# Push-pull operators on convex polytopes

Valentina Kiritchenko<EMAIL_ADDRESS>Laboratory of Algebraic Geometry and
Faculty of Mathematics
National Research University Higher School of Economics, Russian Federation
Usacheva str. 6, 119048 Moscow, Russia  Institute for Information Transmission
Problems, Moscow, Russia

###### Abstract.

A classical result of Schubert calculus is an inductive description of
Schubert cycles using divided difference (or push-pull) operators in Chow
rings. We define convex geometric analogs of push-pull operators and describe
their applications to the theory of Newton–Okounkov convex bodies. Convex
geometric push-pull operators yield an inductive construction of
Newton–Okounkov polytopes of Bott–Samelson varieties. In particular, we
construct a Minkowski sum of Feigin–Fourier–Littelmann–Vinberg polytopes using
convex geometric push-pull operators in type $A$.

###### Key words and phrases:

push-pull operators, Newton–Okounkov polytopes, Cayley sum

The study has been partially funded by the Russian Academic Excellence Project
’5-100’ and by RSF grant 19-11-00056 (Sections 3, 5).

##  1\. Introduction

Let $X$ be a smooth algebraic variety, and $E\to X$ a vector bundle of rank
two on $X$. Define the projective line fibration $Y=\mathbb{P}(E)$ as the
variety of all lines in $E$. The natural projection $\pi:Y\to X$ induces the
pull-back $\pi^{*}:A^{*}(X)\to A^{*}(Y)$ and the push-forward
$\pi_{*}:A^{*}(Y)\to A^{*-1}(X)$ (aka transfer or Gysin map) in the
(generalized) cohomology rings of $X$ and $Y$. The push-pull operator
$\pi^{*}\pi_{*}:A^{*}(Y)\to A^{*-1}(Y)$ is a homomorphism of
$A^{*}(X)$-modules, and can be described explicitly via Quillen–Vishik formula
for any algebraic oriented cohomology theory $A^{*}$ (such as Chow ring,
K-theory or algebraic cobordism). Push-pull operators are used extensively in
representation theory (Demazure operators) and in Schubert calculus (divided
difference operators). We discuss convex geometric counterparts of push-pull
operators and their applications in the theory of Newton–Okounkov convex
bodies and representation theory.

Convex geometric push-pull operators are motivated by the study of Newton-
Okounkov polytopes of flag and Bott–Samelson varieties [Ki18]. Namely, if
$Y=G/B$ is the complete flag variety for a connected reductive group $G$, then
there is a natural projective line fibration $\pi_{i}:Y\to G/P_{i}$ for every
simple root $\alpha_{i}$. Here $P_{i}\subset G$ denotes the minimal parabolic
subgroup associated with $\alpha_{i}$, and $X=G/P_{i}$ the corresponding
partial flag variety. For instance, if $G=GL_{n}(\mathbb{C})$ then points in
$G/B$ can be identified with complete flags $(V^{1}\subset
V^{2}\subset\ldots\subset V^{n-1}\subset\mathbb{C}^{n})$ of subspaces, and the
map $\pi_{i}$ forgets the subspace $V^{i}$. The corresponding push-pull
operator $\partial_{i}:CH^{*}(Y)\to CH^{*-1}(Y)$ for Chow rings is often
called divided difference operator, while the push-pull operator
$D_{i}:K^{*}(Y)\to K^{*-1}(Y)$ for the $K$-theory is usually called Demazure
operator.

A classical result of Schubert calculus [BGG, D] is an inductive description
of Schubert cycles $[X_{w}]\in CH^{*}(Y)$ for all elements $w\in W$ in the
Weyl group of $G$. Namely, if $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell}}$ is a
reduced decomposition of $w$ into the product of simple reflections, then

$None$
$[X_{w}]=\partial_{i_{\ell}}\ldots\partial_{i_{2}}\partial_{i_{1}}[X_{\rm
id}].$

A classical result in representation theory [A] is an inductive description of
the Demazure character $\chi_{w}(\lambda)$ for every Schubert variety $X_{w}$
and a dominant weight $\lambda$ of $G$:

$None$ $\chi_{w}(\lambda)=D_{i_{1}}D_{i_{2}}\ldots D_{i_{\ell}}(e^{\lambda}).$

While formulas (1) and (2) look similar, there is no direct relation between
them since in (1) operators are applied in the order opposite to that of (2).

In [Ki16], we defined convex geometric analogs of Demazure operators. They can
be used to construct inductively polytopes $P_{\lambda}$ such that the sum of
exponentials over lattice points in $P_{\lambda}$ yields the Demazure
character $\chi_{w}(\lambda)$. Recently, Naoki Fujita showed that the
Nakashima–Zelevinsky polyhedral realizations of crystal bases for a special
reduced decomposition of the longest element $w_{0}\in W$ can be constructed
inductively using convex geometric Demazure operators in types $A_{n}$,
$B_{n}$, $C_{n}$, $D_{n}$, and $G_{2}$ [Fu19]. In this setting, the convex
geometric Demazure operators are applied in the same order as in (2).

In the present paper, we define different convex geometric analogs of push-
pull operators that are more natural from the perspective of (1) (Section 3).
For a reduced decomposition $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{\ell}}$, convex
geometric push-pull operators produce inductively polytopes whose volume
polynomials coincide with the degree polynomials of Bott–Samelson varieties
corresponding to collections of simple roots $(\alpha_{i_{1}})$,
$(\alpha_{i_{1}},\alpha_{i_{2}})$,…,
$(\alpha_{i_{1}},\alpha_{i_{2}},\ldots,\alpha_{i_{\ell}})$. The main tool is
the Khovanskii–Pukhlikov ring that can be associated with every convex
polytope (Section 2). We prove an analog of the projective bundle formula for
the Khovanskii–Pukhlikov rings of push-pull polytopes (Section 4, Theorem 4.1)
and describe applications in the theory of Newton–Okounkov convex bodies and
in representation theory (Section 5).

I am grateful to Evgeny Smirnov and Vladlen Timorin for useful discussions.

##  2\. Reminder on convex polytopes and Khovanskii–Pukhlikov rings

In this section, we remind the definition of the polytope ring associated with
a convex polytope $P\subset\mathbb{R}^{n}$. This ring was originally
introduced by Khovanskii and Pukhlikov in [KhP92] to give a convenient
functorial description of the cohomology (or Chow) rings of smooth toric
varieties. Later, Kaveh used Khovanskii–Pukhlikov ring to give a partial
description of cohomology rings of spherical varieties [Ka11]. Recently,
Khovanskii–Pukhlikov rings were applied to cohomology rings of toric bundles
[HKhM20] and Schubert calculus on polyhedral realizations of Demazure crystals
[Fu20].

Let $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$ be the integer lattice. A convex
lattice poytope $P\subset\mathbb{R}^{n}$ is a convex hull of several points
from $\mathbb{Z}^{n}$. Recall that two convex polytopes $P$ and $P^{\prime}$
are called analogous if they have the same normal fan. In particular, there
exist linear functions $h_{1}$,…, $h_{d}$ on $\mathbb{R}^{n}$ such that any
polytope $P^{\prime}$ analogous to $P$ is given by inequalities:

$h_{i}(x)\leq H_{i}(P^{\prime}),\ i=1,\ldots,d$

for some constants $H_{1}(P^{\prime})$,…, $H_{d}(P^{\prime})\in\mathbb{R}$
that depend on $P^{\prime}$. The collection of numbers
$(H_{1}(P^{\prime}),\ldots,H_{d}(P^{\prime}))$ (called support numbers of
$P^{\prime}$) defines uniquely the polytope $P^{\prime}$. If a polytope
$P^{\prime}$ is analogous to $P$ then there is a natural bijection between
faces of $P^{\prime}$ and faces of $P$. In the sequel, we denote by
$F(P^{\prime})$ the face of $P^{\prime}$ that corresponds to a face $F\subset
P$ (in particular, $F=F(P)$).

Denote by $S_{P}$ the set of all lattice polytopes analogous to $P$. This set
can be endowed with the structure of a commutative semigroup using Minkowski
sum

$P_{1}+P_{2}=\\{x_{1}+x_{2}\in\mathbb{R}^{n}\ |\ x_{1}\in P_{1},\ x_{2}\in
P_{2}\\}.$

We can embed the semigroup of convex polytopes into its Grothendieck group
$L_{P}$, which is a lattice in $\mathbb{R}^{d}$. The elements of $V_{P}$ are
called virtual polytopes analogous to $P$. They can be represented by
$d$-tuples $(H_{1},\ldots,H_{d})$ such that $H_{i}=H_{i}(P_{1})-H_{i}(P_{2})$
where $P_{1}$ and $P_{2}$ are analogous to $P$. In general, the rank of
$L_{P}$ is smaller than or equal to $d$. The equality holds if and only if $P$
is simple, that is, all facets of $P$ can be translated independently without
changing the combinatorial type of $P$.

There is a homogeneous polynomial $vol_{P}$ of degree $n$ on the lattice
$L_{P}$, called the volume polynomial. It is uniquely characterized by the
property that its value $vol_{P}(P^{\prime})$ on any convex polytope
$P^{\prime}\in S_{P}$ is equal to the volume of $P^{\prime}$.

The symmetric algebra ${\rm Sym}(L_{P})$ of $\Lambda_{P}$ can be thought of as
the ring of differential operators with constant integer coefficients acting
on $\mathbb{R}[L_{P}]$, the space of all polynomials on $L_{P}$. Let
$(x_{1},\ldots,x_{\ell})$ (where $\ell\leq d$) be coordinates on $L_{P}$.
Denote $\frac{\partial}{\partial x_{i}}$ by $\partial_{x_{i}}$.

######  Definition 1.

The Khovanskii–Pukhlikov ring $R_{P}$ associated with the polytope $P$ is the
quotient ring

$\mathbb{Z}[\partial_{1},\ldots,\partial_{\ell}]/{\rm Ann}(vol_{P}).$

Here ${\rm Ann}(vol_{P})$ denotes the ideal in $R_{P}$ that consists of all
differential operators $D$ such that $D(vol_{P})=0$.

######  Remark 2.1.

In what follows, we use that every (virtual) polytope $P^{\prime}\in L_{P}$
defines a homogeneous element $\partial_{P^{\prime}}$ of degree one in
$R_{P}$. Namely, let $(x_{1}(P^{\prime}),\ldots,x_{\ell}(P^{\prime}))$ be
coordinates of $P^{\prime}$. Put
$\partial_{P^{\prime}}=x_{1}(P^{\prime})\partial_{1}+\ldots+x_{\ell}(P^{\prime})\partial_{\ell}.$

##  3\. Definition of push-pull polytopes

Let $P$, $Q\subset\mathbb{R}^{n}$ be convex polytopes such that $Q$ is
analogous to a codimension two truncation of $P$. More precisely, let
$\mathcal{F}=$ {$F_{1}$, …, $F_{k}$} be a (possibly empty) collection of
codimension two faces of $P$. Let $\widehat{P}$ be a polytope analogous to
$P$. For every $i=1$,…, $k$, choose a linear function $\psi_{i}(x)$ on
$\mathbb{R}^{n}$ such that $\psi_{i}$ takes a constant value
$\Psi_{i}(\widehat{P})$ on $F_{i}(\widehat{P})$, and
$\psi_{i}(x)<\Psi_{i}(\widehat{P})$ for all $x\in\widehat{P}\setminus
F_{i}(\widehat{P})$. Fix a constant $\Psi_{i}(Q)<\Psi_{i}(\widehat{P})$. A
codimension two truncation $Q$ of $\widehat{P}$ is obtained from $\widehat{P}$
by intersecting $\widehat{P}$ with half-spaces
$\\{\psi_{i}(x)\leq\Psi_{i}(Q)\\}$ for $i=1$,…, $k$. In particular, $Q$ has
$k$ extra facets $\Gamma_{i}=\\{\psi_{i}(x)=\Psi_{i}(Q)\\}\cap\widehat{P}$ for
$i=1$,…, $k$ (in addition to facets that come from $\widehat{P}$). We assume
that $\Psi_{i}(Q)$ is sufficiently close to $\Psi_{i}(\widehat{P})$ so that
$\widehat{P}\setminus Q$ does not contain any vertices of $\widehat{P}$ except
for those in $\bigcup_{F_{i}\in\mathcal{F}}F_{i}$.

In what follows, we will need the following description of the volume
polynomial of $Q$. Let $s$ be a positive real number. Define the nonconvex
polytope $\overline{Q}(s)\subset\mathbb{R}^{n}$ as the union of convex
polytopes

$\bigcup_{i=1}^{k}\\{\psi_{i}(x)\geq\Psi_{i}(\widehat{P})+s(\Psi_{i}(Q)-\Psi_{i}(\widehat{P}))\\}\cap\widehat{P}.$

In particular, ${\rm vol}(\widehat{P})={\rm vol}(Q)+{\rm
vol}(\overline{Q}(1))$, as $\widehat{P}=Q\cup\overline{Q}(1)$, and
$Q\cap\overline{Q}(1)=\bigcup_{i=1}^{k}\Gamma_{i}$. In general,

${\rm vol}(\overline{Q}(s))=\left(\sum_{i=1}^{k}a_{i}{\rm
vol}_{n-2}(F_{i}(\widehat{P}))\right)\frac{s^{2}}{2}+\sum_{j=3}^{n}g_{j}(\widehat{P})s^{j}.$

where $a_{1},\ldots,a_{k}\in\mathbb{R}$ and $g_{j}(\widehat{P})$ is a linear
combination of volumes of certain codimension $j$ faces of $\widehat{P}$. This
implies the following formula for the volume of $Q$:

$None$ ${\rm vol}(Q)={\rm vol}(\widehat{P})-\left(\sum_{i=1}^{k}a_{i}{\rm
vol}_{n-2}(F_{i}(\widehat{P}))\right)\frac{s^{2}}{2}-\sum_{j=3}^{n}g_{j}(\widehat{P})s^{j}.$

######  Definition 2.

The push-pull polytope
$\Delta(P,Q)\subset\mathbb{R}^{n+1}=\mathbb{R}^{n}\times\mathbb{R}$ is the
Cayley sum of polytopes $P$ and $Q$. More precisely, $\Delta(P,Q)$ is the
convex hull of the following set:

$(P\times 1)\cup(Q\times 0).$

This definition is slightly different from the usual definition of the Cayley
sum of polytopes. The reason for this modification will be seen from Examples
3.1 and 3.2 below.

In what follows, we mostly consider not the particular polytope $\Delta(P,Q)$
but the whole family of polytopes analogous to $\Delta(P,Q)$. It is easy to
check that if $P^{\prime}$ is analogous to $P$, then $\Delta(P,Q)$ is
analogous to $\Delta(P^{\prime},P^{\prime}-P+Q)$. In particular, the space of
all virtual polytopes analogous to $\Delta(P,Q)$ depends only on the
difference $Q-P$.

Below we consider motivating examples of push-pull polytopes.

###  3.1. Minkowski sum with a segment

A special codimension two truncation of $P$ can be obtained using the
Minkowski sum with a segment. Let $I\subset\mathbb{R}^{n}$ be a segment, and
$P^{\prime}$ a polytope analogous to $P$. It is easy to show that the
Minkowski sum $Q=P^{\prime}+I$ is a codimension two truncation of a polytope
$\widehat{P}$ analogous to $P$. Indeed, the facets of $P^{\prime}+I$ are
either parallel to facets of $P^{\prime}$ or equal to Minkowski sums $F+I$
where $F\subset P^{\prime}$ is a face of codimension two. Then $\Delta(P,Q)$
is a convex hull of

$(P\times 1)\cup((P^{\prime}+I)\times 0).$

######  Example 3.1.

Let $n=2$, and $P\subset\mathbb{R}^{2}$ the triangle with the vertices
$(0,0)$, $(1,0)$, $(0,1)$. Take $I=[(0,0),(0,1)]$, and $P^{\prime}=P$. Then
$Q=P+I$ is analogous to trapezoid obtained from $\widehat{P}=2P$ by truncating
the vertex $(2,0)$ (=codimension two face). Then the push-pull polytope
$\Delta(P,Q)\subset\mathbb{R}^{3}$ is the FFLV polytope in type $A_{2}$
corresponding to the weight $\rho$ (see [FFL] for the definition of FFLV
polytopes in type $A$ and their representation-theoretic meaning). The figure
below shows $P$, $Q$ and $\Delta(P,Q)$, respectively (from left to right).

######  Example 3.2.

cf. [An13, Section 6.4] Let $n=2$, and $P\subset\mathbb{R}^{2}$ the trapezoid
with the vertices $(0,0)$, $(2,0)$, $(0,1)$, $(1,1)$. Take $I=[(0,0),(0,1)]$,
and $P^{\prime}=P$. Then $Q=P+I$ is analogous to a 5-gon obtained from
$\widehat{P}$ by truncating the vertex $(3,0)$. Here $\widehat{P}$ is the
trapezoid with the vertices $(0,0)$, $(3,0)$, $(0,2)$, $(1,2)$. Then the push-
pull polytope $\Delta(P,Q)\subset\mathbb{R}^{3}$ is the Minkowski sum of the
FFLV polytope from Example 3.1 and the segment $J=[(0,0,0),(1,0,0)]$. By
shrinking $J$ we get a degeneration to the FFLV polytope. The figure below
shows $P$, $Q$ and $\Delta(P,Q)$, respectively (from left to right).

###  3.2. Analogous $P$ and $Q$

When $Q=\widehat{P}$ is analogous to $P$ (that is, no codimension two faces
are truncated), the push-pull polytope $\Delta(P,Q)$ coincides with the Newton
polytope of a projective line fibration $Y=\mathbb{P}^{1}(E)$ over the
projective toric variety $X$ corresponding to the polytope $P$. We assume that
$E=\mathcal{O}\oplus\mathcal{L}$ is split so $Y$ is also a toric variety. In
this case, there is a simple relation between the polytope $Q$ and the line
bundle $\mathcal{L}$. Namely, identify the Picard group of $X$ with the
Grothendieck group of lattice polytopes analogous to $P$. Denote by
$[\mathcal{L}]$ the (possibly virtual) polytope corresponding to
$\mathcal{L}$. Then $Q=P+[\mathcal{L}]$. In terms of Khovanskii-Pukhlikov
rings, this can rephrased as follows. Under the isomorphism $CH^{*}(X)\simeq
R_{P}$ the first Chern class $c_{1}(\mathcal{L})$ gets mapped to
$\partial_{Q-P}$. In particular, the construction of Grossberg–Karshon cubes
[GK] motivated by Demazure character formula (2) for Bott–Samelson varieties
can also be reproduced in the spirit of (1) using convex-geometric push-pull
operators. We discuss this approach in more detail in Section 5.1.

While Grossberg–Karshon cubes can be realized as Newton–Okounkov polytopes of
Bott–Samelson varieties for some line bundles [Fu15, HY16, HY17], they do not
degenerate to Newton–Okounkov polytopes of flag varieties (instead they turn
into twisted cubes, which are not true polytopes). More general push-pull
operators (with codimension two truncation) allow us to produce polytopes that
do not break in the limit when passing to flag varieties. For instance,
Example 3.2 produces the Newton–Okounkov polytope of a Bott–Samelson variety
in type $A_{2}$ with the desired degeneration. An example in type $A_{3}$ will
be considered in Section 5.2

##  4\. Khovanskii–Pukhlikov ring of a push-pull polytope

In this section, we show that the Khovanskii–Pukhlikov rings of $P$ and of
$\Delta:=\Delta(P,Q)$ are related in the same manner as cohomology rings
$H^{*}(X)$ and $H^{*}(Y)$ of smooth varieties $X$ and $Y$, whenever $Y$ is a
projectivization of a rank two vector bundle over $X$. We use notation of
Sections 2 and 3.

Let $R_{P}$ and $R_{\Delta}$ denote the Khovanskii–Pukhlikov rings of
polytopes $P$ and $\Delta$, respectively. More precisely, suppose that $R_{P}$
is defined using the lattice $L_{P}$, and $L_{P}$ is spanned by polytopes
$P_{1}$,…, $P_{\ell}$. Define $L_{\Delta}$ as the lattice spanned by
$\Delta(P+P_{i},Q+P_{i})$ for $i=1$,…, $\ell$, and by an extra generator $C$.
The virtual polytope $C$ can be informally thought of as the cone over $Q-P$.
To give a formal definition we will need the following description of
polytopes analogous to $\Delta$.

######  Definition 3.

Let $P^{\prime}$ be a polytope analogous to $P$, and $s$ a positive real
number. Suppose that $P^{\prime}-P=s_{1}P_{1}+\ldots+s_{\ell}P_{\ell}$, where
all $s_{i}$ are nonnegative. Define the polytope
$\Delta(s,P^{\prime})\subset\mathbb{R}^{n}\times\mathbb{R}$ as the convex hull
of

$(P^{\prime}\times s)\cup\left((P^{\prime}+s(Q-P))\times 0\right).$

In particular, $\Delta(1,P)=\Delta$. It is easy to check that the polytopes
$\Delta(s,P^{\prime})$ and $\Delta$ are analogous, and that the following
identity holds:

$\Delta(s,\sum_{i=1}^{\ell}s_{i}P_{i})+\Delta(t,\sum_{i=1}^{\ell}t_{i}P_{i})=\Delta(s+t,\sum_{i=1}^{\ell}(s_{i}+t_{i})P_{i}).$

Let $L_{\Delta}$ be the Grothendieck group of the semigroup of polytopes
$\Delta(s,P^{\prime})$. Then the lattice $L_{\Delta}$ has a unique basis $C$,
$\Delta_{1}$,…, $\Delta_{\ell}$, such that the polytope $\Delta(s,P^{\prime})$
in this basis has coordinates $(s,s_{1},\ldots,s_{\ell})$. In particular, $C$
can be defined as the formal difference of polytopes $\Delta(2,P)$ and
$\Delta$.

Consider the monomorphism of lattices $L_{P}\hookrightarrow L_{\Delta}$, which
sends $P_{i}$ to $\Delta_{i}$ for $i=1$,…, $\ell$. In what follows, we
identify $L_{P}$ with the sublattice of $L_{\Delta}$ using this monomorphism.
Define the polynomial function ${\rm vol}_{\mathcal{F}}$ on $L_{P}$ by the
condition

${\rm vol}_{\mathcal{F}}(P^{\prime})=\sum_{i=1}^{k}a_{i}{\rm
vol}_{n-2}(F_{i}(P^{\prime}))$

for all $P^{\prime}\in L_{P}$ that are analogous to $P$. Similarly, we define
polynomial functions $g_{j}$ on $L_{P}$ for $j=3$,…, $n$ so that formula $(*)$
from Section 3 holds for all $P^{\prime}\in L_{P}$. Define the polynomial
$q(s,P^{\prime})$ on $\mathbb{R}\times L_{P}$ by the formula:

$q(s,P^{\prime})={\rm
vol}_{\mathcal{F}}(P^{\prime})\frac{s^{2}}{2}+\sum_{j=3}^{n}g_{j}(P^{\prime})s^{j}.$

######  Theorem 4.1.

Suppose that there exist a homogeneous element $D_{\mathcal{F}}\in R_{P}$ of
degree two and homogeneous elements $D_{j}\in R_{P}$ of degrees $3$,…, $n$
such that $D_{\mathcal{F}}({\rm vol}_{P})={\rm vol}_{\mathcal{F}}$ and
$D_{j}({\rm vol}_{P})=g_{j}$. Put

$c_{1}=\partial_{\widehat{P}}-\partial_{P},\quad c_{2}=D_{\mathcal{F}}.$

If $q(s,P^{\prime})$ satisfies the condition:

$None$ $\left(\frac{\partial^{2}}{\partial
s^{2}}-c_{1}\frac{\partial}{\partial
s}+c_{2}\right)q(s,P^{\prime}+sc_{1})={\rm
vol}_{\mathcal{F}}(P^{\prime}+sc_{1}),$

then there is a ring isomorphism

$R_{\Delta}\simeq R_{P}[x]/(x^{2}-c_{1}x+c_{2}).$

In particular, $c_{1}$ and $c_{2}$ can be thought of as analogs of the Chern
classes of a rank two vector bundle.

######  Remark 4.2.

Note that if $P$ is simple, then there always exists a homogeneous element
$D_{\mathcal{F}}\in R_{P}$ of degree two such that $D_{\mathcal{F}}({\rm
vol}_{P})={\rm vol}_{\mathcal{F}}$. Indeed, every face $F\subset P$ of
codimension two is the intersection of two facets $\Gamma_{1}$ and
$\Gamma_{2}$. Since $P$ is simple, every facet $\Gamma\subset P$ defines a
homogeneous element $\partial_{\Gamma}\in R_{P}$ of degree one. Put
$D=\partial_{\Gamma_{1}}\partial_{\Gamma_{2}}$. It is easy to check that
$D({\rm vol}_{P})=a{\rm vol}_{F}$ for some constant $a\in\mathbb{R}$ that
depends on the polytope $P$. Similarly, one can show that $D_{3}$,…, $D_{n}$
also exist for simple $P$.

However, if $P$ is not simple then the existence of $D_{\mathcal{F}}$ and
$D_{3}$,…, $D_{n}$ can not be taken for granted.

######  Remark 4.3.

Condition $(**)$ is void if $\mathcal{F}=\varnothing$. Also, it holds
trivially in the case where $g_{3}=\ldots=g_{n}=0$ (that is, $q(s,P^{\prime})$
is a quadratic polynomial in $s$), and $c_{1}c_{2}=0$ in $R_{P}$. This is the
case in Examples 3.1 and 3.2 for dimension reasons.

From a geometric viewpoint, condition $(**)$ is “local”, that is, depends only
on the structure of the polytope $P$ in a sufficiently small neighborhood of
the union of faces $\bigcup_{F_{i}\in\mathcal{F}}F_{i}$. Because of this, it
is sometimes easier to verify $(**)$ than more “global” statements about $P$
and the volume polynomial ${\rm vol}_{P}$.

###### Proof.

First, we construct a ring monomorphism $\varphi:R_{P}\hookrightarrow
R_{\Delta}$. Put $\partial_{s_{i}}=\frac{\partial}{\partial s_{i}}$ for
$i=1$,…, $\ell$, and $\partial_{s}=\frac{\partial}{\partial s}$. Let $D$ be a
polynomial in $\partial_{s_{1}}$,…, $\partial_{s_{\ell}}$. Since $L_{P}\subset
L_{\Delta}$, the differential operator $D$ can also be regarded as the
polynomial $\varphi(D)$ in $\partial_{s}$, $\partial_{s_{1}}$,…,
$\partial_{s_{\ell}}$. It remains to check that the map $\varphi$ is well-
defined, that is, $D({\rm vol}_{P})=0$ implies $\varphi(D)({\rm
vol}_{\Delta})=0$. By definition of $\Delta(s,P^{\prime})$ and formula $(*)$
for the volume of $Q$ from Section 3, we have the following explicit formula
for the volume polynomial ${\rm vol}_{\Delta}$:

${\rm vol}_{\Delta}(\Delta(s,P^{\prime}))=\int_{0}^{s}{\rm
vol}_{P}(x(t))-q(t,x(t))dt,$

where $x(t)=P^{\prime}+t(\widehat{P}-P)$. Since $D({\rm vol}_{P})=0$ and ${\rm
vol}_{\mathcal{F}}=D_{\mathcal{F}}({\rm vol}_{P})$ we have that $D({\rm
vol}_{\mathcal{F}})=D_{\mathcal{F}}D({\rm vol}_{P})=0$. Similarly, we have
that $D(g_{i})=D_{i}D({\rm vol}_{P})=0$. Hence, $\varphi(D)({\rm
vol}_{\Delta})=0$.

Second, we show that

$(\partial_{s}-\partial_{\widehat{P}-P})\partial_{s}{\rm
vol}_{\Delta}=-D_{\mathcal{F}}{\rm vol}_{\Delta}.$

Using the explicit formula for ${\rm vol}_{\Delta}(\Delta(s,P^{\prime}))$ we
get

$(\partial_{s}-\partial_{\widehat{P}-P})\partial_{s}{\rm
vol}_{\Delta}=(\partial_{s}-\partial_{\widehat{P}-P})\left({\rm
vol}_{P}(x(s))-q(s,x(s))\right).$

By definition of the directional derivative $\partial_{\widehat{P}-P}$ we have
that $(\partial_{s}-\partial_{\widehat{P}-P}){\rm vol}_{P}(x(s))=0$. Hence,

$(\partial_{s}-\partial_{\widehat{P}-P})\left({\rm
vol}_{P}(x(s))-q(s,x(s))\right)=-(\partial_{s}-\partial_{\widehat{P}-P})q(s,x(s)).$

By condition $(**)$ the right hand side is equal to $-D_{\mathcal{F}}{\rm
vol}_{\Delta}(\Delta(s,P^{\prime}))$. Indeed,

$D_{\mathcal{F}}{\rm
vol}_{\Delta}(\Delta(s,P^{\prime}))=\int_{0}^{s}D_{\mathcal{F}}{\rm
vol}_{P}(x(t))-D_{\mathcal{F}}q(t,x(t))dt=$ $=\int_{0}^{s}{\rm
vol}_{\mathcal{F}}(x(t))-D_{\mathcal{F}}q(t,x(t))dt=\int_{0}^{s}\partial_{t}(\partial_{t}-\partial_{\widehat{P}-P})q(t,x(t))dt.$

Finally, we establish the isomorphism
$\Phi:R_{P}[x]/(x^{2}-c_{1}x+c_{2})\simeq R_{\Delta}$ by sending $x$ to
$\partial_{s}$. This yields a well-defined ring homomorphism

$\Phi:g+hx\mapsto\varphi(g)+\varphi(h)\partial_{s},$

since we already checked that $\partial_{s}^{2}-c_{1}\partial_{s}+c_{2}=0$ in
$R_{\Delta}$. Clearly, $\Phi$ is surjective since $R_{\Delta}$ is generated by
$\partial_{s}$ and $\varphi(R_{P})$. It is injective since
$\partial_{s}\notin\varphi(R_{P})$. ∎

As a byproduct of the proof of Theorem 4.1 we get the following

######  Corollary 4.4.

The volume polynomial ${\rm vol}_{\Delta}$ of the push-pull polytope satisfies
the differential equation:

$F^{\prime\prime}-(\partial_{\widehat{P}}-\partial_{P})F^{\prime}+D_{\mathcal{F}}F=0,$

where an unknown function $F$ is a function of two variables $s\in\mathbb{R}$
and $x\in L_{P}\otimes\mathbb{R}=\mathbb{R}^{\ell}$, and $F^{\prime}$ is the
derivative of $F$ with respect to $s$.

######  Example 4.5.

We continue Example 3.2. A polytope $P^{\prime}$ is analogous to $P$ if
$P^{\prime}$ is given by inequalities

$0\leq(x-x_{0});\quad 0\leq(y-y_{0})\leq b;\quad(x-x_{0})+(y-y_{0})\leq a+b.$

Choose a basis $P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$ in $L_{P}$ so that
$P^{\prime}$ has coordinates $(a,b,x_{0},y_{0})$ in this basis. In particular,
$\partial_{s_{1}}=\frac{\partial}{\partial a}:=\partial_{a}$ and
$\partial_{s_{2}}=\frac{\partial}{\partial b}:=\partial_{b}$. Since the area
of $P^{\prime}$ does not change under parallel translations we have that
$\partial_{s_{3}}{\rm vol}_{P}=\partial_{s_{4}}{\rm vol}_{P}=0$. In other
words, the volume polynomial ${\rm vol}_{P}=ab+\frac{b^{2}}{2}$ does not
depend on $x_{0}$ and $y_{0}$. It is easy to check that the differential
operators $(\partial_{b}^{2}-\partial_{a}\partial_{b})$ and $\partial_{a}^{2}$
annihilate ${\rm vol}_{P}$. Hence, the Khovanskii–Pukhlikov ring $R_{P}$ is
isomorphic to the quotient of the polynomial ring
$\mathbb{Z}[\partial_{a},\partial_{b}]$ by the ideal
$(\partial_{a}^{2},\partial_{b}^{2}-\partial_{a}\partial_{b})$.

We now compute $c_{1}$ and $c_{2}$. We have that $\widehat{P}-P=P_{2}$, hence,
$\partial_{\widehat{P}-P}=\partial_{b}$. Since $\mathcal{F}$ consists of a
single vertex of $\widehat{P}$, and we may choose any $D_{\mathcal{F}}$ such
that $D_{\mathcal{F}}{\rm vol}_{P}=1$. For instance, take
$D_{\mathcal{F}}=\partial_{a}\partial_{b}$. Hence, the statement of Theorem
4.1 reduces to the following isomorphisms:

$R_{\Delta}\simeq
R_{P}[x]/(x^{2}-\partial_{b}x+\partial_{a}\partial_{b})\simeq\mathbb{Z}[\partial_{s},\partial_{a},\partial_{b}]/(\partial_{a}^{2},\partial_{b}^{2}-\partial_{a}\partial_{b},\partial_{s}^{2}-\partial_{b}\partial_{s}+\partial_{a}\partial_{b}),$

which can be easily checked by direct calculation. Indeed, the volume
polynomial ${\rm vol}_{\Delta}$ is equal to
$s(ab+\frac{b^{2}}{2})+\frac{s^{2}}{2}(a+b)$. Clearly, the differential
operator $\partial_{s}^{2}-\partial_{b}\partial_{s}+\partial_{a}\partial_{b}$
annihilates ${\rm vol}_{\Delta}$.

##  5\. Applications to Schubert calculus and representation theory

###  5.1. Formula for the Schubert cycles and Demazure character formula

We now explain a simple convex geometric relation between formula (1) (for the
Schubert cycles) and formula (2) (for the Demazure characters). Namely, we
construct Grossberg–Karshon cubes for a given reduced sequence
$(\alpha_{i_{1}},\ldots,\alpha_{i_{\ell}})$ of simple roots by two different
methods. Both constructions are inductive but the first method uses terminal
subwords of the word $I=(i_{1},i_{2},\ldots,i_{\ell})$ while the second method
uses initial subwords of $I$.

The first method is the same as in [GK], and can also be described within a
more general framework of convex geometric Demazure operators [Ki16, Section
4]. Namely, we consider a (possibly virtual) polytope $P\subset\mathbb{R}^{n}$
obtained as

$P=D_{i_{1}}D_{i_{2}}\ldots D_{i_{\ell}}(p),$

where $p\in\mathbb{R}^{n}$ and $D_{1}$,…, $D_{\ell}$ are convex geometric
Demazure operators for $G$ (see [Ki16, Section 4.3] for more details). In this
case, the intermediate polytopes $D_{\ell}(p)$, $D_{\ell-1}D_{\ell}(p)$,…, and
$D_{2}\cdots D_{\ell-1}D_{\ell}(p)$ are (possibly twisted) Grossberg-Karshon
cubes for terminal subwords $(i_{\ell})$, $(i_{\ell-1},i_{\ell})$,…,
$(i_{2},\ldots,i_{\ell-1},i_{\ell})$, respectively.

The second method uses push-pull polytopes. Recall that the Bott-Samelson
variety $R_{I}$ can be constructed as a tower of projective line fibrations:

$\\{{\rm pt}\\}=R_{\varnothing}\leftarrow R_{(i_{1})}\leftarrow
R_{(i_{1},i_{2})}\leftarrow\ldots\leftarrow
R_{(i_{1},i_{2},\ldots,i_{\ell-1})}\leftarrow R_{I}.$

This tower is used in one of the proofs of formula (1) (see [D] or [Ma,
Theorem 3.6.18]). Every variety in this tower is obtained form the previous
variety as the projectivization of a rank two vector bundle. Using push-pull
polytopes we will turn a tower of Bott–Samelson varieties into a tower of
convex polytopes:

$\\{p\\}=P_{\varnothing}\leftarrow P_{(i_{1})}\leftarrow
P_{(i_{1},i_{2})}\leftarrow\ldots\leftarrow
P_{(i_{1},i_{2},\ldots,i_{\ell-1})}\leftarrow P_{I}.$

We show that the intermediate polytopes in this tower can be identified with
suitable Grossberg-Karshon cubes for initial subwords $(i_{1})$,
$(i_{1},i_{2})$,…, and $(i_{1},i_{2},\ldots,i_{\ell-1})$, respectively. In
particular, we will show that the resulting polytope $P_{I}$ is analogous to
$P$. This observation exhibits a relation beween formulas (1) and (2).

We now describe construction of $P_{I}$ in more detail. We first construct
$P_{I}$ in convex geometric terms and then explain the relation to
Bott–Samelson varieties and Grossberg–Karshon cubes. Let ($\beta_{1}$,…,
$\beta_{\ell}$) be a collection of vectors in $\mathbb{R}^{\ell}$ (it is
possible that $\beta_{i}=\beta_{j}$ for $i\neq j$). Fix a positive definite
inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{R}^{\ell}$, and define
the linear function $(\cdot,\beta_{i})$ on $\mathbb{R}^{\ell}$ by the formula:

$(\alpha,\beta_{i})=2\frac{\langle\alpha,\beta_{i}\rangle}{\langle\beta_{i},\beta_{i}\rangle}.$

Denote by $s_{\beta_{i}}$ the reflection through the hyperplane orthogonal to
$\beta_{i}$, i.e.,

$s_{\beta_{i}}:\mathbb{R}^{\ell}\to\mathbb{R}^{\ell};\quad
s_{\beta_{i}}:\alpha\mapsto\alpha-(\alpha,\beta_{i})\beta_{i}.$

Let $(x_{1},\ldots,x_{\ell})$ be coordinates on $\mathbb{R}^{\ell}$. Define
$(\ell-1)$ linear functions on $\mathbb{R}^{\ell}$:

$f_{1}=(\beta_{1},\beta_{2})x_{1};\quad
f_{2}=(\beta_{1},\beta_{3})x_{1}+(\beta_{2},\beta_{3})x_{2};\ \ldots\ ;$
$f_{\ell-1}=(\beta_{1},\beta_{\ell})x_{1}+(\beta_{2},\beta_{\ell})x_{2}+\ldots+(\beta_{\ell-1},\beta_{\ell})x_{\ell-1}.$

In particular, $f_{i}$ depends only on $x_{1}$,…, $x_{i}$. We now associate
with a vector $\lambda\in\mathbb{R}^{n}$ the polytope $P_{I}(\lambda)$ defined
by inequalities (cf. [GK, (2.21)]):

$0\leq x_{1}\leq(\lambda,\beta_{1});\quad 0\leq x_{2}\leq-
f_{1}(x_{1})+(\lambda,\beta_{2});\quad 0\leq x_{3}\leq-
f_{2}(x_{1},x_{2})+(\lambda,\beta_{3});\ \ldots\ ;$ $0\leq x_{\ell}\leq-
f_{\ell-1}(x_{1},\ldots,x_{\ell-1})+(\lambda,\beta_{\ell}).$

The polytope $P_{I}(\lambda)$ is a combinatorial cube and can be thought of as
a (possibly virtual) multidimensional version of a trapezoid. There is a
unique vertex $p_{I}(\lambda)$ such that all coordinates of $p_{I}(\lambda)$
are nonzero (for generic $\lambda$). It is easy to find these coordinates:

$None$
$p_{I}(\lambda)=((\lambda,\beta_{1}),(\lambda,s_{\beta_{1}}(\beta_{2})),\ldots,(\lambda,s_{\beta_{1}}\cdots
s_{\beta_{\ell-1}}(\beta_{\ell}))).$

Note that $p_{I}(\lambda)=p(\mu)$ if and only if $P_{I}(\lambda)=P_{I}(\mu)$.

The polytope $P_{I}(\lambda)$ can be constructed inductively using push-pull
polytopes. The induction step is based on the following

######  Lemma 5.1.

Let $I^{1}$ denote the sequence $(\beta_{2},\ldots,\beta_{\ell})$. If
$P_{I^{1}}(\lambda)$ and $P_{I^{1}}(\lambda-\beta_{1})$ are true (not virtual)
polytopes then $P_{I}(\lambda)$ is analogous to the push-pull polytope
$\Delta(P_{I^{1}}(\lambda-\beta_{1}),P_{I^{1}}(\lambda))$.

###### Proof.

Consider two parallel facets $\Gamma_{0}:=P_{I}(\lambda)\cap\\{x_{1}=0\\}$ and
$\Gamma_{1}:=P_{I}(\lambda)\cap\\{x_{1}=(\lambda,\beta_{1})\\}$ of
$P_{I}(\lambda)$. Then $P_{I}(\lambda)=\Delta(\Gamma_{1},\Gamma_{0})$. By
construction
$\Gamma_{0}=P_{I^{1}}(\lambda)\subset\mathbb{R}^{\ell-1}:=\mathbb{R}^{\ell}\cap\\{x_{1}=0\\}$.
It is easy to check that if we identify $\Gamma_{1}$ with its projection to
$\mathbb{R}^{\ell-1}$ along the $x_{1}$-axis, then
$\Gamma_{1}=P_{I^{1}}(s_{\beta_{1}}(\lambda))$. Indeed, the vertex
$p_{I}(\lambda)$ of $\Gamma_{1}$ gets projected to the point

$((\lambda,s_{\beta_{1}}(\beta_{2})),\ldots,(\lambda,s_{\beta_{1}}\cdots
s_{\beta_{\ell-1}}(\beta_{\ell})))\in\mathbb{R}^{\ell-1}.$

It remains to use that
$(\lambda,\mu)=(s_{\beta_{1}}(\lambda),s_{\beta_{1}}(\mu))$ for all $\lambda$,
$\mu\in\mathbb{R}^{\ell}$.

Since $s_{\beta_{1}}(\lambda)=\lambda-(\lambda,\beta_{1})\beta_{1}$, we have
$\Gamma_{1}=\Gamma_{0}-(\lambda,\beta_{1})P_{I^{1}}(\beta_{1})$. By
construction $P_{I}(\lambda)$ is the convex hull of $\Gamma_{0}$ and
$\Gamma_{1}$, hence, $P_{I}(\lambda)$ is analogous to
$\Delta(P_{I^{1}}(\lambda-\beta_{1}),P_{I^{1}}(\lambda))$. Here we use that
$(\lambda,\beta_{1})>0$ since $P_{I}(\lambda)$ is not virtual. ∎

We now consider the case of Bott–Samelson varieties. Let
$(\beta_{1},\ldots,\beta_{\ell})=(\alpha_{i_{\ell}},\ldots,\alpha_{i_{1}})$,
and $\lambda$ a weight of $G$. In what follows, $L_{\lambda}$ denotes the line
bundle on $G/B$ associated with a weight $\lambda$, and $p_{I}:R_{I}\to G/B$
denotes the projection of the Bott–Samelson variety to the flag variety. By
construction $R_{I}=\mathbb{P}(E)$ where $E$ is a rank two vector bundle over
$R_{I^{\ell}}$. The bundle $E$ can be chosen so that there is a short exact
sequence $0\to\mathcal{O}_{R_{I^{\ell}}}\to E\to
p_{I^{\ell}}^{*}L_{\alpha_{i_{\ell}}}\to 0$. In particular,
$c_{1}(E)=c_{1}(p_{I^{\ell}}^{*}L_{\alpha_{i_{\ell}}})$ and $c_{2}(E)=0$.
Hence, if the Khovanskii–Pukhlikov ring $R_{P_{I^{\ell}}}$ is isomorphic to
$CH^{*}(R_{I^{\ell}})$ then Theorem 4.1 together with Lemma 5.1 imply the ring
isomorphism $R_{P_{I}}\simeq H^{*}(R_{I})$.

Note that in order to apply Lemma 5.1 we might have to choose a true polytope
$P_{I^{1}}$ analogous to $P_{I^{1}}(\lambda)$ so that
$P_{I^{1}}-P_{I^{1}}(\beta_{1})$ is also true. In particular, $P_{I}(\lambda)$
itself is often a virtual (twisted) polytope. In other words, we replace the
(not very ample) line bundle $p_{I}^{*}L_{\lambda}$ on $R_{I}$ by a very ample
line bundle $L$ such that $L\otimes L(-\beta_{i})$ is also very ample.

Hence, the inductive construction of the polytope $P_{I}$ via a sequence of
push-pull polytopes repeats the inductive construction of the Bott–Samelson
variety $R_{I}$. On the other hand, the definition of $P_{I}(\lambda)$ by
explicit inequalities together with calculations in [GK, 3.6] exhibit
$P_{I}(\lambda)$ as the Karshon–Grossberg cube for $R_{I}$ and $\lambda$.

######  Remark 5.2.

In the case of Bott–Samelson varieities, coordinates of $p_{I}(\lambda)$
computed in (3) are exactly the coefficients in the Chevalley–Pieri formula in
the Chow ring $CH^{*}(R_{I})$. Recall that the Picard group of $R_{I}$ is
spanned by the classes of hypersurfaces $R_{I^{j}}$ where $I^{j}$ denotes the
sequence $(i_{1},\ldots,i_{j-1},i_{j+1},\ldots,i_{\ell})$. The Chevalley–Pieri
formula for Bott–Samelson varieties yields the following decomposition of
$c_{1}(p_{I}^{*}L_{\lambda})$ in the basis $([R_{I^{j}}])_{1\leq j\leq\ell}$:

$c_{1}(p_{I}^{*}L_{\lambda})=\sum_{j=1}^{\ell}(\lambda,s_{\beta_{1}}\cdots
s_{\beta_{j-1}}(\beta_{j}))[R_{I^{j}}].$

In particular, the induction step of Lemma 5.1 in this setting can be viewed
as a convex geometric counterpart of [D, Proposition 2.1].

On the other hand, there is a simple convex geometric analog of
Chevalley–Pieri formula in the Khovanskii–Pukhlikov ring of $P_{I}$. Let
$\Gamma_{j}$ denote the facet of $P_{I}$ given by equation $x_{j}=0$. The
polytope $P_{I}$ is simple so $\Gamma_{j}$ defines an element
$\partial_{\Gamma_{j}}\in R_{P_{I}}$. Then (3) implies immediately the
following identity:

$\partial_{P_{I}(\lambda)}=\sum_{j=1}^{\ell}(\lambda,s_{\beta_{1}}\cdots
s_{\beta_{j-1}}(\beta_{j}))\partial_{\Gamma_{j}}.$

###  5.2. Minkowski sums of FFLV polytopes

Construction of convex geometric push-pull operators is mainly motivated by
study of Newton–Okounkov convex bodies of Bott-Samelson varieties. Namely,
polytopes that arise as conjectural Newton–Okounkov polytopes can sometimes be
also constructed by iterated application of push-pull operators. In such
cases, properties of Khovanskii–Pukhlikov rings of push-pull polytopes
(Theorem 4.1) allow us to prove that we indeed constructed the full
Newton–Okounkov polytope (the same kind of argument was used in Section 5.1).
Below we illustrate this method in the case of Bott–Samelson variety
$R_{(1,2,1,3,2)}$ in type $A_{3}$. This is a partial case of Bott–Samelson
varieties in type $A_{n}$ considered in [Ki18].

In future, we plan to use the same method to complete [Ki18], and prove
directly that Minkowski sums of FFLV polytopes yield Newton–Okounkov polytopes
of Bott–Samelson varieties in type $A_{n}$. In [Ki18], we used convex
geometric Demazure operators in order to compare volumes, however, this was
quite a roundabout way since FFLV polytopes themselves can not be constructed
using Demazure operators. On the other hand, push-pull operators seem to be a
natural tool to construct FFLV polytopes and their Minkowski sums.

From a geometric viewpoint, points of the Bott–Samelson variety
$R_{5}:=R_{(1,2,1,3,2)}$ can be identified with configurations
$(a_{1},l_{1},a_{2},\Pi,l_{2})$ of subspaces in $\mathbb{P}^{3}$ such that
$a_{1}$, $a_{2}$ are points, $l_{1}$, $l_{2}$ are lines, $\Pi$ is a plane, and

$a_{1}\in l_{0},l_{1};\ a_{2}\in l_{1}\subset\Pi_{0};\ a_{2}\in
l_{2}\subset\Pi,$

where $l_{0}\subset\Pi_{0}$ are fixed line and plane in $\mathbb{P}^{3}$. The
intermediate Bott–Samelson varieties $R_{1}:=R_{(1)}$, $R_{2}:=R_{(1,2)}$,
$R_{3}:=R_{(1,2,1)}$, $R_{4}:=R_{(1,2,1,3)}$ can be identified with the
subvarieties of $R_{5}$, namely, define $R_{4}\subset R_{5}$ as the
hypersurface given by the condition $\\{l_{2}=l_{1}\\}$. Similarly, define
$R_{3}\subset R_{4}$, $R_{2}\subset R_{3}$, and $R_{1}\subset R_{2}$ by the
conditions $\Pi=\Pi_{0}$, $a_{2}=a_{1}$, and $l=l_{0}$, respectively. Clearly,
$R_{i}$ is the projectivization of a quotient tautological rank two bundle
$E_{i-1}$ over $R_{i-1}$ for $i=1$,…, $i=5$. For instance,
$R_{5}=\mathbb{P}(E_{4})$ where the fiber of $E_{4}$ over a point
$(a_{1},l_{1},a_{2},\Pi,l_{2})$ is the quotient $V(\Pi)/V(a_{2})$ (by $V(P)$
we denote the vector subspace whose projectivization is the projective
subspace $P$). In particular, the Chow ring of $R_{5}$ is generated by the
first Chern clasess $\xi_{1}$,…, $\xi_{5}$ of line bundles
$\mathcal{O}_{E_{1}}(1)$,…, $\mathcal{O}_{E_{5}}(1)$. Note also that
$R_{1}\simeq\mathbb{P}^{1}$, and $R_{2}$ is the blow up of $\mathbb{P}^{2}$ at
one point.

We now construct a sequence of polytopes $\Delta_{0}:=\\{\rm 0\\}$,
$\Delta_{1}:=\Delta_{(1)}$, $\Delta_{2}:=\Delta_{(1,2)}$,
$\Delta_{3}:=\Delta_{(1,2,1)}$, $\Delta_{4}:=\Delta_{(1,2,1,3)}$,
$\Delta_{5}:=\Delta_{(1,2,1,3,2)}$ so that every polytope in the sequence is
obtained as a push-pull polytope from the previous one. Using the sequence of
push-pull polytopes $\Delta_{1}$,…, $\Delta_{5}$ and Theorem 4.1 we show that
the Khovanskii–Pukhlikov ring of $\Delta_{5}$ is isomorphic to the Chow ring
of $R_{5}$. More precisely, we construct families of analogous polytopes at
each step. In particular, we consider the family of segments
$\Delta_{1}(a)=\\{u_{1}\in\mathbb{R}\ |\ 0\leq u_{1}\leq a\\}$ and the family
of trapezoids $\Delta_{2}(a)=\\{(u_{1},u_{2})\in\mathbb{R}^{2}\ |\ 0\leq
u_{1}\leq a;\ 0\leq u_{2};\ u_{1}+u_{2}\leq a+b\\}$.

To construct $\Delta_{3}$ first note that the Chow ring of $R_{3}$ is
isomorphic to

$\mathbb{Z}[\xi_{1},\xi_{2},\xi_{3}]/(\xi_{1}^{2},\
\xi_{2}^{2}-\xi_{1}\xi_{2},\ \xi_{3}^{2}+\xi_{2}\xi_{3}+\xi_{2}^{2})$

(this is easy to compute by applying repeatedly projective bundle formula).
Hence, $CH^{*}(R_{3})$ is isomorphic to the Khovanskii–Pukhlikov ring of the
push-pull polytope $\Delta_{3}(a,b,c)\subset\mathbb{R}^{3}$ given by
inequalities

$0\leq u_{1}\leq a+b;\ 0\leq u_{3}\leq c;\ 0\leq u_{2};\ u_{2}+u_{3}\leq b+c;\
u_{1}+u_{2}+u_{3}\leq a+b+c.$

The ring isomorphism takes $\xi_{1}$ to $\partial_{a}$, $\xi_{2}$ to
$\partial_{b}$ and $\xi_{3}$ to $\partial_{c}-\partial_{b}$. Indeed, if we put
$s=c$ in Example 4.5 we get $\Delta_{3}(a,b,c)$.

It remains to construct $\Delta_{4}$ from $\Delta_{3}$ and $\Delta_{5}$ from
$\Delta_{4}$. Again by projective bundle formula we have the ring isomorphism

$CH^{*}(R_{5})\simeq
CH^{*}(R_{3})[\xi_{4},\xi_{5}]/(\xi_{4}^{2}-\xi_{2}\xi_{4},\
\xi_{5}^{2}-\xi_{5}(\xi_{3}+\xi_{4}-\xi_{2})+\xi_{3}(\xi_{4}-\xi_{2})).$

Hence, if we take $\Delta_{4}=\Delta(P,Q)$, where $P=\Delta_{3}(a,b,c)$ and
$Q=\widehat{P}=\Delta_{3}(a,b+1,c))$ we get a polytope whose
Khovanskii–Pukhlikov ring is isomorphic to $CH^{*}(R_{4})$. The ring
isomorphism sends $\xi_{4}$ to $\partial_{d}$. There is a family of polytopes
$\Delta_{4}(a,b,c,d)$ analogous to $\Delta_{4}$ defined by inequalities:

$0\leq u_{1},u_{2},u_{3},u_{4};\ u_{3}\leq c;\ u_{4}\leq d;\ u_{1}+u_{4}\leq
a+b+d;$ $\ u_{2}+u_{3}+u_{4}\leq b+c+d;\ u_{1}+u_{2}+u_{3}+u_{4}\leq a+b+c+d.$

To construct $\Delta_{5}$ as a push-pull polytope over $\Delta_{4}$ we first
rewrite the relation
$\xi_{5}^{2}-\xi_{5}(\xi_{3}+\xi_{4}-\xi_{2})+\xi_{3}(\xi_{4}-\xi_{2}))$
(modulo the other relations) in $CH^{*}(R_{5})$ making the change of variable
$x=\xi_{4}+\xi_{5}$. We get the ring isomorphism

$CH^{*}(R_{5})=R_{\Delta_{4}}[x]/(x^{2}-(\partial_{c}+\partial_{d})x+(\partial_{c}^{2}+\partial_{d}^{2})).$

This gives us a hint on how to choose $\widehat{P}$ and $\mathcal{F}$ in order
to construct $\Delta_{5}$ as $\Delta(P,Q)$ for $P=\Delta_{4}(a,b,c,d)$.
Namely, $\widehat{P}=\Delta(a,b,c+1,d+1)$, and

$\mathcal{F}=\\{\Gamma_{124}\cap\Gamma^{0}_{4},\
\Gamma_{234}\cap\Gamma^{0}_{3},\ \Gamma_{234}\cap\Gamma^{0}_{4},\
\Gamma_{1234}\cap\Gamma^{0}_{3},\ \Gamma_{1234}\cap\Gamma^{0}_{4}\\},$

where $\Gamma^{0}_{i}$ denotes the facet $\\{u_{i}=0\\}$, and $\Gamma_{124}$,
$\Gamma_{234}$ and $\Gamma_{1234}$ denote the facets
$\\{u_{1}+u_{2}+u_{4}=a+b+d\\}$, $\\{u_{2}+u_{3}+u_{4}=b+c+d\\}$ and
$\\{u_{1}+u_{2}+u_{3}+u_{4}=a+b+c+d\\}$, respectively. It is not hard to check
that $\partial_{\widehat{P}-P}=\partial_{c}+\partial_{d}$ and
$D_{\mathcal{F}}=\partial_{c}^{2}+\partial_{d}^{2}$. Applying Theorem 4.1 we
get that the ring $CH^{*}(R_{5})$ is isomorphic to the Pukhlikov–Khovanskii
ring of the polytope $\Delta_{5}(a,b,c,d,e)\subset\mathbb{R}^{5}$ given by
$16$ inequalities:

$u_{1},u_{2},u_{3},u_{4},u_{5}\geq 0;\ u_{1}\leq a+b+d;\ u_{5}\leq e;\
u_{3}+u_{5}\leq c+e;\ u_{4}+u_{5}\leq d+e;$ $\ u_{1}+u_{4}+u_{5}\leq a+b+d+e;\
u_{2}+u_{3}+u_{5}\leq b+c+d+e;\ u_{2}+u_{4}+u_{5}\leq b+c+d+e;$
$u_{1}+u_{2}+u_{3}+u_{5}\leq a+b+c+d+e;\ u_{1}+u_{2}+u_{4}+u_{5}\leq
a+b+c+d+e;$ $u_{2}+u_{3}+u_{4}+2u_{5}\leq b+c+d+2e;\
u_{1}+u_{2}+u_{3}+u_{4}+2u_{5}\leq a+b+c+d+2e.$

The isomorphism takes $\xi_{5}$ to $\partial_{e}-\partial_{d}$.

It is easy to check that the polytope $\Delta_{5}$ is equal to the Minkowski
sum of FFLV polytopes $P_{1}(0,a)$, $P_{2}(0,b,b+c)$, $P_{3}(0,d,d+e,d+e)$
where the FFLV polytope
$P_{n-1}(\lambda_{1},\ldots,\lambda_{n})\subset\mathbb{R}^{\frac{n(n-1)}{2}}$
is defined by the inequalities $u_{k}\geq 0$ and

$\sum_{k\in D}u_{k}\leq\lambda_{j}-\lambda_{i}$

for all Dyck paths $D$ going from $\lambda_{i}$ to $\lambda_{j}$ in table
$(FFLV)$ where $1\leq i<j\leq n$:

$None$
$\begin{array}[]{cccccccccc}\ldots&&\lambda_{4}&&\lambda_{3}&&\lambda_{2}&&\lambda_{1}\\\
&\ldots&&u_{6}&&u_{3}&&u_{1}&\\\ &&\ldots&&u_{5}&&u_{2}&&\\\
&&&\ldots&&u_{4}&&&\\\ &&&&\cdot^{\cdot^{\cdot}}&&&&\\\ \end{array}.$

As a corollary, we get the following

######  Proposition 5.3.

The volume polynomial of the Minkowski sum
$P_{1}(0,a)+P_{2}(0,b,b+c)+P_{2}(0,d,d+e,d+e)$ of FFLV polytopes (considered
as a polynomial in $a$, $b$, $c$, $d$ and $e$) is equal to the self-
intersection index of the divisor
$a\xi_{1}+(b+c)\xi_{2}+c\xi_{3}+(d+e)\xi_{4}+e\xi_{5}$ on $R_{12132}$.

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