varpi^{\vee}_{i}}),w_{0}({\varpi^{\vee}_{i}})}(g)=\Delta_{{\varpi^{\vee}_{\sigma(i)}},u_{\geq
i}w_{0}({\varpi^{\vee}_{\sigma(i)}})}(g^{T}),$

and one can immediately check that the frozen variables introduced below are
exactly equal to those computed in Lemma 3.10. For example, if $u_{\geq
i}=w_{P}$, then we have that

$\Delta^{T}_{w_{P}({\varpi^{\vee}_{i}}),w_{0}({\varpi^{\vee}_{i}})}(u_{-})=\Delta_{{\varpi^{\vee}_{\sigma(i)}},(w^{P})^{-1}({\varpi^{\vee}_{\sigma(i)}})}(u_{-}^{T})$

which is exactly one of the denominators that we computed above.

###  4.4. The Cayley plane

For the Cayley plane, we use the following reduced expression for the longest
word $w_{0}$ of the Weyl group of $E_{6}$ to carry out the construction of an
initial seed:

$w_{0}=(s_{1}s_{2}s_{3}s_{1}s_{4}s_{3}s_{1}s_{2}s_{4}s_{3}s_{5}s_{4}s_{2}s_{3}s_{4}\framebox{\\!$s_{5}s_{1}s_{3}s_{4}s_{2}$\\!})(s_{6}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}\underline{s_{6}}s_{1}s_{3}s_{4}\underline{s_{5}s_{2}s_{4}s_{3}s_{1}}).$

The last $16$ indices correspond to our previous choice of $(w^{P})^{-1}$. We
have boxed the indices $t_{i}$ for $1\leq i<6$ and underlined the last
occurrence of each $s_{i}$ ($1\leq i\leq 6$). Thus, the set of
$w_{0}$-exchangeable indices is given by the complement of the underlined
indices:

$e(w_{0})=\\{1,\dots,36\\}\setminus\\{1,2,3,4,5,9\\}.$

This gives the mutable cluster variables

$\begin{array}[]{lllll}\phi(6,{\varpi^{\vee}_{4}})&\phi(7,{\varpi^{\vee}_{3}})&\phi(8,{\varpi^{\vee}_{1}})&\phi(10,{\varpi^{\vee}_{5}})&\phi(11,{\varpi^{\vee}_{4}})\\\
\phi(12,{\varpi^{\vee}_{3}})&\phi(13,{\varpi^{\vee}_{2}})&\phi(14,{\varpi^{\vee}_{4}})&\phi(15,{\varpi^{\vee}_{5}})&\phi(16,{\varpi^{\vee}_{6}})\end{array}$
(4.1)

and the frozen variables

$\begin{array}[]{lllll}\phi(t_{1}=20,{\varpi^{\vee}_{1}})&\phi(t_{2}=17,{\varpi^{\vee}_{2}})&\phi(t_{3}=19,{\varpi^{\vee}_{3}})&\phi(t_{4}=18,{\varpi^{\vee}_{4}})\\\
\phi(t_{5}=21,{\varpi^{\vee}_{5}})&\phi(t_{6}=-6,{\varpi^{\vee}_{6}})&\Delta^{T}_{{\varpi^{\vee}_{6}},{\varpi^{\vee}_{6}}}.\end{array}$
(4.2)

For ease of notation, we define the following polynomials in Plücker
coordinates, in addition to the ones appearing in (3.1) and (3.2):

$\displaystyle q_{8}$ $\displaystyle=p_{1}p_{7}^{\prime}-p_{0}p_{8},\quad
q_{8}^{\prime}=p_{2}p_{6}^{\prime}-p_{1}p_{7}^{\prime}+p_{0}p_{8},\quad
q_{10}=p_{2}p_{8}^{\prime\prime}-p_{1}p_{9}^{\prime\prime}+p_{0}p_{10}^{\prime},$
$\displaystyle q_{15}$
$\displaystyle=p_{6}^{\prime}(p_{1}p_{8}^{\prime}-p_{0}p_{9}^{\prime})-p_{7}^{\prime\prime}(p_{1}p_{7}^{\prime}+p_{0}p_{8})$

######  Lemma 4.3.

The mutable cluster variables in equation 4.1 can be expressed on
$U_{-}^{\circ}$ in terms of Plücker coordinates as:

$\begin{array}[]{llll}\phi(6,{\varpi^{\vee}_{4}})=p_{3},&\phi(7,{\varpi^{\vee}_{3}})=p_{2},&\phi(8,{\varpi^{\vee}_{1}})=p_{1},&\phi(10,{\varpi^{\vee}_{5}})=p_{4}^{\prime\prime},\\\
\phi(11,{\varpi^{\vee}_{4}})=q_{8}^{\prime},&\phi(12,{\varpi^{\vee}_{3}})=q_{8},&\phi(13,{\varpi^{\vee}_{2}})=p_{4}^{\prime},&\phi(14,{\varpi^{\vee}_{4}})=q_{15},\\\
\phi(15,{\varpi^{\vee}_{5}})=q_{10},&\phi(16,{\varpi^{\vee}_{6}})=p_{5}^{\prime\prime},\end{array}$
(4.3)

whereas the frozen cluster variables in equation 4.2 can be expressed as:

$\begin{array}[]{llll}\phi(17,{\varpi^{\vee}_{2}})=q_{12},&\phi(18,{\varpi^{\vee}_{4}})=q_{24},&\phi(19,{\varpi^{\vee}_{3}})=q_{16},&\phi(20,{\varpi^{\vee}_{1}})=p_{8},\\\
\phi(21,{\varpi^{\vee}_{5}})=q_{20},&\phi(-6,{\varpi^{\vee}_{6}})=p_{16},&\Delta_{{\varpi^{\vee}_{6}},{\varpi^{\vee}_{6}}}=p_{0}.\end{array}$
(4.4)

###### Proof.

The cluster variables, being generalized minors, are calculated using
Algorithm 3.11 (after applying the change of looking for paths going down from
$u_{\geq k}({\varpi^{\vee}_{i}})$ to $w_{0}({\varpi^{\vee}_{i}})$ as mentioned
in Remark 4.2), while we use Algorithm 3.12 to compute the torus expansions of
the Plücker coordinate polynomials; afterwards, it is a straightforward check
that each equality claimed above holds. ∎

We provide in Figure 4.1 (left) the cluster quiver as defined by [GLS08],
which we described in Subsection 4.2. We have labelled the vertices by the
cluster variables above in Lemma 4.3.

$1$$2$$3$$4$$5$$6$$p_{16}$$q_{20}$$q_{24}$$q_{12}$$q_{16}$$p_{8}$$q_{15}$$q_{8}$$q_{10}$$q_{8}^{\prime}$$p_{5}^{\prime\prime}$$p_{4}^{\prime}$$p_{4}^{\prime\prime}$$p_{3}$$p_{2}$$p_{1}$$p_{0}$

$1$$2$$3$$4$$5$$6$

Figure 4.1. The cluster quiver (left) as defined in [GLS08] and quiver (right)
defined in [CMP08] mentioned in Remark 4.4 for the Cayley plane. The frozen
variables are blue squares in the cluster quiver.

######  Remark 4.4.

The cluster quiver is remarkably similar to a quiver defined in [CMP08], which
we have drawn to the right in Figure 4.1 for comparison, although this quiver
is not known to be a cluster quiver. It is defined using a fixed reduced
expression for $w^{P}$ (denoted $w_{X}$ there), having as vertices the simple
reflections of the expression, and an arrow from each simple reflection to the
first occurrence of each non-commuting simple reflection to the right of it.
See Definition 2.1 of [CMP08] for more details.

This quiver is shown to provide a combinatorial tool used to determine
Poincaré dual Schubert classes that generalizes Young diagrams for
Grassmannians to general (co-) minuscule homogeneous spaces. They also use the
quivers to formulate a quantum Chevalley formula, and show that they can be
used to determine “higher quantum Poincaré duality”, as well as to determine
the smallest power of the quantum parameter appearing in the quantum product
of two Schubert classes. The quivers thus play an important role in the
Schubert calculus of (co-) minuscule homogeneous spaces, but are not shown to
be related to any cluster structure in [CMP08]. Note that these quivers are
also used in Algorithm 3.12 as well as in Corollary 8.12 of [Spa21] as
combinatorial tools to find subexpressions of a given Weyl group element
inside the fixed reduced expression of another element.

We have formulated an algorithm to convert the quiver of [CMP08] into the
cluster quiver as defined in [GLS08], and we have verified the algorithm in
the cases of the Cayley plane, the Freudenthal variety and a number of cases
of Grassmannians and quadrics. We intend to verify this more generally and
explore this relation between the cluster structure and quantum cohomology
further.

Although it is simple to check that the generalized minor is equal to the
given Plücker coordinate expression restricted to $U_{-}^{\circ}$ using
Algorithms 3.11 and 3.12, similar to the proof of Lemma 3.10, it is difficult
to obtain these Plücker coordinate expressions. We were able to perform the
computations directly in this case using a computer algebra program, but the
computations for the Freudenthal variety require a more sophisticated
approach, which we describe in the following subsection.

###  4.5. The Freudenthal variety

For the Freudenthal variety, we use the following reduced expression for the
longest word $w_{0}$ of the Weyl group of $E_{7}$ to construct an initial
seed:

$\displaystyle w_{0}=$ $\displaystyle
s_{1}s_{2}s_{3}s_{1}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{2}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}\framebox{\\!$s_{2}s_{6}s_{5}s_{4}s_{3}s_{1}$\\!}$
$\displaystyle
s_{7}s_{6}s_{5}s_{4}s_{3}s_{2}s_{4}s_{5}s_{6}s_{1}s_{3}s_{4}s_{7}s_{5}s_{2}s_{4}s_{3}\underline{s_{1}}s_{6}s_{5}s_{4}\underline{s_{2}s_{3}s_{4}s_{5}s_{6}s_{7}}.$

We have boxed the indices $t_{i}$ for $1\leq i<7$ and underlined the last
occurrence of each $s_{i}$ ($1\leq i\leq 7$). Thus, the set of
$w_{0}$-exchangeable indices is given by the complement of the underlined
indices:

$e(w_{0})=\\{1,\dots,63\\}\setminus\\{1,2,3,4,5,6,10\\}.$

This gives the mutable cluster variables:

$\begin{array}[]{lllll}\phi(7,{\varpi^{\vee}_{4}})&\phi(8,{\varpi^{\vee}_{5}})&\phi(9,{\varpi^{\vee}_{6}})&\phi(11,{\varpi^{\vee}_{3}})&\phi(12,{\varpi^{\vee}_{4}})\\\
\phi(13,{\varpi^{\vee}_{2}})&\phi(14,{\varpi^{\vee}_{5}})&\phi(15,{\varpi^{\vee}_{7}})&\phi(16,{\varpi^{\vee}_{4}})&\phi(17,{\varpi^{\vee}_{3}})\\\
\phi(18,{\varpi^{\vee}_{1}})&\phi(19,{\varpi^{\vee}_{6}})&\phi(20,{\varpi^{\vee}_{5}})&\phi(21,{\varpi^{\vee}_{4}})&\phi(22,{\varpi^{\vee}_{2}})\\\
\phi(23,{\varpi^{\vee}_{3}})&\phi(24,{\varpi^{\vee}_{4}})&\phi(25,{\varpi^{\vee}_{5}})&\phi(26,{\varpi^{\vee}_{6}})&\phi(27,{\varpi^{\vee}_{7}})\end{array}$
(4.5)

and the frozen variables:

$\begin{array}[]{llll}\phi(t_{1}=28,{\varpi^{\vee}_{1}})&\phi(t_{2}=33,{\varpi^{\vee}_{2}})&\phi(t_{3}=29,{\varpi^{\vee}_{3}})&\phi(t_{4}=30,{\varpi^{\vee}_{4}})\\\
\phi(t_{5}=31,{\varpi^{\vee}_{5}})&\phi(t_{6}=32,{\varpi^{\vee}_{6}})&\phi(t_{7}=-7,{\varpi^{\vee}_{7}})&\Delta^{T}_{{\varpi^{\vee}_{7}},{\varpi^{\vee}_{7}}}.\end{array}$
(4.6)

As before, we define the following polynomials in Plücker coordinates in
addition to the ones appearing in (3.4) and (3.5), for ease of notation:

$\displaystyle q_{10}$ $\displaystyle=p_{1}p_{9}-p_{10},\qquad
q_{10}^{\prime}=p_{2}p_{8}^{\prime}-p_{1}p_{9}+p_{10},\qquad
q_{10}^{\prime\prime}=p_{3}p_{7}^{\prime}-p_{2}p_{8}^{\prime}+p_{1}p_{9}-p_{10},$
$\displaystyle q_{12}$
$\displaystyle=p_{5}^{\prime\prime}p_{7}^{\prime\prime}-p_{6}^{\prime}p_{6}^{\prime\prime},\qquad
q_{14}=p_{2}p_{12}^{\prime\prime}-p_{1}p_{13}^{\prime\prime}+p_{14}^{\prime}\qquad
q_{19}^{\prime}=p_{9}^{\prime}(p_{10}-p_{1}p_{9})-p_{8}^{\prime}(p_{11}-p_{1}p_{10}^{\prime})$
$\displaystyle q_{20}$
$\displaystyle=p_{5}^{\prime}p_{15}^{\prime\prime}-p_{4}p_{16}^{\prime\prime}+p_{3}p_{17}^{\prime\prime}-p_{2}p_{18}^{\prime\prime},\qquad
q_{23}=p_{5}^{\prime\prime}(p_{18}-p_{1}p_{17})-p_{6}^{\prime\prime}(p_{17}^{\prime}-p_{1}p_{16})$
$\displaystyle q_{18}^{\prime}$
$\displaystyle=p_{2}(p_{5}^{\prime}p_{11}^{\prime\prime}-p_{4}p_{12}^{\prime\prime})+p_{1}(p_{4}p_{13}^{\prime\prime}-p_{5}^{\prime}p_{12}^{\prime})+p_{5}^{\prime}p_{13}-p_{4}p_{14}^{\prime}$
$\displaystyle q_{28}^{\prime}$
$\displaystyle=(p_{1}p_{15}-p_{16}^{\prime})(p_{5}^{\prime\prime}p_{7}^{\prime\prime}-p_{6}^{\prime}p_{6}^{\prime\prime})-(p_{1}p_{16}-p_{17}^{\prime})(p_{4}p_{7}^{\prime\prime}-p_{5}^{\prime}p_{6}^{\prime\prime})+(p_{1}p_{17}-p_{18})(p_{4}p_{6}^{\prime}-p_{5}^{\prime}p_{5}^{\prime\prime})$
$\displaystyle q_{30}$
$\displaystyle=p_{18}^{\prime\prime}(p_{3}p_{9}^{\prime\prime}-p_{2}p_{10}^{\prime\prime}+p_{1}p_{11}^{\prime}-p_{12})-p_{19}^{\prime\prime}(p_{3}p_{8}^{\prime\prime}-p_{2}p_{9}^{\prime}+p_{1}p_{10}^{\prime}-p_{11})$
$\displaystyle\quad-
p_{20}^{\prime}(-p_{3}p_{7}^{\prime}+p_{2}p_{8}^{\prime}-p_{1}p_{9}+p_{10})+p_{3}q_{27}$
$\displaystyle q_{40}$
$\displaystyle=(p_{2}p_{12}^{\prime\prime}-p_{1}p_{13}^{\prime\prime}+p_{14}^{\prime})(p_{6}^{\prime\prime}p_{20}^{\prime\prime}-p_{5}^{\prime\prime}p_{21}^{\prime}+p_{4}p_{22}^{\prime})$
$\displaystyle\quad+(p_{2}p_{11}^{\prime\prime}-p_{1}p_{12}^{\prime}+p_{13})(-p_{5}^{\prime}p_{22}^{\prime}+p_{6}^{\prime}p_{21}^{\prime}-p_{7}^{\prime\prime}p_{20}^{\prime\prime}+q_{27}^{\prime\prime})$

######  Lemma 4.5.

The mutable cluster variables in equation 4.5 can be expressed on
$U_{+}^{\circ}$ in terms of Plücker coordinates as:

$\begin{array}[]{llll}\phi(7,{\varpi^{\vee}_{4}})=p_{4},&\phi(8,{\varpi^{\vee}_{5}})=p_{3},&\phi(9,{\varpi^{\vee}_{6}})=p_{2},&\phi(11,{\varpi^{\vee}_{3}})=p_{5}^{\prime\prime},\\\
\phi(12,{\varpi^{\vee}_{4}})=q_{10}^{\prime\prime},&\phi(13,{\varpi^{\vee}_{2}})=p_{5}^{\prime},&\phi(14,{\varpi^{\vee}_{5}})=q_{10}^{\prime},&\phi(15,{\varpi^{\vee}_{7}})=p_{1},\\\
\phi(16,{\varpi^{\vee}_{4}})=q_{18}^{\prime},&\phi(17,{\varpi^{\vee}_{3}})=q_{12},&\phi(18,{\varpi^{\vee}_{1}})=p_{6}^{\prime\prime},&\phi(19,{\varpi^{\vee}_{6}})=q_{10},\\\
\phi(20,{\varpi^{\vee}_{5}})=q_{19}^{\prime},&\phi(21,{\varpi^{\vee}_{4}})=q_{28}^{\prime},&\phi(22,{\varpi^{\vee}_{2}})=q_{14},&\phi(23,{\varpi^{\vee}_{3}})=q_{23},\\\
\phi(24,{\varpi^{\vee}_{4}})=q_{40},&\phi(25,{\varpi^{\vee}_{5}})=q_{30},&\phi(26,{\varpi^{\vee}_{6}})=q_{20},&\phi(27,{\varpi^{\vee}_{7}})=p_{10},\end{array}$
(4.7)

whereas the frozen cluster variables can be expressed as:

$\begin{array}[]{llll}\phi(28,{\varpi^{\vee}_{1}})=q_{18},&\phi(29,{\varpi^{\vee}_{3}})=q_{36}^{\prime},&\phi(30,{\varpi^{\vee}_{4}})=q_{54},&\phi(31,{\varpi^{\vee}_{5}})=q_{45},\\\
\phi(32,{\varpi^{\vee}_{6}})=q_{36},&\phi(33,{\varpi^{\vee}_{2}})=q_{27},&\phi(-7,{\varpi^{\vee}_{7}})=p_{27},&\Delta^{T}_{{\varpi^{\vee}_{7}},{\varpi^{\vee}_{7}}}=p_{0}\end{array}$
(4.8)

###### Proof.

The methods are the same as in the proof of Lemma 4.3: use Algorithm 3.11
(with the direction change of Remark 4.2) for the cluster variables, and use
Algorithm 3.12 for the Plücker coordinate polynomials, and verify that the
equalities hold on $U^{\circ}_{-}$. ∎

In Figure 4.2 we provide the quiver, with vertices labelled by the cluster
variables above in Lemma 4.5. As we noted in Remark 4.4 before, the cluster
quiver is remarkably similar to a quiver defined in [CMP08], which we have
drawn to the right for comparison.

$1$$2$$3$$4$$5$$6$$7$$p_{27}$$p_{10}$$p_{1}$$q_{36}$$q_{20}$$q_{10}$$p_{2}$$q_{45}$$q_{30}$$q_{19}^{\prime}$$q_{10}^{\prime}$$p_{3}$$q_{54}$$q_{40}$$q_{28}^{\prime}$$q_{18}^{\prime}$$q_{10}^{\prime\prime}$$p_{4}$$q_{36}^{\prime}$$q_{23}$$q_{12}$$p_{5}^{\prime\prime}$$q_{27}$$q_{14}$$p_{5}^{\prime}$$q_{18}$$p_{6}^{\prime\prime}$$p_{0}$

$1$$2$$3$$4$$5$$6$$7$

Figure 4.2. The cluster quiver (left) as defined in [GLS08] and quiver (right)
defined in [CMP08] mentioned in Remark 4.4 for the Freudenthal variety. The
frozen variables are blue squares in the cluster quiver.

It is straightforward to check that the generalized minor is equal to the
given Plücker coordinate expression when restricted to $U_{-}^{\circ}$ using
Algorithms 3.11 and 3.12. However, it is difficult to obtain the Plücker
coordinate expressions themselves, and they are sufficiently complicated that
we could not obtain them by brute force. Instead, we used the following
algorithm. In a preprocessing step, we first compute the expansions of the
Plücker coordinates in torus coordinates on $U_{-}^{\circ}$, yielding a list
of homogeneous polynomials in the torus coordinates. Because we work in a
minuscule representation, the highest power appearing in any Plücker
coordinate expansion is $1$, see Theorem 2.10.

######  Algorithm 4.6.

  1. (1)

For a given generalized minor $\phi(i,{\varpi^{\vee}_{r_{i}}})$, first compute
its expansion in torus coordinates using Algorithm 3.11 with the direction
change of Remark 4.2.

  2. (2)

Fix the monomial order $<$ on $\mathbb{C}[a_{1},\dots,a_{\ell}]$ induced by
the degree-lexicographic order using the ordering $a_{1}>a_{2}>\dots>a_{\ell}$
of the torus coordinates.

  3. (3)

Let $L=\mathrm{in}_{<}(\phi(i,{\varpi^{\vee}_{r_{i}}}))$ denote the minimal
term of $\phi(i,{\varpi^{\vee}_{r_{i}}})$ with respect to $<$.

  4. (4)

Let $m$ denote the highest power of any torus coordinate appearing in $L$, and
set $L_{j}$ to be the product of all torus coordinates appearing in $L$ with
power at least $j$ for $1\leq j\leq m$.

  5. (5)

Next, let $p_{a_{j}}$ denote the Plücker coordinate with $L_{j}$ as leading
term for $1\leq j\leq m$.

  6. (6)

If $\phi(i,{\varpi^{\vee}_{r_{i}}})\pm\prod_{j=1}^{m}p_{a_{j}}\neq 0$ (the
sign is chosen to cancel the leading term of $L$), then repeat the above steps
(3)-(5) with the leading term of this new expression.

  7. (7)

Else if we obtain $0$, then we have found our desired Plücker coordinate
expression.

While it is not clear whether step (5) is always possible, we were able to
perform it in every case that we computed. We now give an example of this
reduction procedure for $\phi(23,{\varpi^{\vee}_{3}})$.

######  Example 4.7.

Consider $f=\phi(23,{\varpi^{\vee}_{3}})$. We compute its torus expansion
(including only the last few terms for brevity, but see Appendix A for the
full expansion):

$\displaystyle\dots+a_{5}a_{7}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}$
$\displaystyle+a_{5}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}$
$\displaystyle+a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}.$

Its $<$-minimal term is

$\mathrm{in}_{<}(\phi(23,{\varpi^{\vee}_{3}}))=a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3},$

which has highest exponent $m=3$. Thus we have

$\displaystyle L_{1}=a_{27}$ $\displaystyle
L_{2}=a_{18}a_{23}a_{24}a_{25}a_{26}a_{27}$ $\displaystyle
L_{3}=a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}a_{27}$

These are the minimal terms of $p_{1},p_{6}^{\prime\prime},p_{16}$
respectively. $f-p_{1}p_{6}^{\prime\prime}p_{16}\neq 0$, and we find that its
minimal term is

$-a_{10}a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3},$

which has highest exponent $m=3$. Thus we have

$\displaystyle L_{1}=a_{27}$ $\displaystyle
L_{2}=a_{23}a_{24}a_{25}a_{26}a_{27}$ $\displaystyle
L_{3}=a_{10}a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}a_{27}$

which are the minimal terms of $p_{1},p_{5}^{\prime\prime},p_{17}$
respectively.
$f-p_{1}p_{6}^{\prime\prime}p_{16}+p_{1}p_{5}^{\prime\prime}p_{17}\neq 0$, and
we find that its minimal term is

$-a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{2},$

which has highest exponent $m=2$. Thus we have

$\displaystyle L_{1}=a_{18}a_{23}a_{24}a_{25}a_{26}a_{27}$ $\displaystyle
L_{2}=a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}a_{27}$

which are the minimal terms of $p_{6}^{\prime\prime},p_{17}^{\prime}$
respectively.
$f-p_{1}p_{6}^{\prime\prime}p_{16}+p_{1}p_{5}^{\prime\prime}p_{17}+p_{6}^{\prime\prime}p_{17}^{\prime}\neq
0$, and we find that its minimal term is

$a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{2}$

which has highest exponent $m=2$. Thus we have

$\displaystyle L_{1}=a_{23}a_{24}a_{25}a_{26}a_{27}$ $\displaystyle
L_{2}=a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}a_{27}$

which are the minimal terms of $p_{5}^{\prime\prime},p_{18}$ respectively. Now
$f-p_{1}p_{6}^{\prime\prime}p_{16}+p_{1}p_{5}^{\prime\prime}p_{17}+p_{6}^{\prime\prime}p_{17}^{\prime}-p_{5}^{\prime\prime}p_{18}=0$,
so we obtain the expression

$\phi(23,{\varpi^{\vee}_{3}})=p_{5}^{\prime\prime}(p_{18}-p_{1}p_{17})-p_{6}^{\prime\prime}(p_{17}^{\prime}-p_{1}p_{16})$

exactly as given in Lemma 4.5. $\lozenge$

##  5\. Valuations, Khovanskii bases, and Newton-Okounkov bodies

Algorithm 4.6 is very reminiscent of the _subduction algorithm_ for
_Khovanskii bases_. In this section we will show that the Plücker coordinates
form a Khovanskii basis for $\mathbb{C}[\mathbb{X}^{\vee}]$, and we then use
this to give a _Newton-Okounkov body_ for $\mathbb{X}^{\vee}$. Throughout this
section, we denote by $A$ a domain that is also a finitely generated algebra
over an algebraically closed field $k$, and by $\Gamma$ a group with an
ordering $\succ$ of its elements that respects the group operation.

######  Definition 5.1 ([KM19], Definition 2.1).

A _valuation_ is a function $\nu:A\setminus\\{0\\}\rightarrow\Gamma$ such that

  1. (1)

For all $f,g\in A\setminus\\{0\\}$ with $f+g\neq 0$, we have
$\nu(f+g)\succeq\min_{\succ}(\nu(f),\nu(g))$.

  2. (2)

For all $f,g\in A\setminus\\{0\\}$, we have $\nu(fg)=\nu(f)+\nu(g)$.

  3. (3)

For all $f\in A\setminus\\{0\\}$ and $0\neq c\in k$, we have $\nu(cf)=\nu(f)$.

For $g\in\Gamma$, let $F_{\succeq g}$ be the vector space $\\{f\in
A\setminus\\{0\\}\mid\nu(f)\succeq g\\}\cup\\{0\\}$, and define $F_{\succ g}$
analogously. We then define the _associated graded algebra_

$\mathrm{gr}_{\nu}(A)=\bigoplus_{g\in\Gamma}F_{\succeq g}/F_{\succ g}.$

If for all $g\in\Gamma$, the quotient $F_{\succeq g}/F_{\succ g}$ is at most
one-dimensional, then we say that $\nu$ has _one-dimensional leaves_.

The following lemma gives an important family of valuations with one-
dimensional leaves.

######  Lemma 5.2.

Let $S\subset k[x_{1},\dots,x_{n}]$ be a finitely generated subalgebra of a
polynomial ring, and let $\nu$ be the valuation sending $f\in S$ to the
exponent vector of its degree-lexicographically minimal term. Then $\nu$ has
one-dimensional leaves.

###### Proof.

Let $g\in\mathbb{Z}^{n}$, and suppose $a,b\in F_{\succeq g}\setminus F_{\succ
g}$. (If $a,b\in F_{\succ g}$, then $a,b=0$ in the quotient $F_{\succeq
g}/F_{\succ g}$.) Then $\nu(a)=\nu(b)=g$, so the minimal terms of $a$ and $b$
have the same exponent vector. Thus there is some constant $c\in k$ such that
the minimal terms of the expansions of $a$ and $cb$ are equal, so that
$\nu(a-cb)\succ g$, and hence $a-cb\in F_{\succ g}$. Thus $a=cb$ in
$F_{\succeq g}/F_{\succ g}$, and we conclude that $F_{\succeq g}/F_{\succ g}$
has dimension at most $1$. ∎

We next define _Khovanskii bases_ and the _Newton-Okounkov body_ associated to
a valuation.

######  Definition 5.3 ([KM19], Definition 2.5).

A subset $B\subset A$ is called a _Khovanskii basis_ for $A$ with respect to
the valuation $\nu$ if the image of $B$ in $\mathrm{gr}_{\nu}(A)$ is a set of
algebra generators for $\mathrm{gr}_{\nu}(A)$. Khovanskii bases give a
_subduction algorithm_ for writing elements of $A$ as polynomials in the
elements of $B$.

######  Definition 5.4 ([KM19], Definition 2.21).

Let $A$ further be positively graded by $\mathbb{Z}_{>0}$. The _Newton-
Okounkov body_ $\Delta(A,\nu)$ associated to $A$ and a valuation $\nu$ is the
closed convex set

$\Delta(A,\nu)=\overline{\mathrm{conv}\\!\left(\bigcup_{i>0}\tfrac{1}{i}{\nu(A_{i}\setminus\\{0\\})}\right)}$

where $A_{i}$ denotes the set of homogeneous elements of $A$ of degree $i$.

Although the Newton-Okounkov body associated to a valuation $\nu$ is generally
very far from being a polytope, the following proposition gives an important
family of examples where $\Delta(A,\nu)$ is a polytope.

######  Proposition 5.5 ([KM19], Corollary 2.24, Remark 2.25).

Let $A$ be the homogeneous coordinate ring of a projective variety $Y$, and
let $\nu$ be a valuation on $A$ with one-dimensional leaves. Then
$\deg(Y)=\mathrm{Vol}(\Delta(A,\nu))$, where $\mathrm{Vol}$ refers to the
normalized lattice volume so that standard lattice $d$-simplex has volume
$\frac{1}{d!}$. If furthermore $A$ has a finite Khovanskii basis with respect
to $\nu$, then $\Delta(A,\nu)$ is a rational polytope and $\mathrm{Proj}(A)$
has a degeneration to a toric variety whose normalization is the toric variety
associated to $\Delta(A,\nu)$.

Now let $A=\mathbb{C}[\mathbb{X}^{\vee}]$ be the coordinate ring of the Cayley
plane ($n=6$) or the Freudenthal variety ($n=7$), with
$\mathbb{Z}_{>0}$-grading given by the total degree in Plücker coordinates. We
use the degree-lexicographic term order $<$ on
$\mathbb{C}[a_{1},\dots,a_{\ell}]$ from Algorithm 4.6 to define a valuation
$\nu_{n}:A\rightarrow\mathbb{Z}^{\ell}$ as follows. Given a polynomial $f$ in
the Plücker coordinates, compute its expansion in the coordinates of
$U_{+}^{\circ}$ using Algorithm 3.11, and define $\nu_{n}(f)$ to be the
exponent vector of the degree-lexicographic minimal term $\mathrm{in}_{<}(f)$
of $f$. It is straightforward to check that $\nu_{n}$ is a valuation.
Furthermore, because the image of $\nu_{n}$ in
$\mathbb{C}[a_{1},\dots,a_{\ell}]$ is a finitely generated subalgebra, Lemma
5.2 implies that $\nu_{n}$ has one-dimensional leaves. Proposition 5.5 thus
shows that the normalized volume of $\Delta(A,\nu_{n})$ is equal to the degree
of $\mathbb{X}^{\vee}$. By Definition 5.4, the suitably scaled valuation of
any homogeneous element of $A$ is contained in the Newton-Okounkov body
$\Delta(A,\nu_{n})$, and thus so is the convex hull of the set of suitably
scaled valuations of any finite subset of homogeneous elements of $A$. In
particular, if we can find $B\subset A$ such that

$\delta(B,\nu_{n})=\mathrm{conv}\\!\left(\bigcup_{i>0}\tfrac{1}{i}{\nu_{n}(B\cap
A_{i}\setminus\\{0\\})}\right)$

has normalized volume equal to the degree of $\mathbb{X}^{\vee}$, then
$\delta(B,\nu_{n})=\Delta(A,\nu_{n})$.

######  Proposition 5.6.

The Newton-Okounkov body $\Delta(A,\nu_{n})$ is equal to the convex hull of
the valuations of the Plücker coordinates on $\mathbb{X}^{\vee}$.

###### Proof.

The proof is by direct computation. We give the valuations of Plücker
coordinates in Appendix B, and we used Polymake ([GJ00, AGH+17]) to verify
that their convex hulls have the correct volumes.

For the Cayley plane ($n=6$), we obtain a $16$-dimensional polytope with
volume $78$, which is equal to the degree of the Cayley plane. For the
Freudenthal variety ($n=7$), we obtain a $27$-dimensional polytope with volume
$13110$, which is equal to the degree of the Freudenthal variety. We conclude
that in both cases, the respective Newton-Okounkov bodies are given by the
convex hulls of the valuations of the respective Plücker coordinates. ∎

In particular, $\Delta(A,\nu_{n})$ is a rational polytope (in fact a
$0/1$-polytope, which follows from the fact that we used a minuscule
representation) with vertices given by the valuations of Plücker coordinates.
Moreover, we computed in each case that each polytope contains no lattice
points other than its vertices.

At this point, we would like to conclude that the Plücker coordinates form
Khovanskii bases. However, the converse direction of Proposition 5.5 requires
some further assumptions, e.g. as in [RW19, Proposition 17.4]. These are
satisfied in our case because we work with the projectively normal minimal
(Plücker) embeddings of homogeneous spaces $P\backslash G$ into the
projectivizations of the corresponding irreducible representations
$V({\varpi^{\vee}_{n}})$, and because our Newton-Okounkov bodies do not
contain any extra lattice points. We summarize this discussion in the
following proposition.

######  Proposition 5.7.

The Plücker coordinates give Khovanskii bases for the coordinate rings
$\mathbb{C}[\mathbb{X}^{\vee}]$ with respect to the valuations $\nu_{n}$.

######  Remark 5.8.

Even though the Plücker coordinates give a finite Khovanskii basis for
$\mathbb{C}[\mathbb{X}^{\vee}]$ with respect to the valuation $\nu_{n}$, this
is not sufficient to guarantee that Algorithm 4.6 terminates for any input. If
this is the case, then in the language of Definition 3.8 and Section 5 of
[KM19], $\nu_{n}$ is called a _subductive valuation_ and corresponds to a
_prime cone_ of the associated tropical variety. While it would be interesting
to investigate this further, we do not pursue this here.

None of the above statements depended on the exceptional types in a crucial
way. For any homogeneous space $P\backslash G$, we may choose a reduced
expression for $w^{P}$ which thus gives a parametrization of the corresponding
$U_{+}^{\circ}$. We then define a valuation $\nu$ on $A=\mathbb{C}[P\backslash
G]$ by expanding any element in the coordinates of $U_{+}^{\circ}$ and taking
its degree-lexicographic minimal term, and this valuation will have one-
dimensional leaves by an analogous proof to the proof of Lemma 5.2. We have
not verified it ourselves, but we believe the following statement holds
_mutatis mutandis_ :

######  Conjecture 5.9.

Let $X=P\backslash G$ be a minuscule homogeneous space with coordinate ring
$A$ and valuation $\nu$ associated to a choice of reduced expression for
$w^{P}$ as defined above. Then the Plücker coordinates of $X$ form a
Khovanskii basis for $A$, and the associated Newton-Okounkov body
$\Delta(A,\nu)$ is a rational polytope which is equal to the convex hull of
the valuations of Plücker coordinates.

When $X$ is not minuscule, the Plücker coordinates are more difficult to
understand, the valuations of the Plücker coordinates will no longer be
$0/1$-vectors, and the Newton-Okounkov body may be less well-behaved. However,
it would still be interesting to investigate to what extent the above
conjecture holds. Moreover if $X$ is neither minuscule nor cominuscule,
$w^{P}$ is no longer _fully commutative_ , so that different reduced
expressions may not be related by swapping adjacent transpositions, and it
would be interesting to investigate the relationship between the Newton-
Okounkov bodies obtained for different reduced expressions, and whether there
is any relation to cluster mutation.

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##  Appendix A Torus expansion example

We give an example torus expansion computation for the Freudenthal variety,
computing the expansion of the generalized minor
$\Delta_{{\varpi^{\vee}_{3}},u({\varpi^{\vee}_{3}})}$, where $u=(w_{0})_{\geq
23}w_{0}$ and $(w_{0})_{\geq 23}=s_{{\ell_{0}}}\dots s_{23}$. We represent the
$8645$-dimensional representation $V({\varpi^{\vee}_{3}})$ as a
subrepresentation of the alternating square of the adjoint representation
$\bigwedge^{\\!2}V({\varpi^{\vee}_{1}})\cong V({\varpi^{\vee}_{3}})\oplus
V({\varpi^{\vee}_{1}})$.

The action of $u$ on ${\varpi^{\vee}_{3}}$ gives the coweight vector
$\mu^{\vee}_{u}=(0,1,0,0,0,1,-3)$ (written in the basis of fundamental
coweights), and the corresponding action on $v_{{\varpi^{\vee}_{3}}}^{+}$
gives $v_{\mu^{\vee}_{u}}\in V({\varpi^{\vee}_{3}})$. In principle, we then
compute all sequences $(i_{1},\dots,i_{k})$ such that
$e^{\vee}_{i_{1},\dots,i_{k}}\cdot v_{\mu^{\vee}_{u}}$ is a multiple of
$v_{{\varpi^{\vee}_{3}}}^{+}$, check which sequences appear in the expansion
of $u_{+}$, and then collect the coefficients. However, there are too many
sequences to compute directly and most of them do not appear in $u_{+}$. In
our computations, we use this to facilitate computations, and we determine
that only $43$ sequences have nonzero contributions (we suppress the commas
and brackets, i.e. we write $345\ldots$ instead of $(3,4,5,\ldots)$):

$\begin{array}[]{lll}34561342574315423456677&34561327443155423456677&34561342743155423456677\\\
34561274331544234556677&34561327431544234556677&34561327443154234556677\\\
34561342743154234556677&34561274331542344556677&34561327431542344556677\\\
34562743115423344556677&34561274315423344556677&34561342543154234566777\\\
34561342431554234566777&34561324431554234566777&34513424316554234566777\\\
34513244316554234566777&34132544316554234566777&34561243315442345566777\\\
34561324315442345566777&34512433165442345566777&34125433165442345566777\\\
31425433165442345566777&34513243165442345566777&34132543165442345566777\\\
34561342431542345566777&34561324431542345566777&34513424316542345566777\\\
34513244316542345566777&34132544316542345566777&34561243315423445566777\\\
34561324315423445566777&34512433165423445566777&34125433165423445566777\\\
31425433165423445566777&34513243165423445566777&34132543165423445566777\\\
34562431154233445566777&34561243154233445566777&34524311654233445566777\\\
\\!\framebox{\\!$34254311654233445566777$\\!}\\!&34512431654233445566777&34125431654233445566777\\\
31425431654233445566777.\end{array}$

As an example, we consider the sequence
$(3,4,2,5,4,3,1,1,6,5,4,2,3,3,4,4,5,5,6,6,7,7,7)$, which we boxed above, to
determine which monomials this sequence contributes to the torus expansion.
Corresponding to our choice of $w^{P}$, we have $u_{+}$ of the form:

$\\{x^{\vee}_{r_{1}}(a_{1})x^{\vee}_{r_{2}}(a_{2})\cdots
x^{\vee}_{r_{27}}(a_{27})~{}|~{}a_{i}\in\mathbb{C}^{*}\\},$

where we fixed
$(r_{1},r_{2},\ldots,r_{27})=(7,6,5,4,3,2,4,5,6,1,3,4,2,5,7,4,3,1,6,5,4,2,3,4,5,6,7)$.
From this we find by inspection one term of $u_{+}$ which sends
$v_{\mu^{\vee}_{u}}$ to $v_{{\varpi^{\vee}_{3}}}^{+}$. The coefficient of this
term is the following polynomial in torus coordinates:

$\displaystyle
a_{5}a_{7}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}$
$\displaystyle+a_{5}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}$
$\displaystyle+a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}$
$\displaystyle=(a_{5}a_{7}+a_{5}a_{12}+a_{11}a_{12})a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}.$

This polynomial is obtained as follows from $u_{+}$. First, we box those terms
in the expansion of each $x^{\vee}_{r_{i}}(a_{i})$ which must be taken to
obtain the desired $e^{\vee}_{i_{1}\dots i_{k}}$.

$\displaystyle(\framebox{$1$}+a_{1}e^{\vee}_{7}+\dots)(\framebox{$1$}+a_{2}e^{\vee}_{6}+\dots)(\framebox{$1$}+a_{3}e^{\vee}_{5}+\dots)(\framebox{$1$}+a_{4}e^{\vee}_{4}+\dots)(1+a_{5}e^{\vee}_{3}+\dots)$
$\displaystyle(\framebox{$1$}+a_{6}e^{\vee}_{2}+\dots)(1+a_{7}e^{\vee}_{4}+\dots)(\framebox{$1$}+a_{8}e^{\vee}_{5}+\dots)(\framebox{$1$}+a_{9}e^{\vee}_{6}+\dots)(\framebox{$1$}+a_{10}e^{\vee}_{1}+\dots)$
$\displaystyle(1+a_{11}e^{\vee}_{3}+\dots)(1+a_{12}e^{\vee}_{4}+\dots)(1+\framebox{$a_{13}e^{\vee}_{2}$}+\dots)(1+\framebox{$a_{14}e^{\vee}_{5}$}+\dots)(\framebox{$1$}+a_{15}e^{\vee}_{7}+\dots)$
$\displaystyle(1+\framebox{$a_{16}e^{\vee}_{4}$}+\dots)(1+\framebox{$a_{17}e^{\vee}_{3}$}+\dots)(1+a_{18}e^{\vee}_{1}+\framebox{$\tfrac{1}{2!}{a_{18}^{2}(e^{\vee}_{1})^{2}}$}+\dots)$
$\displaystyle(1+\framebox{$a_{19}e^{\vee}_{6}$}+\dots)(1+\framebox{$a_{20}e^{\vee}_{5}$}+\dots)(1+\framebox{$a_{21}e^{\vee}_{4}$}+\dots)(1+\framebox{$a_{22}e^{\vee}_{2}$}+\dots)$
$\displaystyle(1+a_{23}e^{\vee}_{3}+\framebox{$\tfrac{1}{2!}{a_{23}^{2}(e^{\vee}_{3})^{2}}$}+\dots)(1+a_{24}e^{\vee}_{4}+\framebox{$\tfrac{1}{2!}{a_{24}^{2}(e^{\vee}_{4})^{2}}$}+\dots)$
$\displaystyle(1+a_{25}e^{\vee}_{5}+\framebox{$\tfrac{1}{2!}{a_{25}^{2}(e^{\vee}_{5})^{2}}$}+\dots)(1+a_{26}e^{\vee}_{6}+\framebox{$\tfrac{1}{2!}{a_{26}^{2}(e^{\vee}_{6})^{2}}$}+\dots)$
$\displaystyle(1+a_{27}e^{\vee}_{7}+\tfrac{1}{2!}{a_{27}^{2}(e^{\vee}_{7})^{2}}+\framebox{$\tfrac{1}{3!}{a_{27}^{3}(e^{\vee}_{7})^{3}}$}+\dots)$

Removing the unused terms gives:

$\displaystyle(1)(1)(1)(1)(1+a_{5}e^{\vee}_{3}+\dots)(1)(1+a_{7}e^{\vee}_{4}+\dots)(1)(1)(1)(1+a_{11}e^{\vee}_{3}+\dots)$
$\displaystyle(1+a_{12}e^{\vee}_{4}+\dots)(a_{13}e^{\vee}_{2})(a_{14}e^{\vee}_{5})(1)(a_{16}e^{\vee}_{4})(a_{17}e^{\vee}_{3})\Bigl{(}\tfrac{1}{2!}{a_{18}^{2}(e^{\vee}_{1})^{2}}\Bigr{)}(a_{19}e^{\vee}_{6})(a_{20}e^{\vee}_{5})$
$\displaystyle(a_{21}e^{\vee}_{4})(a_{22}e^{\vee}_{2})\Bigl{(}\tfrac{1}{2!}{a_{23}^{2}(e^{\vee}_{3})^{2}}\Bigr{)}\Bigl{(}\tfrac{1}{2!}{a_{24}^{2}(e^{\vee}_{4})^{2}}\Bigr{)}\Bigl{(}\tfrac{1}{2!}{a_{25}^{2}(e^{\vee}_{5})^{2}}\Bigr{)}\Bigl{(}\tfrac{1}{2!}{a_{26}^{2}(e^{\vee}_{6})^{2}}\Bigr{)}\Bigl{(}\tfrac{1}{3!}{a_{27}^{3}(e^{\vee}_{7})^{3}}\Bigr{)},$

and we simplify to obtain:

$\displaystyle=(1+a_{5}e^{\vee}_{3}+\dots)(1+a_{7}e^{\vee}_{4}+\dots)(1+a_{11}e^{\vee}_{3}+\dots)(1+a_{12}e^{\vee}_{4}+\dots)$
$\displaystyle\left(\tfrac{1}{96}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}(e^{\vee}_{25431^{2}65423^{2}4^{2}5^{2}6^{2}7^{3}})\right).$

Expanding the first four terms above up to second order gives:

$\displaystyle 1$
$\displaystyle+(a_{5}+a_{11})e^{\vee}_{3}+(a_{7}+a_{12})e^{\vee}_{4}+(a_{5}a_{11})e^{\vee}_{3^{2}}+(a_{7}a_{12})e^{\vee}_{4^{2}}$
$\displaystyle+\tfrac{1}{4}\framebox{$\\!(a_{5}a_{7}+a_{5}a_{12}+a_{11}a_{12})\\!$}e^{\vee}_{34}+\tfrac{1}{4}(a_{7}a_{11})e^{\vee}_{43}+\dots,$

and we have boxed the relevant term in this last formula. Altogether, we
obtain:

$\Bigl{(}\tfrac{1}{384}{a_{5}a_{7}+a_{5}a_{12}+a_{11}a_{12})a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}}\Bigr{)}e^{\vee}_{3425431^{2}65423^{2}4^{2}5^{2}6^{2}7^{3}},$

and we compute
$\frac{1}{384}e^{\vee}_{3425431^{2}65423^{2}4^{2}5^{2}6^{2}7^{3}}v_{\varpi}^{+}=v_{{\varpi^{\vee}_{3}}}^{+}$,
so that the contribution of this term is

$\displaystyle\Bigl{\langle}(a_{5}a_{7}+a_{5}a_{12}+a_{11}a_{12})a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}v_{{\varpi^{\vee}_{3}}}^{+},v_{{\varpi^{\vee}_{3}}}^{+}\Bigr{\rangle}$
$\displaystyle=(a_{5}a_{7}+a_{5}a_{12}+a_{11}a_{12})a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}.$

The remaining $42$ sequences yield analogous computations, though they only
contribute one term each to the resulting polynomial which thus has in total
45 terms (the final three terms are the ones we computed explicitly above):

$\begin{array}[]{l}a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{14}a_{15}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{15}a_{16}a_{17}a_{18}a_{20}^{2}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{15}a_{16}^{2}a_{17}a_{18}a_{20}^{2}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{15}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{15}a_{16}^{2}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{15}a_{16}a_{17}a_{18}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{13}a_{15}a_{16}a_{17}^{2}a_{18}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{15}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{13}a_{15}a_{16}a_{17}^{2}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{13}a_{15}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{13}a_{15}a_{16}a_{17}a_{18}^{2}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{2}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{16}a_{17}a_{18}a_{20}^{2}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{16}^{2}a_{17}a_{18}a_{20}^{2}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{11}a_{12}a_{13}a_{16}a_{17}a_{18}a_{19}a_{20}^{2}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{11}a_{13}a_{16}^{2}a_{17}a_{18}a_{19}a_{20}^{2}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{10}a_{11}a_{13}a_{14}a_{16}^{2}a_{17}a_{18}a_{19}a_{20}^{2}a_{21}a_{22}a_{23}a_{24}a_{25}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{12}a_{13}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{16}^{2}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{11}a_{12}a_{13}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{11}a_{13}a_{16}^{2}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{10}a_{11}a_{13}a_{14}a_{16}^{2}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{16}a_{17}a_{18}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{13}a_{16}a_{17}^{2}a_{18}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{11}a_{13}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{10}a_{11}a_{13}a_{14}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{13}a_{16}a_{17}^{2}a_{18}a_{19}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{10}a_{13}a_{14}a_{16}a_{17}^{2}a_{18}a_{19}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{10}a_{12}a_{13}a_{14}a_{16}a_{17}^{2}a_{18}a_{19}a_{20}a_{21}^{2}a_{22}a_{23}a_{24}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{11}a_{13}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{13}a_{16}a_{17}^{2}a_{18}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{11}a_{13}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{10}a_{11}a_{13}a_{14}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{13}a_{16}a_{17}^{2}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{10}a_{13}a_{14}a_{16}a_{17}^{2}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{10}a_{12}a_{13}a_{14}a_{16}a_{17}^{2}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{10}a_{13}a_{16}a_{17}a_{18}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{9}a_{13}a_{16}a_{17}a_{18}^{2}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{10}a_{13}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{10}a_{13}a_{14}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{10}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{8}a_{13}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{7}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{5}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}\\\
{}+a_{11}a_{12}a_{13}a_{14}a_{16}a_{17}a_{18}^{2}a_{19}a_{20}a_{21}a_{22}a_{23}^{2}a_{24}^{2}a_{25}^{2}a_{26}^{2}a_{27}^{3}.\end{array}$

##  Appendix B Plücker coordinate valuations

Below are the valuations of the Plücker coordinates for the Cayley plane (we
again write $001\ldots$ for $(0,0,1,\ldots)$):

$\begin{array}[]{lll}\nu_{6}(p_{0\hphantom{1}}\\!\\!)\,\,=0000000000000000,&\nu_{6}(p_{1\hphantom{1}}\\!\\!)\,\,=0000000000000001,&\nu_{6}(p_{2\hphantom{1}}\\!\\!)\,\,=0000000000000011,\\\
\nu_{6}(p_{3\hphantom{1}}\\!\\!)\,\,=0000000000000111,&\nu_{6}(p_{4\hphantom{1}}^{\prime}\\!\\!)\,\,=0000000000001111,&\nu_{6}(p_{4\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=0000000000010111,\\\
\nu_{6}(p_{5\hphantom{1}}^{\prime}\\!\\!)\,\,=0000000000011111,&\nu_{6}(p_{5\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=0000000100010111,&\nu_{6}(p_{6\hphantom{1}}^{\prime}\\!\\!)\,\,=0000000000111111,\\\
\nu_{6}(p_{6\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=0000000100011111,&\nu_{6}(p_{7\hphantom{1}}^{\prime}\\!\\!)\,\,=0000000001111111,&\nu_{6}(p_{7\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=0000000100111111,\\\
\nu_{6}(p_{8\hphantom{1}}\\!\\!)\,\,=0000000011111111,&\nu_{6}(p_{8\hphantom{1}}^{\prime}\\!\\!)\,\,=0000000101111111,&\nu_{6}(p_{8\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=0000001100111111,\\\
\nu_{6}(p_{9\hphantom{1}}^{\prime}\\!\\!)\,\,=0000000111111111,&\nu_{6}(p_{9\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=0000001101111111,&\nu_{6}(p_{10}^{\prime})=0000001111111111,\\\
\nu_{6}(p_{10}^{\prime\prime})=0000011101111111,&\nu_{6}(p_{11}^{\prime})=0000011111111111,&\nu_{6}(p_{11}^{\prime\prime})=0001011101111111,\\\
\nu_{6}(p_{12}^{\prime})=0000111111111111,&\nu_{6}(p_{12}^{\prime\prime})=0001011111111111,&\nu_{6}(p_{13})=0001111111111111,\\\
\nu_{6}(p_{14})=0011111111111111,&\nu_{6}(p_{15})=0111111111111111,&\nu_{6}(p_{16})=1111111111111111.\end{array}$

The convex hull of the above points forms
$\Delta(\mathbb{C}[\mathbb{X}^{\vee}],\nu_{6})$. We compute with Polymake
([GJ00, AGH+17]) that this is a $16$-dimensional polytope with normalized
volume $78$ and f-vector:

$(27,297,1858,7598,21884,46415,74521,92095,88372,65979,38160,16900,5612,1349,221,22).$

Below are the valuations of the Plücker coordinates for the Freudenthal
variety:

$\begin{array}[]{ll}\nu_{7}(p_{0\hphantom{1}}\\!\\!)\,\,=000000000000000000000000000,&\nu_{7}(p_{1\hphantom{1}}\\!\\!)\,\,=000000000000000000000000001,\\\
\nu_{7}(p_{2\hphantom{1}}\\!\\!)\,\,=000000000000000000000000011,&\nu_{7}(p_{3\hphantom{1}}\\!\\!)\,\,=000000000000000000000000111,\\\
\nu_{7}(p_{4\hphantom{1}}\\!\\!)\,\,=000000000000000000000001111,&\nu_{7}(p_{5\hphantom{1}}^{\prime}\\!\\!)\,\,=000000000000000000000101111,\\\
\nu_{7}(p_{5\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=000000000000000000000011111,&\nu_{7}(p_{6\hphantom{1}}^{\prime}\\!\\!)\,\,=000000000000000000000111111,\\\
\nu_{7}(p_{6\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=000000000000000001000011111,&\nu_{7}(p_{7\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=000000000000000001000111111,\\\
\nu_{7}(p_{7\hphantom{1}}^{\prime}\\!\\!)\,\,=000000000000000000001111111,&\nu_{7}(p_{8\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=000000000000000001001111111,\\\
\nu_{7}(p_{8\hphantom{1}}^{\prime}\\!\\!)\,\,=000000000000000000011111111,&\nu_{7}(p_{9\hphantom{1}}^{\prime\prime}\\!\\!)\,\,=000000000000000011001111111,\\\
\nu_{7}(p_{9\hphantom{1}}^{\prime}\\!\\!)\,\,=000000000000000001011111111,&\nu_{7}(p_{9\hphantom{1}}\\!\\!)\,\,=000000000000000000111111111,\\\
\nu_{7}(p_{10}^{\prime\prime})=000000000000000011011111111,&\nu_{7}(p_{10}^{\prime})=000000000000000001111111111,\\\
\nu_{7}(p_{10})=000000000000100000111111111,&\nu_{7}(p_{11}^{\prime\prime})=000000000000000111011111111,\\\
\nu_{7}(p_{11}^{\prime})=000000000000000011111111111,&\nu_{7}(p_{11})=000000000000100001111111111,\\\
\nu_{7}(p_{12}^{\prime\prime})=000000000000001111011111111,&\nu_{7}(p_{12}^{\prime})=000000000000000111111111111,\\\
\nu_{7}(p_{12})=000000000000100011111111111,&\nu_{7}(p_{13}^{\prime\prime})=000000000000001111111111111,\\\
\nu_{7}(p_{13}^{\prime})=000000000000010111111111111,&\nu_{7}(p_{13})=000000000000100111111111111,\\\
\nu_{7}(p_{14})=000000000000011111111111111,&\nu_{7}(p_{14}^{\prime})=000000000000101111111111111,\\\
\nu_{7}(p_{14}^{\prime\prime})=000000000000110111111111111,&\nu_{7}(p_{15})=000000000001011111111111111,\\\
\nu_{7}(p_{15}^{\prime})=000000000000111111111111111,&\nu_{7}(p_{15}^{\prime\prime})=000000001000110111111111111,\\\
\nu_{7}(p_{16})=000000000011011111111111111,&\nu_{7}(p_{16}^{\prime})=000000000001111111111111111,\\\
\nu_{7}(p_{16}^{\prime\prime})=000000001000111111111111111,&\nu_{7}(p_{17})=000000000111011111111111111,\\\
\nu_{7}(p_{17}^{\prime})=000000000011111111111111111,&\nu_{7}(p_{17}^{\prime\prime})=000000001001111111111111111,\\\
\nu_{7}(p_{18})=000000000111111111111111111,&\nu_{7}(p_{18}^{\prime})=000000001011111111111111111,\\\
\nu_{7}(p_{18}^{\prime\prime})=000000011001111111111111111,&\nu_{7}(p_{19}^{\prime})=000000001111111111111111111,\\\
\nu_{7}(p_{19}^{\prime\prime})=000000011011111111111111111,&\nu_{7}(p_{20}^{\prime})=000000011111111111111111111,\\\
\nu_{7}(p_{20}^{\prime\prime})=000000111011111111111111111,&\nu_{7}(p_{21}^{\prime})=000000111111111111111111111,\\\
\nu_{7}(p_{21}^{\prime\prime})=000001111011111111111111111,&\nu_{7}(p_{22}^{\prime\prime})=000001111111111111111111111,\\\
\nu_{7}(p_{22}^{\prime})=000010111111111111111111111,&\nu_{7}(p_{23})=000011111111111111111111111,\\\
\nu_{7}(p_{24})=000111111111111111111111111,&\nu_{7}(p_{25})=001111111111111111111111111,\\\
\nu_{7}(p_{26})=011111111111111111111111111,&\nu_{7}(p_{27})=111111111111111111111111111.\end{array}$

The convex hull of the above points forms
$\Delta(\mathbb{C}[\mathbb{X}^{\vee}],\nu_{7})$. We compute with Polymake
([GJ00, AGH+17]) that this is a $27$-dimensional polytope with volume $13110$.
We were unable to compute the f-vector due to computational limitations (our
computations exceeded 100GB of RAM).