TWO PROBLEMS ON LATTICE POINT ENUMERATION OF RATIONAL

POLYTOPES

A thesis presented to the faculty ofSan Francisco State University

In partial fulfilment ofThe Requirements for

The Degree

Master of ArtsIn

Mathematics

by

Andres R. Vindas Melendez

San Francisco, California

August 2017

Copyright byAndres R. Vindas Melendez

2017

CERTIFICATION OF APPROVAL

I certify that I have read TWO PROBLEMS ON LATTICE POINT

ENUMERATION OF RATIONAL POLYTOPES by Andres R. Vindas

Melendez and that in my opinion this work meets the criteria for ap-

proving a thesis submitted in partial fulfillment of the requirements for

the degree: Master of Arts in Mathematics at San Francisco State Uni-

versity.

Federico Ardila-Mantilla Professor of Mathematics

Matthias Beck Professor of Mathematics

Serkan Ho§ten Professor of Mathematics

TVVO PROBLEMS ON LATTICE POINT ENUMERATION OF RATIONAL

POLYTOPES

Andres R. Vindas Melendez San Francisco State University

2017

Motivated by the generalization of Ehrhart theory with group actions, the first part

of this thesis makes progress towards obtaining the equivariant Ehrhart theory of the

permutahedron. The subset that is fixed by a group action on the permutahedron

is itself a rational polytope. We prove that these fixed polytopes are combinato-

rially equivalent to lower dimensional permutahedra. Furthermore, we show that

these fixed polytopes are zonotopes, i.e., Minkowski sum of line segments. This

part is joint work with Anna Schindler. The second part of this thesis provides a

decomposition of the h*-polynomial for rational polytopes. This decomposition is ·

an analogue to the decomposition proven by Ulrich Betke and Peter McMullen for

lattice polytopes.

I certify that the Abstract is a correct representation of the content of this thesis.

Chair, Thesis Committee Date

ACKNOWLEDGMENTS

Thank you to everyone who has crossed paths with me and has

helped me redefine what it means to be a mathematician.

Matt : Thank you for always being patient with me and recognizing that

everyone works at their own pace. Also, for introducing me to Ehrhart

theory my first semester at SFSU.

Federico: You saw my potential as a mathematician even when I did

not. Thank you for teaching me that there is beauty in counting and

that with mathematics you can have a life that is both academically and

socially fulfilling.

Thank you to all the friends that I have made at the SFSU Math

Department. To Serkan Hosten for serving on my committee and teach-

ing me how to communicate mathematics e↵ectively. To my academic

siblings, Anastasia Chavez, John Guo, and Anna Schindler for strug-

gling and succeeding with me. To Kim Seashore for caring and showing

me that education and mathematics are not exclusive. To my friends

Karly Behncke, Alejandra Chaidez, Maria Flores, and Paola Flores for

your artistic and moral support.

I also thank the Creator and my creators: Mami and Papi. To my

sisters Alejandra and Sarah, tıa Ana, tıo Julio, my cousins, and the rest

of my family: gracias por su amor y por apoyar mis metas. Este logro

es de todos. ¡Pura vida!

v

TABLE OF CONTENTS

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Ehrhart Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Ehrhart Theory in Dimension 2 . . . . . . . . . . . . . . . . . 15

2.2.2 Ehrhart Theory in General Dimension . . . . . . . . . . . . . 18

2.2.3 Ehrhart Theory for Rational Polytopes . . . . . . . . . . . . . 30

2.3 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Equivariant Ehrhart Theory . . . . . . . . . . . . . . . . . . . . . . . 38

3 Fixed Subpolytopes of the Permutahedron . . . . . . . . . . . . . . . . . . 42

4 A Decomposition of the h⇤-Polynomial for Rational Polytopes . . . . . . . 63

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

vi

Figure Page

LIST OF FIGURES

2.1 From left to right: A 1-simplex, 2-simplex, and 2-simplex. . . . . . . . 8

2.2 Two triangulations of the regular hexagon. . . . . . . . . . . . . . . . 9

2.3 ⇧3

is a 2-polytope in 3-space. . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Minkowski sum of a square and a triangle. . . . . . . . . . . . . . . . 13

2.5 Examples of di↵erent permutahedron subdivided into parallelepipeds. 14

2.6 S = conv{(�1, 1), (�1,�1), (1, 1), (1,�1)}. . . . . . . . . . . . . . . . 15

2.7 Dilations of S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 A hexagon on a triangular lattice. . . . . . . . . . . . . . . . . . . . . 17

2.9 A hexagon and its dilates. . . . . . . . . . . . . . . . . . . . . . . . . 18

2.10 Cone over a square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.11 The 3 + 3 + 1 labeled forests on three nodes. . . . . . . . . . . . . . . 25

2.12 link(7) = {1, 2, 3, 5, 6, e1, e2, e3, e4, e5}. . . . . . . . . . . . . . . . . . 27

2.13 A triangulations of S. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.14 P = conv{��1

2

, 1�

,��1

2

,�1�

,�

1

2

, 1�

,�

1

2

,�1�}. . . . . . . . . . . . . . . 32

3.1 ⇧3

(hexagon) and (⇧3

)(12)

(line segment). . . . . . . . . . . . . . . . . 44

3.2 ⇧4

(truncated octahedron) and (⇧4

)(12)

(hexagon). . . . . . . . . . . . 47

4.1 ⇧(�, p)� = �⇧(�, p)� + pw1

+ pw2

. . . . . . . . . . . . . . . . . . . . 65

4.2 Triangulated P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

vii

1

Chapter 1

Introduction

Ehrhart theory is a well-established field that studies lattice points in dilations of

polytopes. There are several excellent sources that can serve as exposure or further

reading to this area, including, but certainly not limited to [2, 13, 29].

Much of the structure connecting the discrete volume of a dilated polytope to

the number of lattice points it contains is encoded in its Ehrhart polynomial. An

object of interest is the discrete volume ehrP(m) := #(mP\Zd) of the mth dilate of

a polytope P that is invariant under the action of a group G. The counting function

that arises in this work for polytopes is known as the Ehrhart quasipolynomial of P .

The Ehrhart quasipolynomial is encoded in the Ehrhart series, written as a rational

function, where we call the numerator the h⇤-polynomial :

EhrP(z) =P

m�0

ehrP(m)zm =h⇤P(z)

(1� zp)dim(P)+1

.

2

This thesis is a result of two di↵erent projects within the realm of Ehrhart theory

and discrete geometry. The first part of this thesis was done with a goal of obtaining

an equivariant analogue of Ehrhart theory for the permutahedron. The second part

of this thesis was motivated by finding rational analogues to the works of Betke and

McMullen in [4] and Alan Stapledon in [30].

1.1 Part I

The first part of this thesis expands the work of Stapledon, where we make progress

towards answering one of his open problems in [31], specifically, determining the

equivariant Ehrhart theory of the permutahedron. This project is joint work with

Anna Maria Schindler.

Alan Stapledon described an equivariant analogue to Ehrhart theory as an ex-

tension of the theory with group actions. For the purposes of this thesis, we consider

our group to be the symmetric group Sd and our polytope P to be the permuta-

hedron ⇧d. The permutahedron is a (d � 1)-dimensional polytope embedded in an

d-dimensional space, the vertices of which are formed by permuting the coordinates

of the vector (1, 2, 3, . . . , d). An element � 2 Sd acts on Zd by permuting the coordi-

nates. Let (⇧d)� denote the set of lattice points of ⇧d that are fixed by a permutation

� 2 Sd; we call (⇧d)� the fixed sub-polytope of ⇧d. Stapledon proved that P� is a

rational polytope. Our work expands this theory where P = ⇧d and P� = (⇧d)� is

the fixed sub-polytope of the d-permutahedron. We prove the following theorem:

3

Theorem 3.8. Let ⇧d denote the d-permutahedron, and � 2 Sd, where � is com-

posed of disjoint cycles, i.e, � = �1

�2

· · · �m. Let (⇧d)� be the sub-polytope of ⇧d

fixed by �. Let V (�) denote the set of vertices v = (x1

, . . . , xd) of ⇧d that satisfy the

following property: for each cycle �j = (j1

j2

. . . jr) of �, xj1 , . . . , xjr is a consecutive

sequence of integers. Let e�j =P

i2�jei.

The following are equal:

(A) (⇧d)� = {x 2 ⇧d : � · x = x},

(B) V� = conv

(

1

|�|

|�|P

i=1

�iv : v 2 V (�)

)

,

(C) M� =P

j 6=k

⇥

lje�k, lke�j

⇤

+P

k

lk+1

2

e�k.

Theorem 3.8 establishes that (⇧d)� is a zonotope, the translation of a Minkowski

sum of line segments. Also, it shows that (⇧d)� = V�, providing a vertex description

of (⇧d)�. Stapledon’s results tell us that (⇧d)� = conv(W (�)), where W is the set of

all vertices of ⇧d and W (�) =n

1

|�|P|�|

i=1

�iw : w 2 Wo

. As V (�) is a proper subset

of W (�), our theorem provides a refinement of Stapledon’s definition. Additionally,

the fixed subpolytopes (⇧d)� are combinatorially equivalent to ⇧m, where m is the

number of cycles of �, and its vertex set is V (�).

The theory of lattice zonotopes provides a formula for the Ehrhart series of a

zonotope. Therefore, by proving that the fixed sub-polytopes are zonotopes we

hoped to determine its Ehrhart series, where we would have concluded our goal of

4

determining the equivariant Ehrhart theory of the permutahedron. Unfortunately,

our fixed sub-polytopes are not integral, and it is not known whether a similar

formula holds for rational zonotopes, which our fixed sub-polytopes are.

1.2 Part II

The second part of this thesis introduces a decomposition formula of the h⇤-polynomial

for rational polytopes. In 1985 Ulrich Betke and Peter McMullen [4] proved that

the h⇤-polynomial of a lattice polytope can be decomposed as

X

�2T

hlink(�)

(z)B�

(z),

where hlink(�)

and B�

(z) are defined in Section 2.2. This result is particularly inter-

esting because it brings together arithmetic data from the simplices of the triangu-

lation and combinatorial information from the face structure of the triangulation.

In Chapter 4 we prove the analogue of this theorem for rational polytopes:

Theorem 4.3. Fix a triangulation T of the rational d-polytope P such that p is the

least common multiple of all the vertex coordinate denominators of every � 2 T .

Then

EhrP(z) =

P

�2T hlink(�)

(zp)B�,p(z)

(1� zp)d+1

.

5

Chapter 2

Background

In this chapter we discuss the foundational topics that are required to reach our

main results.

2.1 Polytopes

We begin with a discussion of convex polytopes. For a more thorough introduction

to polytopes, see [2], [12], and [34]. For our pursuits, we focus only on objects that

are convex, so we begin with defining convexity.

Definition 2.1. A subset X of Rn is convex if {�x+(1��)y : 0 � 1} ✓ X for

every x, y 2 X. If we consider a finite set C in Rn, the convex hull of C, which we

denote conv(C), is the intersection of all convex sets containing C. Equivalently,

conv(C) =

(

X

c2C

�cc : �c 2 R�0

,X

c2C

�c = 1

)

.

6

The d-dimensional analogue of a convex polygon is a convex polytope, i.e., poly-

topes are geometric objects with flat faces and extreme points called vertices.

Definition 2.2. One way to define a convex polytope P is as the convex hull of

points, v1

, v2

, . . . , vn 2 Rn:

P = conv{v1

, v2

, . . . , vn} :=

(

�1

v1

+ · · ·+ �kvn : �j � 0,nX

j=1

�j = 1

)

.

This is known as the vertex-description (V-description) of P , and we consider P to

be a V-polytope.

Definition 2.3. The set {x 2 Rd : Ax = b} is called a hyperplane. The sets

{x 2 Rd : Ax b} and {x 2 Rd : Ax � b} are called closed halfspaces. We say that

the hyperplane H = {x 2 Rd : Ax = b} is a supporting hyperplane of P if P lies

entirely on one side of H.

Definition 2.4. A polytope can be also defined as the bounded intersection of

finitely many half-spaces and hyperplanes. This is known as the hyperplane descrip-

tion (H-description) of a polytope, where we consider P to be a H-polytope.

Theorem 2.5. A polytope P is a V-polytope if and only if it is a H-polytope.

For a proof of this theorem see [34].

Now that we know that these descriptions are equivalent, we need not specify

whether a polytope is an H-polytope or V-polytope.

7

Definition 2.6. The dimension of a polytope P is the dimension of the a�ne space

a↵spanP := {x+ �(y � x) : x, y 2 P ,� 2 R}

spanned by P . If P has dimension d, we denote it as dimP = d and refer to P a

d-polytope.

The following are several useful definitions regarding polytopes.

Definition 2.7. The integer points Zd form a lattice in Rd and we refer to integer

points as lattice points. Let P be a d-polytope, then

• a face of P is a set of the form P \H, where H is a supporting hyperplane of

P ,

• the (d� 1)-dimensional faces are called facets,

• the 1-dimensional faces are called edges,

• the 0-dimensional faces are called vertices of P and are the “extreme points”

of P ,

• a polytope P is called integral if all of its vertices have integer coordinates

(also called a lattice polytope),

• a polytope P is called rational if all of its vertices have rational coordinates,

8

• we are also interested in the dilations of P , which we denote mP := {mx :

x 2 P}.

One particular group of polytopes that will be helpful are simplices, which we

now define:

Definition 2.8. A convex d-polytope with exactly d+1 vertices is called a d-simplex

and is denoted �.

Figure 2.1: From left to right: A 1-simplex, 2-simplex, and 2-simplex.

Observe that a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex

is a tetrahedron.

Definition 2.9. A triangulation of a convex d-polytope P is a finite collection Tof simplices with the following properties:

• P = [�2T �

• For every �1

,�2

2 T ,�1

\�2

is a face common to both �1

and �2

.

9

Theorem 2.10. [2] Every convex polytope can be triangulated using no new vertices,

i.e., there exists a triangulation T such that the vertices of every � 2 T are vertices

of P.

Figure 2.2: Two triangulations of the regular hexagon.

Definition 2.11. A subdivision D of a polytope P is a non-empty, finite collection

of polytopes such that

i) Q 2 D implies all faces of Q are also in D,

ii) Q1

,Q2

2 D implies Q1

\Q2

is a common face of both Q1

and Q2

,

iii) [Q2DQ = P .

Definition 2.12. Consider a polytope P ⇢ Rd.

• Given a vector w 2 Rd, let facew(P) = {x 2 P : x · w y · w, for all y 2

10

P}. Then facew(P) is a polytope because it is the intersection of P with the

hyperplane x · w = min{x · w : x 2 P}. Such a polytope is called a face of P .

• For F a face of polytope P , we define the set

NP(F ) = {w 2 Rd : facew(P) = F},

the normal cone of P at F .

• The normal fan of P is the collection of all the normal cones of P ,

N(P) = {NP(F ) : F is a face of P}.

• Two polytopes are normally equivalent if their normal fans coincide.

A polytope of particular interest to us is the permutahedron. We will see that the

numerous combinatorial properties of the permutahedron make it a natural object

to study.

Definition 2.13. The permutahedron

⇧d := conv{(⇡(1), ⇡(2), ..., ⇡(d)) : ⇡ 2 Sd}

is the convex hull of (1, ..., d) and all points formed by permuting its entries.

The permutahedron has the following properties [32]:

11

i) ⇧d is a (d� 1)-dimensional polytope in Rd.

ii) ⇧d has d! vertices.

iii) All the vectors v� lie on the hyperplaneP

xi =d(d+1)

2

, where v� is the vertex

vector whose coordinates are the elements of �.

iv) Each v� is a vertex of ⇧d, and two vertices v� and v�0 are adjacent (i.e., con-

nected by an edge), if and only if the permutations � and �0 are adjacent (i.e.,

di↵er by the exchange of two consecutive numbers).

Note that a polytope P ⇢ Rd does not necessarily have dimension d, as is the

case with the permutahedron. As of now, we have seen the V-description of the

permutahedron. The following defines the H-description of the permutahedron.

Proposition 2.14. [5, 32, 34] The H-description of the permutahedron is

x1

+ · · ·+ xd =d(d+ 1)

2,

xi1 + · · ·+ xin � n(n+ 1)

2

for ; ( {i1

, . . . , in} ( [d].

The permutahedron is a nice polytope to study because it falls under a special

class of polytopes called zonotopes. Zonotopes are special polytopes that have sev-

eral equivalent definitions; they are Minkowski sums of line segments and projections

12

of cubes, see below. The following figure depicts the 3-permutahedron as concrete

examples of the aforementioned theory.

Figure 2.3: ⇧3

is a 2-polytope in 3-space.

Definition 2.15. Given d linearly independent vectors v1

, . . . , vd 2 Zm, we define

the d-dimensional lattice parallelepiped as

P (v1

, . . . , vd) := {�1

v1

+ �2

v2

+ · · ·+ �dvd : 0 �i 1} .

Definition 2.16. Consider the polytopes P1

,P2

, . . . ,Pn ⇢ Rd. We define the

Minkowski sum of the n polytopes as

P1

+ P2

+ · · ·+ Pn := {x1

+ x2

+ · · ·+ xn : xj 2 Pj for 1 j n}.

13

Figure 2.4: Minkowski sum of a square and a triangle.

Definition 2.17. The unit d-cube Cd := [0, 1]d is a d-dimensional polytope that

takes the standard unit basis vectors e1

, e2

, . . . , ed as its generating vectors:

Cd = {�1

e1

+ �2

e2

+ · · ·+ �ded : 0 �i 1}.

A zero-dimensional parallelepiped is a point, a line segment is a one-dimensional

parallelepiped, and a two-dimensional parallelepiped is a parallelogram. We consider

a parallelepiped as a higher-dimensional parallelogram. The unit d-cube is probably

the simplest example of a d-dimensional parallelepiped.

Definition 2.18. Take n line segments in Rd, where each line segment has one

endpoint at the origin and other endpoint is located at the vector uj 2 Rd, for

j = 1, . . . , n. We then define a zonotope as the Minkowski sum of these line segments:

Z(u1

, u2

, . . . , un) = {�1

u1

+ �2

u2

+ · · ·+ �nun : 0 �j 1}.

Note that any translation of a zonotope is itself a zonotope.

An interesting property of zonotopes is observed in the following theorem, which

14

was proved by Geo↵rey Shephard [25].

Theorem 2.19. [25] Every zonotope admits a subdivision into parallelepipeds.

(a) ⇧

3

tiled by parallelograms.

(b) ⇧

4

tiled by parallelepipeds [23].

Figure 2.5: Examples of di↵erent permutahedron subdivided into parallelepipeds.

We stated earlier that the permutahedron is a zonotope, this means it can be

written as a Minkowski sum of line segments. That is what the following theorem

states.

Theorem 2.20. [2] Let ⇧d be the d-permutahedron. Then

⇧d =X

j 6=k

[ej, ek] +X

j

ej ,

i.e., ⇧d is the Minkowski sum of the line segments between each pair of unit vectors

in Rd and a translation vector.

15

2.2 Ehrhart Theory

2.2.1 Ehrhart Theory in Dimension 2

The power of Ehrhart theory is best illustrated by examples.

Example 2.21. Let S = conv{(�1, 1), (�1,�1), (1, 1), (1,�1)}. This gives the

following square in R2.

Figure 2.6: S = conv{(�1, 1), (�1,�1), (1, 1), (1,�1)}.

The following properties of S can be easily verified from the image:

1. S has area 4.

2. S has 8 lattice boundary points.

3. S has a total of 9 lattice points including the boundary and interior.

These three properties are tied together by the expression

9 = 4 +8

2+ 1.

16

That is, the number of lattice points in S is equal to the sum of the area of S, one-

half the number of boundary points and one. In 1899, the Austrian mathematician

Georg Pick proved that this relationship holds for any convex lattice polygon [21].

Theorem 2.22 (Pick 1899). For any convex lattice polygon P

LP = AP +BP

2+ 1

where AP , BP , and LP denote the area of P , number of lattice points on the boundary

of P , and total number of lattice points of P , respectively.

If we dilate the polygon by a factor of m, is there a polynomial that counts the

number of lattice points? The answer to this is yes, and is the result of Ehrhart’s

theorem [10].

Theorem 2.23 (Ehrhart’s Theorem in Dimension 2). Let P be a convex lattice

polygon and let m be a positive integer. The following equality holds:

LP (m) = APm2 +

BP

2m+ 1.

Example 2.24. Taking S as defined in the previous example, we dilate by a factor

of m to obtain

LS(m) = 4m2 + 4m+ 1.

17

Figure 2.7: Dilations of S.

Recall that the permutahedron is a (d� 1)-dimensional polytope, so let us look

at another example, the 3-permutahedron, which is a hexagon.

Example 2.25. Let ⇧3

denote the following hexagon on a triangular lattice:

Figure 2.8: A hexagon on a triangular lattice.

Similar to the other examples, we can gather that the lattice points in dilations

of the hexagon can be counted by the following polynomial:

L⇧3 = 3m2 + 3m+ 1.

18

Figure 2.9: A hexagon and its dilates.

Next, we provide the tools to explore Ehrhart theory in higher dimensions.

2.2.2 Ehrhart Theory in General Dimension

We have seen in dimension two that the Ehrhart polynomial of a polygon is a

counting function for the number of integer points in the polygon. In general, we

consider a lattice polytope P ⇢ Rd and its Ehrhart function

ehrP(m) :=�

�mP \ Zd�

� .

We call the corresponding generating function

EhrP(z) := 1 +X

m�1

ehrP(m)zm

19

the Ehrhart series of P . The Ehrhart function turns out to be a polynomial in any

dimension. Before we continue detailing facts about the Ehrhart polynomial and its

Ehrhart series, we continue by stating some facts about triangulations, cones, and

other topics, which will become useful as we continue.

Definition 2.26. A pointed cone K ✓ Rd is a set of the form

K = {v + �1

w1

+ �2

w2

+ · · ·+ �mwm : �1

, . . . ,�m � 0},

where v, w1

, w2

, . . . , wm 2 Rd are such that there exists a hyperplane H for which

H \K = {v}, i.e., K\{v} lies strictly on one side of H. In this setting

• the vector v is called the apex of K,

• the wk are the generators of K,

• the cone is rational if v, w1

, . . . , wm 2 Qd,

• the dimension of K is the dimension of the a�ne space spanned by K,

• if K is of dimension d it is called a d-cone,

• a d-cone K is simplicial if K has precisely d linearly independent generators.

We study the arithmetic of a set S ✓ Rd by computing a multivariate gener-

ating function, which we call the integer-point transform. We want to encode the

information contained by the lattice points in the set S.

20

Definition 2.27. If S is a rational cone or polytope and ~z = (z1

, . . . , zd), then the

integer-point transform is

�S(~z) = �S(z1, z2, . . . , zd) :=X

m2S\Zd

~z m,

the generating function that lists all integer points in S as a formal sum of Laurent

monomials. Here ~zm = zm11

· · · zmdd .

Generating functions of this form are extremely helpful for lattice-point problems.

When we are interested in the lattice-point count, we evaluate �S at z = (1, 1, . . . , 1):

�S(1, 1, . . . , 1) =X

m2S\Zd

~1m =X

m2S\Zd

1 = #(S \ Zd),

where ~1 denotes the vector whose components are all 1.

Theorem 2.28. Suppose

K := {�1

w1

,�2

w2

+ · · ·+ �dwd : �1

,�2

, . . . ,�d � 0}

is a simplicial d-cone, where w1

, . . . , wd 2 Zd. Then for v 2 Rd, the integer-point

transform �v+K of the shifted cone v +K is the rational function

�v+K =�v+⇧

(~z)

(1� ~z w1)(1� ~z w2) · · · (1� ~z wd),

21

where ⇧ is the half-open parallelepiped

⇧ := {�1

w1

+ �2

w2

+ · · ·+ �dwd : 0 �1

,�2

, . . . ,�d < 1}.

The half-open parallelepiped ⇧ is called the fundamental parallelepiped of K.

Corollary 2.29. For a pointed cone

K = {v + �1

w1

+ �2

w2

+ · · ·+ �1

wm : �1

,�2

, . . . ,�m � 0}

with v 2 Rd, w1

, w2

, . . . , wm 2 Zd, the integer-point transform �K(z) evaluates to a

rational function in the coordinates of z.

For a proof of Theorem 2.28 and Corollary 2.29, refer to [2].

Now, you may be wondering: why are cones important? The answer to that question

in our setting is that we can cone over a polytope. If we are given a polytope P ⇢ Rd

with vertices v1

, v2

, . . . , vn, we lift these vertices into Rd+1 by adding a 1 to their

last coordinate. We then proceed by letting the generators of cone(P) be

w1

= (v1

, 1), w2

= (v2

, 1), . . . , wn = (vn, 1).

With this, now we define our cone over a polytope.

22

Definition 2.30. The cone over P = conv(v1

, . . . , vn) is defined to be

cone(P) = {�1

w1

+ �2

w2

+ · · ·+ �nwn : �1

,�2

, . . . ,�n � 0} ⇢ Rd+1,

where wi = (vi, 1) for i = 1, . . . , n

This pointed cone has the origin as its apex, and we can obtain our original

polytope P (to be more precise, its translated copy {(x, 1) : x 2 P}) by cutting

cone(P) with the hyperplane xd+1

= 1, as can be seen in Figure 2.10. We can then

compute the Ehrhart series of P as

EhrP(z) =X

m�0

#(lattice points in cone(P) at height m)zm.

Figure 2.10: Cone over a square.

This leads us to the following theorem.

23

Theorem 2.31. [2] For a lattice polytope P

�cone(P)

(1, 1, . . . , 1, z) = 1 +X

t�1

ehrP(m)zm = EhrP(z).

Definition 2.32. The normalized volume of a d-polytope P on a lattice M is its

d-dimensional volume divided by the volume of any fundamental parallelepiped of

M , i.e., the parallelepiped generated by any lattice basis. The surface volume of P

is the sum of (d� 1)-dimensional volumes of its facets.

The following result is due to the French mathematician Eugene Ehrhart [10].

Theorem 2.33 (Ehrhart 1962). Let P be a d-polytope with integer vertices in Rn.

Then

ehrP(m) = cdmd + cd�1

md�1 + · · ·+ c1

m+ c0

is a polynomial in m of degree d, which we call the Ehrhart polynomial. Furthermore,

cd = volume of P , cd�1

=1

2(surface volume of P) , and c

0

= 1.

Now we encode the Ehrhart polynomial in a generating function. As mentioned

earlier, this is the Ehrhart series.

24

Corollary 2.34. For a lattice polytope P in Rn, the Ehrhart series of P is

EhrP(z) =X

m�0

ehrP(m)zm = ehrP(0)z0 + ehrP(1)z

1 + ehrP(2)z2 + · · ·

=h⇤dz

d + h⇤d�1

zd�1 + · · ·+ h⇤1

z + h⇤0

(1� z)d+1

,

where h⇤k equals the number of integer points in the fundamental parallelepiped ⇧ of

cone(P) that are on the hyperplane xd+1

= k.

For proofs and further reading, refer to [2, 3, 6, 10]

We call the numerator of EhrP the h⇤-polynomial, and the vector (h⇤0

, . . . , h⇤d) of its

coe�cients is called the h⇤-vector for P . One theorem about this numerator is due

to Richard Stanley and is sometimes called Stanley’s non-negativity theorem [26].

Theorem 2.35 (Stanley 1980). If P is a lattice d-polytope, then (h⇤0

, . . . , h⇤d) 2 Zd+1

�0

.

Definition 2.36. A graph is a pair of sets (V,E), where V is the set of nodes and

E is the set of edges, formed by pairs of vertices. Furthermore, a forest is a graph

with no cycles.

Theorem 2.37 (Ehrhart Polynomial of the Permutahedron). The coe�cient ck of

the Ehrhart polynomial

ehr⇧d(m) = cd�1

md�1 + · · ·+ c1

m+ c0

25

of the permutahedron ⇧d equals the number of labeled forests on d nodes with k edges.

For more on this theorem including its proof, refer to [2, 29].

Going back to our original example, we can compute the Ehrhart polynomial

seen in Figure 2.11 for the hexagon using the information from the above theo-

rem. We see that there are 3 labeled forests with 2 edges, 3 with 1 edge, and 1

with 0 edges. This is consistent with the Ehrhart polynomial in Example 2.25:

ehr⇧3(m) = 3m2 + 3m+ 1.

Figure 2.11: The 3 + 3 + 1 labeled forests on three nodes.

We can also use this method to find the Ehrhart polynomial for ⇧4

. On 4 vertices

there are 16 labeled forests wth 3 edges, 15 with 2 edges, 6 with one edge, and 1

with 0 edges, giving us

ehr⇧4(m) = 16m3 + 15m2 + 6m+ 1.

26

Let us now look at a theorem that shows a decomposition of the h⇤-polynomial

of a lattice polytope which we later generalize for rational polytopes in Chapter 4.

First, we provide some set-up and definitions. Given a lattice d-polytope P , we fix

a lattice triangulation T . Given a simplex � 2 T , let ⇧(�)� be the fundamental

parallelepiped of cone(�) and let B�

(z) := �⇧(�)

�(1, 1, . . . , 1, z). Thus, B�

(z) is

an “open”variant of h⇤�

(z) = �⇧(�)

(1, 1, . . . , 1, z). We include Ø in the collection of

faces of a triangulation of a polytope and by convention we define dim(Ø) := �1;

we consider Ø the empty simplex and define BØ

(z) := 1. We need one more concept

before introducing the theorem:

Definition 2.38. Given a simplex � 2 T , let

linkT (�) := {⌦ 2 T : ⌦ \� = Ø,⌦ ✓ � for some � 2 T with � ✓ �},

the link of �. In words, this is the set of simplices that are disjoint from �, yet are

still a face of a simplex in T containing �. We then encode the combinatorial data

coming from the simplices in the h-polynomial (to not deviate too much, for further

reading on h-polynomials, see [2, Theorem 10.2] or [29]) of the link as

hlink(�)

(z) = (1� z)d�dim(�)

X

�◆�

✓

z

1� z

◆

dim(�)�dim(�)

, (2.1)

where the sum is over all simplices � 2 T containing �, and d is the dimension of

the polytope P .

27

Figure 2.12: link(7) = {1, 2, 3, 5, 6, e1, e2, e3, e4, e5}.

In Figure 2.12 the hexagon is a simplicial complex. The link of 7 in the simplicial

complex is link(7) = {1, 2, 3, 5, 6, e1, e2, e3, e4, e5}. The aforementioned set-up was

necessary to introduce the following theorem. It decomposes the h⇤-polynomial and

is due to Ulrich Betke and Peter McMullen [4].

Theorem 2.39 (Betke & McMullen 1985). Fix a lattice triangulation T of the

lattice d-polytope P. Then

EhrP(z) =

P

�2T hlink(�)

(z)B�

(z)

(1� z)d+1

.

Consider an example we have seen before, S = conv{(�1, 1), (�1,�1), (1, 1), (1,�1)}.The following are two ways to compute its Ehrhart series: from its Ehrhart polyno-

mial and then by using Theorem 2.39.

Example 2.40. Take the polytope S defined in Example 2.21. Then the Ehrhart

28

Series of S is

EhrS(z) =X

m�0

ehrS(m)zm

=X

m�0

(4m2 + 4m+ 1)zm

=z2 + 6z + 1

(1� z)3.

Next we use Theorem 2.39. Fix the following triangulation.

Figure 2.13: A triangulations of S.

Through careful computation, we obtain:

29

Applying Theorem 2.39, we obtain

EhrS(z) =z2 + 6z + 1

(1� z)3,

which agrees with the Ehrhart series above.

To see how to compute B�

(z), you may refer to Example 4.4.

From the decomposition of the h⇤-polynomial, provided by Betke and McMullen,

Alan Stapledon was able to prove the following relationship involving the h⇤-polynomial.

30

Theorem 2.41 (Stapledon 2009). [30] If h⇤P(z) has degree s, then there exist unique

polynomials a(z) and b(z) with nonnegative coe�cients such that

(1 + z + · · ·+ zd�s)h⇤P(z) = a(z) + zd+1�sb(z),

a(z) = zda(1z ), b(z) = zd�lb(1z ), and, writing a(z) = adzd + ad�1

zd�1 + · · ·+ a0

,

1 = a0

a1

aj for 2 j d� 1.

2.2.3 Ehrhart Theory for Rational Polytopes

For this subsection, let P be a d-dimensional convex polytope in Rn whose vertices

have rational coordinates. As stated earlier, we call P a rational polytope. Here we

see some results that parallel the results for lattice polytopes.

Definition 2.42. A quasipolynomial Q is an expression of the form

Q(m) = cn(m)mn + · · ·+ c1

(m)m+ c0

(m),

where c0

, . . . , cn are periodic functions in m. The degree of Q is n and the least

common multiple of c0

, . . . , cn is the period of Q. An alternate way of looking

at a quasipolynomial Q is that there exist a positive integer k and polynomials

p0

, p1

, . . . , pk�1

such that

31

Q(m) =

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

p0

(m) if m ⌘ 0mod k,

p1

(m) if m ⌘ 1mod k,

...

pk�1

(m) if m ⌘ k � 1mod k.

The minimal such k is the period of Q, and for this minimal k, the polynomials

p0

, p1

, . . . , pk�1

are the constituents of Q.

Theorem 2.43 (Ehrhart’s Theorem for Rational Polytopes). If P is a rational d-

polytope, then ehrP(m) is a quasipolynomial in m of degree d. Its period divides the

least common multiple of the denominators of the coordinates of the vertices of P.

Corollary 2.44. If P is a rational d-polytope in Rn and p 2 Z>0

is such that pPis a lattice polytope, the Ehrhart series of P is

EhrP(z) = 1 +X

m�1

ehrP(m)zm =h⇤P,p(z)

(1� zp)d+1

,

where the numerator is called the h⇤-polynomial and is of degree less than p(d+ 1).

Note that the h⇤-polynomial depends on the choice of p.

One possible p is any multiple of the denominators of the vertex coordinates of

P . The proofs for the previous theorems can be found in [2, 16, 27]. The following

theorem is in the class of reciprocity theorems and is thanks to Ian Macdonald [16].

32

Theorem 2.45 (Ehrhart–Macdonald Reciprocity Theorem). Suppose P is a convex

rational d-polytope and m 2 Z�0

. Then the evaluation of the quasipolynomial ehrP

at negative integers yields

ehrP(�m) = (�1)d ehrP�(m),

where P� is the relative interior of P.

Now that we have this information about rational polytopes, let us look at an

example.

Example 2.46. Let P = conv{��1

2

, 1�

,��1

2

,�1�

,�

1

2

, 1�

,�

1

2

,�1�}.

Figure 2.14: P = conv{��1

2

, 1�

,��1

2

,�1�

,�

1

2

, 1�

,�

1

2

,�1�}.

33

The Ehrhart polynomial of P can be shown to be

ehr(m) =

8

>

>

<

>

>

:

2m2 + 3m+ 1 when m is even,

2m2 +m when m is odd.

We now compute the Ehrhart series:

Ehr(z) =X

m�0

ehr(m)zm

=X

m�0

m even

(2m2 + 3m+ 1)zm +X

m�1

m odd

(2m2 +m)zm

=X

k�0

[2(2k)2 + 3(2k) + 1]z2k +X

k�0

[2(2k + 1)2 + (2k + 1)]z2k+1

=3z4 + 12z2 + 1

(1� z2)3+

z5 + 12z3 + 3z

(1� z2)3

=z5 + 3z4 + 12z3 + 12z2 + 3z + 1

(1� z2)3,

which is in agreement with Corollary 2.44

2.3 Representation Theory

Before we get to equivariant Ehrhart theory, we recall in this section some fundamen-

tal facts from representation theory. For an introduction to this field of mathematics

and proofs of the statements made, we refer to [11] and [24].

34

Definition 2.47. Let G be a finite group with identity e, and let S be a finite set.

A group action of G on S is an operation · : G⇥ S ! S such that, for all g, h 2 G

and all s 2 S:

1. g · s 2 S

2. (gh) · s = g · (h · s)

3. e · s = s.

Definition 2.48. Let V be a d-dimensional vector space over C. Let Matd be the

set of all d ⇥ d matrices with complex entries and define GLd to be the complex

general linear group of degree d,

GLd = {X : X = (xi,j)d⇥d 2 Matd is invertible}.

Definition 2.49. A representation of G is a multiplication · : G ⇥ V ! V such

that, for all g, h 2 G, all v, w 2 V , and all c, d 2 C:

1. g · v 2 V

2. g · (cv + dw) = c(g · v) + d(g · w)

3. (gh) · v = g · (h · v)

4. e · v = v.

35

Observe that each group element is associated with a linear transformation of

V . If we choose a basis for V , we can consider the representation as a group

homomorphism X : G ! GLd mapping each group element to the matrix associated

with this linear transformation. We summarize this idea in the following definition.

Definition 2.50. A matrix representation of a group G is a group homomorphism

X : G ! GLd .

From this definition it follows that X(e) = I is the identity matrix, where e is

the identity element in the group G, and X(gh) = X(g)X(h), for all g, h 2 G.

Let us look at some examples of representations on the symmetric group Sd.

Example 2.51. The trivial representation of Sd:

X : Sd ! GL1

X(g) = 1 8g 2 Sd

Example 2.52. The sign representation of Sd:

X : Sd ! GL1

X(g) = sign(g) 8g 2 Sd

36

where sign(g) is the function yielding �1 if g is an odd permutation, and 1 if g is an

even permutation. Any permutation may be written as a product of transpositions.

If the number of transpositions is even then it is an even permutation, otherwise it

is an odd permutation.

Definition 2.53. Let S be a finite set and G a group. Let CS denote the vector

space over C generated by S. If G has a group action on S, then this action can be

extended to CS, giving us a representation called a permutation representation.

Example 2.54. Let X be a permutation representation. Then, for all g 2 G, X(g)

has all entries equal to either 0 or 1. If we let X(g) = (xi,j)d⇥d, where the rows and

columns are indexed with basis elements s1

, . . . , sd of CS, then

xi,j =

8

>

>

<

>

>

:

1 if g · sj = si,

0 otherwise.

Example 2.55. Consider S3

and the permutation representation

X : S3

! GL3

.

37

Then

X(e) =

2

6

6

6

6

4

1 0 0

0 1 0

0 0 1

3

7

7

7

7

5

, X((1, 2)) =

2

6

6

6

6

4

0 1 0

1 0 0

0 0 1

3

7

7

7

7

5

,

X((1, 3)) =

2

6

6

6

6

4

0 0 1

0 1 0

1 0 0

3

7

7

7

7

5

, X((2, 3)) =

2

6

6

6

6

4

1 0 0

0 0 1

0 1 0

3

7

7

7

7

5

,

X((1, 2, 3)) =

2

6

6

6

6

4

0 0 1

1 0 0

0 1 0

3

7

7

7

7

5

, X((3, 2, 1)) =

2

6

6

6

6

4

0 1 0

0 0 1

1 0 0

3

7

7

7

7

5

.

Definition 2.56. The group character of a representation X is the map � : G ! C

defined by �(g) = tr(X(g)) =Pd

i=1

xi,i.

If X(g) is a permutation representation, then xi,i = 1 if g · si = si, and xi,i = 0

otherwise. Thus �(g) counts the number of basis elements of CS that are fixed

under the action of g. We will call these fixed points.

Definition 2.57. A class function is a function on a group G that is constant on

the conjugacy classes of G.

Note that the character of a representation of G over a field is always a class function.

Example 2.58. One important example of a permutation representation is that

38

associated with the left action of G on itself, called the standard representation of

G, or Xstd

. Note that, if g 6= e, {h 2 G : g · h = h} = ;, i.e., there are no points

fixed by g. On the other hand, all points in G are fixed by e. Thus, if �std

denotes

the character associated with the standard representation,

�std

(g) =

8

>

>

<

>

>

:

|G| if g = e,

0 otherwise.

2.4 Equivariant Ehrhart Theory

In 2011, Alan Stapledon described an equivariant analogue to Ehrhart theory; refer

to [31] for further reading and proofs of theorems stated in this section. We will see

that equivariant Ehrhart theory allows us to encode the action of a group G on a

polytope P . In particular, the theory organizes and quantifies the points of P that

are fixed by the action of the various elements of G. Let

• G be a finite group,

• P be a lattice polytope that is invariant under the action of G,

• mP be the mth dilate of P ,

• XmP denote the permutation representation induced by the action of G on the

lattice points mP \ Zd.

39

Our first object of interest is the character associated with XmP , which we denote

by �mP . For a fixed element g 2 G, recall that a permutation character evaluated at

g counts the number of points fixed by the group action. It is the case that �mP(g)

is the character that counts the number of lattice points in mP that are fixed by

the group action.

Theorem 2.59 (Stapledon 2011). Let Pg denote the set of lattice points of P that

are fixed by g, i.e., Pg = {x 2 P : g · x = x}. Then

Pg = conv

8

<

:

1

|g||g|X

i=1

gi · v : v is a vertex of P9

=

;

is a rational polytope.

Corollary 2.60. Let m 2 Z be the dilation factor of P, then mPg is a rational

polytope.

Computing �mP is equivalent to enumerating the lattice points contained in mPg

for each g 2 G. We can look to Ehrhart theory to address this enumeration.

Corollary 2.61. If ehrPg(m) denotes the Ehrhart quasi-polynomial for Pg, then

�mP(g) = ehrPg(m).

Since mPg is a rational polytope, it follows that the characters �mP(g) are quasi-

polynomials, i.e.,

�mP(g) = ehrPg(m) = cn(m)mn + · · ·+ c1

(m)m+ c0

(m),

40

where c0

, . . . , cn are periodic functions in m.

Recall that, as a class function, �mP is constant on conjugacy classes (for a fixed

m). Thus we study �mP by looking at the character individually on each conjugacy

class. Taken together, these characters give us a full picture of how G acts on the

dilates of P .

Note that if g = e, then �mP(g) is simply the ordinary Ehrhart quasi-polynomial

ehrP(m). Furthermore, if G is the trivial group, equivariant Ehrhart theory reduces

to ordinary Ehrhart theory.

Corollary 2.62. The leading coe�cient cn(m) of �mP(g) is equal tovolP|G| �std

.

The second leading coe�cient cn�1

(m) has similar properties:

Corollary 2.63.

cn�1

(m)(g) =

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

surface volume P2

if g = e and P is a lattice polytope,

volume Pg if g is a transposition and m ⌘ 0mod ind(Pg),

0 otherwise,

where the index of Pg, denoted ind(Pg), is the smallest positive integer m such that

the a�ne span of mPg contains a lattice point.

Now, we turn our attention to the generating series for �mP (g).

41

Definition 2.64. Let ⇢ denote the permutation representation of G on the lattice

M , i.e., ⇢ : G ! GL(M). Then, we define the equivariant Ehrhart series as

X

m�0

�mP (g)zm =

�(z)

(1� z) det[I � ⇢(g)z]

for each g 2 G.

Thus, �(z) is the equivariant analog to the h⇤ polynomial of ordinary Ehrhart theory;

�(z) is a rational function [31].

Just as �mP restricts to the Ehrhart quasi-polynomial for P when G is the

trivial group, in this case �(z) restricts to the h⇤-polynomial. This can be verified

by noting that for a d-dimensional lattice, I � ⇢z is a d ⇥ d diagonal matrix with

diagonal entries. For a lattice polytope, this means that its determinant is (1� z)d.

Comparing with the closed form of the Ehrhart series, we see that �(z) = h⇤(z). In

the next chapter we compute some examples.

42

Chapter 3

Fixed Subpolytopes of the Permutahedron

In the following, we compute examples and prove certain theorems, which lead us

closer to describing an equivariant Ehrhart theory for the permutahedron.

First, we compute �mP(g) for each conjugacy class of S3

and S4

as well as compute

the closed form of the generating series. By comparing our result to

X

m�0

�mP (g)zm =

�(z)

(1� z) det[I � ⇢(g)z]

we will recover �(z).

Example 3.1. Let P = ⇧3

be the 3-permutahedron and letG = S3

act on the lattice

points of m⇧3

by permuting their coordinates. Note that S3

has three conjugacy

classes: that of e, (ij), and (ijk). Take � 2 S3

.

43

e:

For � = e, (⇧3

)e = ⇧3

, so �m⇧3(e) is the ordinary Ehrhart polynomial for the

3-permutahedron, which we saw in Example 2.25:

�m⇧3(e) = ehr⇧3(m) = 3m2 + 3m+ 1.

The corresponding Ehrhart series is

X

m�0

�

3m2 + 3m+ 1�

zm =z2 + 4z + 1

(1� z)3.

In this case the denominator of the equivariant series will be (1�z)3, so we conclude

that �(z) = z2 + 4z + 1, which is, as expected, the ordinary h⇤-polynomial for ⇧3

.

(ij ):

For � = (ij), (⇧3

)� is the intersection of ⇧3

with the plane xi = xj, which is the axis

of reflection for the transformation enacted by (ij). Thus, (⇧3

)� is a line segment

with rational vertices, see Figure 3.1. By counting lattice points in dilates of (⇧3

)�,

it can be checked that its quasipolynomial is given by

�m⇧3(�) =

8

>

>

<

>

>

:

m+ 1 if m is even,

m if m is odd.

44

For � 2 {(ij)} we then have

X

m�0

m even

(m+ 1) zm +X

m�0

m odd

mzm =z3 + z2 + z + 1

(1� z2)2.

Comparing with

�(z)

(1� z) det [I � ⇢((ij))z]=

�(z)

(1� z)(1� z2),

we see that �(z) = z2 + 1.

Figure 3.1: ⇧3

(hexagon) and (⇧3

)(12)

(line segment).

(ijk):

Finally, for � = (ijk), (⇧3

)(ijk) is the intersection of ⇧

3

with the line given by

x1

= x2

= x3

. This consists only of one lattice point. Thus

�m⇧3(�) = 1.

45

Then for the associated generating function for � 2 {(ijk)}, we have

X

m�0

zm =1

1� z=

�(z)

(1� z) det [I � ⇢((ijk))z]=

�(z)

1� z3

where �(z) = z2 + z + 1.

Example 3.2. Let P = ⇧4

be the 4-permutahedron and let G = S4

act on the

lattice points of m⇧4

by permuting their coordinates. S4

has five conjugacy classes:

those of e, (ij), (ijk), (ij)(kl), (ijkl). Take � 2 S4

.

e:

For � = e, we have the ordinary Ehrhart polynomial for the 4-permutahedron:

�m⇧4(e) = 16m3 + 15m2 + 6m+ 1.

Computing the generating series for �m⇧4(e), we find that

X

m�0

(16m3 + 15m2 + 6m+ 1)zm =6z3 + 55z2 + 34z + 1

(1� z)4.

Thus, �(z) = 6z3 + 55z2 + 34z + 1 is once again the h⇤-polynomial for the Ehrhart

series.

(ij ):

Next, we find �m⇧4((ij)). Note that (⇧4

)(ij) is the intersection of ⇧

4

with the

hyperplane xi = xj. This yields a two-dimensional cross section of ⇧4

. We observe

46

that this is a hexagon whose edges bisect the facets of ⇧4

, see Figure 3.2. This

is a rational polytope with denominator 2, and thus we expect a degree 2 quasi-

polynomial with period 2. Enumerating the lattice points in several dilates, we

deduce that

�m⇧4((ij)) =

8

>

>

<

>

>

:

4m2 + 3m+ 1 if m is even,

4m2 + 2m if m is odd.

To recover �(z) we compute

X

m�0

m odd

�

4m2 + 2m�

zm +X

m�0

m even

�

4m2 + 3m+ 1�

zm

=X

m�0

�

4m2 + 2m�

zm +X

m�0

m even

(m+ 1) zm

=2z4 + 9z3 + 15z2 + 5z + 1

(1� z2)3=

�(z)

(1� z) det [I � ⇢((ij))z]=

�(z)

(1� z)3(1 + z).

where �(z) = 2z3 + 7z2 + 8z � 3 +4

1 + z.

Remark 3.3. This example demonstrates that unlike for integral polytopes �(z) is

not always a polynomial.

47

Figure 3.2: ⇧4

(truncated octahedron) and (⇧4

)(12)

(hexagon).

(ijk):

Next we consider �m⇧4((ijk)). Observe that (⇧4

)(ijk) is the intersection of ⇧

4

with

the plane xi = xj = xk. This intersection yields a lattice polytope: a line seg-

ment through the center of ⇧4

with endpoints on two of its hexagonal facets. One

can check that the number of lattice points on (⇧4

)(ijk) is given by the degree-one

polynomial

�m⇧4((ijk)) = m+ 1.

We compute the generating series and find �(z):

X

m�0

(m+ 1)zm =1

(1� z)2=

�(z)

(1� z)(1� z3)

where �(z) = z2 + z + 1.

48

(ij )(kl):

Consider �m⇧4((ij)(kl)). By similar reasoning as above, (⇧4

)(ij)(kl) is also a line

segment through the center of ⇧4

, but in this case it is the rational polytope with

endpoints on two of the square facets of ⇧4

. (⇧4

)(ij)(kl) has denominator 2, so once

again we should have a quasi-polynomial of period 2 and degree one. By counting

lattice points we conclude that

�m⇧4((ij)(kl)) =

8

>

>

<

>

>

:

2m+ 1 if m is even,

2m if m is odd.

Once again we find the generating series and solve for �(z):

X

m�0

m even

(2m+ 1) zm +X

m�0

m odd

(2m) zm =2z2 + z + 1

(1� z)2(1 + z)=

�(z)

(1� z)2(1 + z)2

where �(z) = 2z3 + 3z2 + 2z + 1.

(ijkl):

Finally, for permutations (ijkl), (⇧4

)(ijkl) consists only of the point in ⇧

4

such that

x1

= x2

= x3

= x4

. This point has coordinates (2.5, 2.5, 2.5, 2.5), so we can see that

(m⇧4

)(ijkl) is a lattice point if and only if m is even. Thus

49

�m⇧4((ijkl)) =

8

>

>

<

>

>

:

1 if m is even,

0 if m is odd.

Here, we have the Ehrhart series

X

m�0

m even

zm =1

1� z2=

�(z)

(1� z)(1 + z)(z2 + 1)

where �(z) = z2 + 1.

We can determine the dimensions of the fixed sub-polytopes as a result of the

following theorem.

Proposition 3.4. Let � = �1

· · · �m 2 Sd act on ⇧d by permuting coordinates, then

(⇧d)� has dimension m� 1, where m equals the number of disjoint cycles.

Proof. ⇧d has dimension d � 1 and points in ⇧d are determined by d parameters.

Suppose � permutes the coordinates of ⇧d via k transpositions. Then (⇧d)� is the

intersection of ⇧d with the set of points satisfying xi = xj for k pairs i, j (not

necessarily disjoint). This means that the number of parameters in the points of

⇧d has been reduced by k, yielding a polytope (⇧d)� of dimension d � 1 � k. It

is known that the minimum number of transpositions that can be used to write a

permutation as a product of transpositions is precisely d�m, where m is number of

disjoint cycles in the given permutation on d letters [17]. Letting k be the minimum

50

number of transpositions, we have that k = d�m. Equivalently, m = d� k. Thus,

d� 1� k = d� k� 1 = m� 1, the result follows. Note that there is no redundancy

because each � = �1

· · · �m 2 Sd is a product of disjoint cycles and each cycle has

distinct elements from 1, . . . , d.

The following theorem establishes a vertex description of the fixed sub-polytopes

and that they are zonotopes. Furthermore, it establishes that three representations

of the fixed sub-polytope are equal.

Definition 3.5. Let ⇧d denote the d-permutahedron, and � 2 Sd with disjoint

cycles �1

, �2

, . . . , �m. If W is the set of all vertices of ⇧d then

W� =

8

<

:

1

|�||�|X

i=1

�iw : w 2 W

9

=

;

is the set of points of W fixed by �. Using Theorem 2.59, it follows that (⇧d)� =

conv(W�).

Lemma 3.6. For any vertex w = (w1

, w2

, . . . , wd) of ⇧d, let w� be the corresponding

vertex of (⇧d)�, where w� = 1

|�|

|�|P

i=1

�iw. Then

w� =mX

j=1

P

i2�jwi

lje�j

!

.

Proof. Let w�i be the component wise product of w with e�i , and |�j| = lj. Since

�1

, �2

, . . . �m are disjoint cycles a↵ecting only w�1 , w�2 , . . . , w�m , respectively, we can

51

break up the sum as follows:

w� =1

|�||�|X

i=1

�iw

=1

|�||�|X

i=1

�i(w�1 + w�2 + · · ·+ w�m)

=1

|�|mX

j=1

0

@

0

@

|�|X

i=1

�iw�j

1

A

=1

|�|mX

j=1

0

@

0

@

|�||�j|

|�j |X

i=1

�iw�j

1

A

=1

|�|mX

j=1

0

@

|�||�j|

0

@

X

i2�j

wi

1

A e�j

1

A

=mX

j=1

0

@

1

|�j|

0

@

X

i2�j

wi

1

A e�j

1

A

=mX

j=1

P

i2�jwi

lje�j

!

.

Note that the fourth equality follows from the fact that each time a cycle of � acts

on w�j |�| times the permutations, w�j , �w�j , �2w�j , . . . , �

|�|w�j cycle through the

same values |�||�j | times.

52

Lemma 3.7.

X

j>k

[0, lke�j�lje�k]+

lm+···+l1P

i=lm+···+l2+1

i

l1

e�1+· · ·+

lmP

i=1

i

lme�m =

X

j 6=k

⇥

lje�k, lke�j

⇤

+X

j

lj + 1

2e�j .

Proof. Let us start by noting that the numerator of the coe�cient of e�kis a sum of

consecutive integers. In particular, it is the sum of the consecutive integers starting

with 1 +P

j>klj and ending with

P

j�klk. Thus the numerator of the coe�cient of e�k

is equal to

mP

j=k+1

lj +lk + 1

2

!

lk.

This means that the coe�cient of e�kis

mP

j=k+1

lj +lk + 1

2

!

lk

lk=

mP

j=k+1

lj +lk + 1

2.

Furthermore, M� can be expressed as

M� =X

j>k

[0, lke�j � lje�k] +

mX

k=1

mX

j=k+1

lj +lk + 1

2

!

e�k

!

.

Summing over k,

X

j>k

[0, lke�j � lje�k] +X

k

✓✓

lm + · · ·+ lk+1

+lk+1

2

◆

e�k

◆

=X

j>k

[0, lke�j � lje�k] +X

j>k

lje�k+X

k

lk+1

2e�k

=X

j 6=k

⇥

lje�k, lke�j

⇤

+X

k

lk + 1

2e�k

.

53

In the following theorem, we provide three di↵erent descriptions for the sub-

polytopes fixed by an action of the symmetric group, and prove that they are in

fact the same set. The three descriptions, which are detailed below, are denoted

(⇧d)�, conv(V�), and M�. This proof provides a vertex description of the fixed

sub-polytopes and establishes that they are also zonotopes.

Theorem 3.8. Let ⇧d denote the d-permutahedron, and � 2 Sd, where � is com-

posed of disjoint cycles, i.e, � = �1

�2

· · · �m. Let (⇧d)� be the sub-polytope of ⇧d

fixed by �. Let V (�) denote the set of vertices v = (x1

, . . . , xd) of ⇧d that satisfy the

following property: for each cycle �j = (j1

j2

. . . jr) of �, xj1 , . . . , xjr is a consecutive

sequence of integers. Let e�j =P

i2�jei.

The following are equal:

(A) (⇧d)� = {x 2 ⇧d : � · x = x},

(B) V� = conv

(

1

|�|

|�|P

i=1

�iv : v 2 V (�)

)

,

(C) M� =P

j 6=k

⇥

lje�k, lke�j

⇤

+P

k

lk+1

2

e�k.

Proof. We prove that these three items are equal through containment as follows:

i) (B) ✓ (A)

ii) (C) = (B)

iii) (A) ✓ (C)

54

We begin:

i) (B) ✓ (A):

The first containment follows from refining the definitions that Stapledon pro-

vided in his work [31]. He originally defined the fixed sub-polytopes as the

convex hull of a larger set, W (�). Since V (�) is contained in W (�), the

containment straightforwardly follows. That is, it was originally defined that

(⇧d)� = conv(W (�)). Note that by definition V (�) ✓ W (�). Therefore, it

follows that V� ✓ conv(W (�)) = (⇧d)�.

ii) (C) = (B):

Here we actually prove that (C) = (B). The goal of this equality is to show

that the vertices of M� are in bijection with the points in V (�). This part of

the proof is developed in two parts. First, we take a direction such that in that

direction there is a vertex of M�. Then we show that the vertex is in V (�).

Secondly, we choose a vertex in V� and then produce a direction such that in

that direction we obtain a maximal point of M�, which is equal to the chosen

vertex of V (�).

We let c be the linear function defined by

c(x) = c(x1

, . . . , xn) = c1

x1

+ c2

x2

+ · · ·+ cnxn

for x = (x1

, . . . , xn) 2 Rn. Take a direction c such that (M�)c is a vertex. We

55

now proceed by showing that (M�)c 2 V (�).

Let c�k=

P

i2�k

ci

lk. We show that in order to find (M�)c it is enough to know the

ordering of c�1 , c�2 , . . . , c�p . By the properties of Minkowski sums,

(M�)c =X

j 6=k

[lje�k, lke�j ]c +

mX

j=1

lj + 1

2e�j .

To find the point of M� that maximizes c, we find the point in each Minkowski

summand of M� that maximizes c. Furthermore, the point that maximizes a

line segment will always be one of its endpoints. Thus, to find (M�)c it su�ces

to find the endpoint of each interval that maximizes c.

We show that c(lke�j) < c(lje�k) if and only if c�j < c�k

. Suppose that c�j < c�k.

Then

c(lke�j) = lkc(e�j) = lk

0

@

X

i2�j

ci

1

A = lkljc�j

< lkljc�k= lj

X

i2�k

ci

!

= ljc(e�k) = c(lje�k

).

Reversing our steps we can see that the converse also holds. So, to find which

endpoint of each interval maximizes c, and as a result, which point of M�

maximizes c, it is enough to know the ordering of c�1 , c�2 , . . . , c�m .

Now, suppose c has c�1 , c�2 , . . . , c�m pairwise distinct. Since c(lke�j) < c(lje�k)

56

if and only if c�j < c�k, this also means that [lje�k

, lke�j ]c = lje�kif and only if

c�j < c�k. Hence,

(M�)c =mX

k=1

0

B

B

@

X

jc�j<c�k

lj

1

C

C

A

e�k+

mX

k=1

lk + 1

2e�j

=mX

k=1

0

B

B

@

0

B

B

@

X

jc�j<c�k

lj

1

C

C

A

+lk + 1

2

1

C

C

A

e�k.

Lastly, we show that (M�)c corresponds to a point v� in V�.

Recall that V� =

⇢

1

|�|P|�|

i=1

�iv : v 2 V (�)

�

, where V (�) is the set of vertices

v = (x1

, . . . , xd) of ⇧n that satisfy the following property: for each cycle �j =

(j1

j2

· · · jr) of �, xj1 , . . . , xjr is a consecutive sequence of integers. Observe

that each vertex in V (�) corresponds to a linear ordering of �1

, . . . , �m. This

ordering determines which set of consecutive integers is placed in the coordinates

corresponding to each cycle. In particular, if �j = (j1

j2

· · · jr) is the smallest

cycle in the order, the corresponding vertex v = (x1

, x2

, . . . , xd) 2 ⇧d will have

xj1 = 1, xj2 = 2, . . . , xjr = lj. If �k = (k1

k2

· · · ks) is the next cycle in the

order, v will have xk1 = lj + 1, xk2 = lj + 2, . . . , xks = lj + lk. Continuing in

57

this manner, every coordinate of v is determined. Recalling that

(l1

+ · · ·+ lj + 1) + (l1

+ · · ·+ lj + 2) + · · ·+ (l1

+ · · ·+ lj + lk)

lk

= l1

+ · · ·+ lj +lk + 1

2,

it follows that v� is equal to (M�)c if and only if �1

, . . . , �m have the same

ordering as c�1 , . . . , c�m . Therefore, the vertices of M� are precisely the points

in V�, proving that M� = V�.

iii) (A) ✓ (C):

The last and third containment demonstrates that (⇧d)� is a subset of M�.

Knowing that (⇧d)� is defined to be the convex hull of the set W (�), we prove

that the set W (�) is contained in M�, which implies that the convex hull of the

set is also in M�. This is done by showing that any element of the fixed sub-

polytope is a part of the Minkowski sum. We know that (⇧d)� = conv(W (�)),

where W (�) is defined in Definition 3.5. As mentioned earlier, for any vertex

w of ⇧d we have w� = 1

|�|

|�|P

i=1

�iw.

This part of the proof is done in three main parts. First, we start with a vertex of

⇧d with coordinates (d, d�1, . . . , 2, 1). We prove that if we average the points on

the �-orbit of that vertex, we obtain a point t� onM�. Secondly, we demonstrate

that if any two vertices of ⇧d di↵er by a transposition of two coordinates, the

corresponding fixed points with respect to � are only a small translation, which

58

we compute, away from one another. Lastly, using the previous idea and that

any vertex of ⇧d can be obtained by a number of transpositions of consecutive

integers on the vertex with coordinates (d, d � 1, . . . , 2, 1), we prove that any

point fixed by � acting on a vertex of ⇧d is equal to t� plus a translation we

have control over; this fixed point is shown to be in M�.

Let t denote the vertex of ⇧d with coordinates (d, d� 1, . . . , 2, 1). Then

t� =1

|�||�|X

i=1

�it =

ld+···+l2+l1P

i=ld+···+l2+1

i

l1

e�1 + · · ·+

ld+ld�1P

i=ld+1

i

lde�d�1

+

ldP

i=1

i

lde�d

, (3.1)

where by Lemma 3.7

M� =X

j>k

[0, lke�j � lje�k] +

ld+···+l2+l1P

i=ld+···+l2+1

i

l1

e�1 + · · ·+

ld+ld�1P

i=ld+1

i

lde�d�1

+

ldP

i=1

i

lde�d

=X

j>k

[0, lke�j � lje�k] + t�.

Hence, it follows that t� 2 M�. We now show that if two vertices w =

(w1

, w2

, . . . , wd), w0 = (w01

, w02

, . . . , w0d) of ⇧d di↵er by a transposition of two

coordinates wa and wb, i.e., w0a = wb and w0

b = wa, where a 2 �j, b 2 �k with

j � k, then

59

w0� = w� +

8

>

>

<

>

>

:

wb � wa

ljlk(lke�j � lje�k

) if j > k,

0 if j = k.

. (3.2)

Let w = (w1

, w2

, . . . , wn) be a vertex of ⇧d. Then

w� =

P

i2�1wi

l1

e�1 +

P

i2�2wi

l2

e�2 + · · ·+P

i2�dwi

lde�d

.

The coe�cient of e�j is an average of the coordinates of w whose positions

are in �j. Let w0 = (w01

, w02

, . . . , w0d), and suppose that w and w0 di↵er by a

transposition of two coordinates wa and wb, i.e. w0a = wb and w0

b = wa, where

a 2 �j, b 2 �k, with j � k. Then w0� di↵ers from w� only in the coe�cients of

e�j and e�k. If j = k, then the only di↵erence is in the ordering of the terms

in the numerator of the coe�cient of e�j = e�k. Thus w0

� = w�. If j > k, then

the coe�cient of e�j in w0� di↵ers from that of w� by wb�wa

ljand the coe�cient

of e�kin w0

� di↵ers from that of w� by wa�wblk

. This means that

w0� = w� +

wb � wa

lje�j +

wa � wb

lke�k

= w� +wb � wa

ljlk(lke�j � lje�k

).

Now, let w be any vertex of ⇧d. Then we can consider the coordinates of w

as an element ⇡w 2 Sd. Similarly, let ⇡t 2 Sd be the permutation associated

60

with the vertex t defined above. Recall that the number of transpositions of

consecutive integers required to change the identity permutation to ⇡w is equal

to the number of inversions in ⇡w [14]. Similarly, the number of transpositions

of consecutive integers required to turn ⇡t into ⇡w is the number of correctly

ordered pairs of integers in ⇡w. By (3.2) each transposition of consecutive

integers ta, tb (where ta = tb + 1) to ⇡t corresponds to adding 1

lj lk(lke�j � lje�k

)

to t� if j > k, and no change if j = k. Thus, if oj,k is the number of correctly

ordered pairs of integers xa, xb in w, where a 2 �j and b 2 �k,

w� = t� +X

j>k

oj,kljlk

(lke�j � lje�k)

Now, note that for any choice of j and k, the number of pairs of integers

xa and xb with a 2 �j and b 2 �k is ljlk. Thus the number of correctly

ordered pairs of such integers is at most ljlk, and we have 0 oj,kljlk

1. This

means that, for each choice of j and k,oj,kljlk

�

lke�j � lje�k

�

is a point in the

segment [0, lke�j � lje�k], and thus

P

j>k

oj,kljlk

�

lke�j � lje�k

�

is an element of the

Minkowski sumP

j>k[0, lke�j � lje�k]. Recall that t� +

P

j>k[0, lke�j � lje�k] =

M�. Thus, we conclude that w� 2 M�.

Observe that in the examples of the fixed sub-polytopes of (⇧d)� are all similar

in shape to lower dimensional permutahedron. That observation is made precise by

the following theorem.

61

Theorem 3.9. The fixed subpolytopes (⇧d)� are combinatorially equivalent to an

m-permutahedron, where m is the number of disjoint cycles of �.

Proof. Consider the Minkowski sum of line segmentsP

j>k[0, ej �ek]. This Minkowski

sum is a translation of ⇧m, implying that it is combinatorially equivalent to ⇧m.

Now consider the map f : Rm ! (Rd)� ✓ Rd defined by f(ei) =1

lie�i and extended

linearly. Note that f is bijective since it does not have elements getting mapped

to the same image because the map takes and ei and maps it to a factor of a unit

vector. Also, any element in the range is attained. Moreover,

f

X

j>k

[0, ej � ek]

!

=X

j>k

0,1

lje�j �

1

lke�k

�

.

This implies thatP

j>k[0, ej�ek] is linearly equivalent, and thus combinatorially equiv-

alent, toP

j>k[0, 1

lje�j � 1

lke�k

].

If we scale each summand in the Minkowski sum, the normal fan of the resulting

polytope does not change [1]. This tells us thatP

j>k[0, 1

lje�j � 1

lke�k

] is normally

equivalent toX

j>k

ljlk[0,1

lje�j �

1

lke�k

] =X

j>k

[0, lke�j � lje�k].

Furthermore, normal equivalence implies combinatorial equivalence [1], soP

j>k[0, 1

lje�j�

1

lke�k

] is combinatorially equivalent toP

j>k[0, lke�j � lje�k

]. Bringing together these

equivalences, we conclude that (⇧d)� is combinatorially equivalent to ⇧m.

62

We had hoped to determine the Ehrhart theory of the fixed sub-polytopes, which

are rational polytopes. The theory of rational zonotopes is not as largely researched

and no formula is known that provides the Ehrhart series for rational zonotopes.

In the near future, we plan to determine a formula for the volume of the fixed

sub-polytopes in hopes that it will lead us to developing the rest of the equivariant

Ehrhart theory of the permutahedron.

63

Chapter 4

A Decomposition of the h⇤-Polynomial for

Rational Polytopes

For this section we adjust the set-up for Theorem 2.39 to satisfy rational polytopes.

Definition 4.1. Consider a rational d-simplex � with vertices v1

, v2

, . . . , vd+1

2 Qd.

Fix any integer p such that p� is a lattice simplex. We let w1

= (v1

, 1), w2

=

(v2

, 1), . . . , wd+1

= (vd+1

, 1) and define

cone(�) := {�1

w1

+ �2

w2

+ · · ·+ �d+1

wd+1

: �1

,�2

, . . . ,�d+1

� 0} ⇢ Rd+1

and

⇧(�, p)� := {�1

pw1

+ �2

pw2

+ · · ·+ �d+1

pwd+1

: 0 < �1

,�2

, . . . ,�d+1

< 1}.

We call this open parallelepiped ⇧(�, p)� the fundamental open parallelepiped of

64

cone(�).

Given a rational d-polytope P , we fix a triangulation T . Given a simplex � 2 T ,

let ⇧(�, p)� be the open fundamental parallelepiped of cone(�) and let B�,p(z) :=

�⇧(�,p)�(1, 1, . . . , 1, z). We include Ø in the collection of faces of a triangulation of

a polytope; we consider Ø the empty simplex.

Proposition 4.2. B�,p(z) = (zp)dim(�)+1B

�,p

�

1

z

�

.

Proof. Note that ⇧(�, p)� = �⇧(�, p)� + pw1

+ pw2

+ · · ·+ pwd+1

:

�⇧(�, p)� + pw1

+ · · ·+ pwd+1

= {�µ1

pw1

� · · ·� µd+1

pwd+1

: 0 < µi < 1}

+ pw1

+ · · ·+ pwd+1

=

(

�d+1

X

i=1

µipwi : �1 < µi � 1 < 0

)

+ pd+1

X

i=1

wi

=

(

pd+1

X

i=1

wi (1� µi) : 0 < 1� µi < 1

)

=

(

pd+1

X

i=1

�1

wi : 0 < �i < 1

)

= ⇧(�, p)�.

65

Figure 4.1: ⇧(�, p)� = �⇧(�, p)� + pw1

+ pw2

.

We take the integer-point transform of the open parallelepiped,

�⇧(�,p)�(~z) = ��⇧(�,p)�(~z)~z

pw1~z pw2 · · · ~z pwd+1

= �⇧(�,p)�

✓

1

z1

,1

z2

, . . . ,1

zd+1

◆

~z pw1~z pw2 · · · ~z pwd+1

Now, recall that we want to specialize at ~z = (1, 1, . . . , z). Therefore, we arrive at

�⇧(�,p)�(1, 1, . . . , z) = �

⇧(�,p)��

1, 1, . . . , 1z�

(zp)d+1. This is equivalent to saying that

B�,p(z) = (zp)dim(�,p)+1B

�,p

�

1

z

�

.

Theorem 4.3. Fix a triangulation T of the rational d-polytope P such that p is the

least common multiple of all the vertex coordinate denominators of every � 2 T .

Then

EhrP(z) =

P

�2T hlink(�)

(zp)B�,p(z)

(1� zp)d+1

.

66

Proof. We write P as the disjoint union of all open nonempty simplices in T and

use Ehrhart-Macdonald reciprocity,

EhrP(z) = 1 +X

�2T \{Ø}

Ehr�

�(z)

= 1 +X

�2T \{Ø}

(�1)dim(�)+1 Ehr�

✓

1

z

◆

= 1 +X

�2T \{Ø}

(�1)dim(�)+1

h⇤�,p(

1

z )

(1� 1

zp )dim(�)+1

= 1 +X

�2T \{Ø}

(zp)dim(�)+1(1� zp)d�dim(�)h⇤�,p(

1

z )

(1� zp)d+1

Next, we use the fact that

�⇧(�,p)(~z) = 1 +

X

Ø 6=⌦✓�

�⇧(⌦,p)�(~z),

where the sum is over all nonempty faces of � [2]. Taking ~z = (1, 1, . . . , 1, z), the

identity specializes to

h⇤�,p(z) =

X

⌦✓�

B⌦,p(z),

where this sum is over all faces of �, including Ø. Using this, we can rewrite our

67

series as

EhrP(z) = 1 +X

�2T \Ø

(zp)dim(�)+1(1� zp)d�dim(�)

P

⌦✓�

B⌦,p(

1

z )

(1� zp)d+1

=

P

�2T (zp)dim(�)+1(1� zp)d�dim(�)

P

⌦✓�

B⌦,p(

1

z )

(1� zp)d+1

.

By Proposition 4.2,

h⇤P(z) =

X

�2T

(zp)dim(�)+1(1� zp)d�dim(�)

X

⌦✓�

B⌦,p

✓

1

z

◆

=X

�2T

(zp)dim(�)+1(1� zp)d�dim(�)

X

⌦✓�

(zp)� dim(⌦)�1B⌦,p(z)

=X

⌦2T

X

�◆⌦

(zp)dim(�)�dim(⌦)(1� zp)d�dim(�)B⌦,p(z)

=X

⌦2T

(1� zp)d�dim(⌦)B⌦,p(z)

X

�◆⌦

✓

zp

1� zp

◆

dim(�)�dim(⌦)

.

Using Definition 2.38, (2.1), the theorem follows.

Example 4.4. Let P be as defined in Example 2.46. We compute the Ehrhart

series of P using Theorem 4.3.

Fix the following triangulation of P :

68

Figure 4.2: Triangulated P .

Here we see a computation of a 2-simplex, �1

= conv{��1

2

, 1�

,�

1

2

, 1�

, (0, 0)}.

⇧(�1

, 2)� = {2�1

✓�1

2, 1, 1

◆

+ 2�2

✓

1

2, 1, 1

◆

+ 2�3

(0, 0, 1) : 0 < �1

,�2

,�3

< 1}

= {(��1

, 2�1

, 2�1

) + (�2

, 2�2

, 2�2

) + (0, 0, 2�3

) : 0 < �1

,�2

,�3

< 1}

= {(�2

� �1

, 2(�1

+ �2

), 2(�1

+ �2

+ �3

)) : 0 < �1

,�2

,�3

< 1}

= {(A,B,C)}

(A): For (A) to be an integer, it must equal 0. ) �2

� �1

= 0 ) �2

= �1

.

(B): 2(�1

+ �2

) = 2(2�1

) = 4�1

= m for m 2 Z. Then m = �14

. Thus,

�1

= �2

= 1

4

, 24

, 34

.

(C): 2(�1

+�2

+�3

) = 2(2�1

+�3

) = n for n 2 Z. So, 1+2�3

= n or 2+2�3

= n

or 3 + 2�3

= n, which all imply that �3

= 1

2

.

69

Therefore, the integer points are: (0, 1, 2), (0, 2, 3), (0, 3, 4). This makes the integer-

point transform

�⇧(�1,2)�(1, 1, z) = z0

1

z12

z23

+ z01

z22

z33

+ z01

z22

z43

= (1)0(1)1z2 + (1)0(1)2z3 + (1)0(1)2z4 = z2 + z3 + z4.

Through further careful computation, we obtain:

70

Applying Theorem 4.3,

EhrP(z) =z5 + 3z4 + 12z3 + 12z2 + 3z + 1

(1� z2)3,

which agrees with Example 2.25.

This is only the beginning to exploring the h⇤-polynomial for rational polytopes.

One direction to take this work is to prove a rational analogue to Theorem 2.41.

71

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