ALGEBRAIC AND COMBINATORIAL ASPECTS OF TWO SYMMETRIC POLYTOPES AA thesis

86 Pages·2017·4.21 MB·English

ALGEBRAIC AND COMBINATORIAL ASPECTS OF TWO SYMMETRIC
POLYTOPES
A thesis presented to the faculty of
San Francisco State University
In partial fulfillment of
A
The Requirements for
56 The Degree
c30l^
Master of Arts
. '■* i-,? In
Mathematics
by
Anna Maria Schindler
San Francisco, California
August 2017
Copyright by
Anna Maria Schindler
2017
CERTIFICATION OF APPROVAL
I certify that I have read ALGEBRAIC AND COMBINATORIAL AS­
PECTS OF TWO SYMMETRIC POLYTOPES by Anna Maria Schindler
and that in ray opinion this work meets the criteria for approving a thesis
submitted in partial fulfillment of the requirements for the degree: Master
of Arts in Mathematics at San Francisco State University.
Federico Ardila
Professor of Mathematics
ikl
Matthias Beck
Professor of Mathematics
ALGEBRAIC AND COMBINATORIAL ASPECTS OF TWO SYMMETRIC
POLYTOPES
Anna Maria Schindler
San Francisco State University
2017
Equivariant Ehrhart theory is an extension of Ehrhart theory that considers lattice
polytopes under group actions. Ehrhart theory tells us that the lattice points in a
lattice (or rational) polytope are counted by polynomials (or quasi-polynomials). In
the equivariant analog, we consider the Ehrhart theory of the subsets of the polytope
fixed by the action. This first part of this thesis, a joint project with Andres Vindas,
focuses on the equivariant Ehrhart theory of IIn under the action of Sn. We prove
that the fixed sub-polytopes of IIn are zonotopes and are combinatorially equivalent
to permutahedra. We provide vertex and hyperplane descriptions. We also compute
the equivariant Ehrhart theory for 111,112,113, and II4. This thesis also includes work
in spectral graph theory. Motivated by a tropical approach to the Hodge conjecture,
we compute the spectrum of the tropical Laplacian matrix of the root polytope An,
confirming a special case of a result of Babaee and Huh.
I certify that the Abstract is a correct representation of the content of this thesis.
Q J JU ___________JJ n . W l
a u
Chair, Thesis Committee Date
ACKNOWLEDGMENTS
First, I would like to thank my thesis advisor, Federico Ardila. I can truly
say that the geometry class I took with Federico my very first semester at
San Francisco State University is a big part of the reason that I applied
to the master’s program here. Federico has provided me with so much
insight and encouragement over the past few years, for which I am very
grateful.
Federico’s research meetings were collaborative in nature, so this thesis
has been touched by many, dxl would like to thank all of the fellow
students who listened to my thoughts and shared their own insights. In
particular, I would like to thank my research partner, Andres Vindas, with
whom part of this thesis is joint work. Andres excitement and motivation
about the project was invaluable. I would also like to thank our dear
friend John Guo, who was there for support and insight throughout the
whole thesis process.
Finally, I would like to thank my thesis committee, Serkan Hosten and
Matthias Beck, whose input was essential to the finished product.
TABLE OF CONTENTS
1 Introduction.............................................................................................................. 1
1.1 Equivariant Ehrhart theory ........................................................................ 1
1.2 The tropical Laplacian of An ......................................................................... 6
2 The fixed sub-polytopes of the permutahedron................................................ 9
2.1 Background..................................................................................................... 9
2.1.1 Discrete geometry............................................................................... 9
2.1.2 Ehrhart theory.................................................................................. 14
2.1.3 The symmetric group........................................................................ 19
2.1.4 The permutahedron............................................................................ 21
2.1.5 The Ehrhart theory of the permutahedron......................................24
2.1.6 Representation theory.........................................................................26
2.1.7 Equivariant Ehrhart theory................................................................29
2.2 Results...................................................................................................................32
2.2.1 The set-up................................................................................................33
2.2.2 The fixed sub-polytopes of Iln ............................................................34
2.2.3 The equivariant Ehrhart theory of 111,112,113, and II4 ....................53
3 The tropical Laplacian of the root polytope An ...................................................60
3.1 Background..................................................................................................... 61
3.1.1 Results......................................................................................................63
vi
Bibliography
LIST OF TABLES
Table Page
viii
LIST OF FIGURES
Figure Page
1.1 (n4)(i3)........................................................................................ 4
1.2 n3.................................................................................................................... 4
1.3 n3............................................................................................... 7
2.1 n3 ............................................................................................ 15
2.2 n3, 2II3, and 3II3 ................................................................................................16
2.3 Lattice points in II3, 2II3, and 3II3 ........................................................... 17
2.4 n4.............................................................................................................................22
2.5 Labeled forests on three vertices........................................................................25
2.6 The fixed sub-polytope (Il4)(13).........................................................................35
2.7 The fixed sub-polytope (Il4)(i3) and its vertices.............................................38
2.8 (II4) (12)....................................................................................................................54
2.9 (n4)(i23)...................................................................................................................59
3.1 The polytope A4...................................................................................................69
3.2 The graph of A\....................................................................................................70
3.3 The directed graph A'n associated with A4......................................................70
1
Chapter 1
Introduction
1.1 Equivariant Ehrhart theory
Equivariant Ehrhart theory was first established by Alan Stapledon in his 2011
paper. [13] The theory extends Ehrhart theory by studying lattice polytopes together
with the action of groups.
Ehrhart theory is concerned with enumerating the lattice points in the dilates of
a polytope P. Let mP denote the dilation of P by a positive integer m, i.e,
mP — {mx : x E P}.
Eugene Ehrhart’s remarkable result is that, if P is a lattice polytope, the number
of lattice points in mP is counted by a polynomial in m, called the Ehrhart polynomial
of P. If P is a rational polytope, then the number of lattice points in mP is counted
by a quasi-polynomial. The generating function, called the Ehrhart series of a