THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE ...

106 Pages·2016·0.84 MB·English

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University of Kentucky Doctoral Dissertations Graduate School
2008
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Erik Stokes
University of Kentucky, [email protected]
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Stokes, Erik, "THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF SQUAREFREE
STRONGLY STABLE IDEALS" (2008). University of Kentucky Doctoral Dissertations. 636.
https://uknowledge.uky.edu/gradschool_diss/636
This Dissertation is brought to you for free and open access by the Graduate School at UKnowledge. It has been
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ABSTRACT OF DISSERTATION
Erik Stokes
The Graduate School
University of Kentucky
2008
THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF
SQUAREFREE STRONGLY STABLE IDEALS
ABSTRACT OF DISSERTATION
A dissertation submitted in partial
fulfillment of the requirements for
the degree of Doctor of Philosophy
in the College of Arts and Sciences
at the University of Kentucky
By
Erik Stokes
Lexington, Kentucky
Director: Dr. Uwe Nagel, Professor of Mathematics
Lexington, Kentucky 2008
Copyright(cid:13)c Erik Stokes 2008
ABSTRACT OF DISSERTATION
THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF
SQUAREFREE STRONGLY STABLE IDEALS
Making use of algebraic and combinatorial techniques, we study two topics: the
arithmetic degree of squarefree strongly stable ideals and the h-vectors of matroid
complexes.
For a squarefree monomial ideal, I, the arithmetic degree of I is the number of
facets of the simplicial complex which has I as its Stanley-Reisner ideal. We consider
the case when I is squarefree strongly stable, in which case we give an exact formula
for the arithmetic degree in terms of the minimal generators of I as well as a lower
bound resembling that from the Multiplicity Conjecture. Using this, we can produce
an upper bound on the number of minimal generators of any Cohen-Macaulay ideals
with arbitrary codimension extending Dubreil’s theorem for codimension 2.
A matroid complex is a pure complex such that every restriction is again pure. It
isalong-standingopenproblemtoclassifyallpossibleh-vectorsofsuchcomplexes. In
thecasewhenthecomplexhasdimension1wecompletelyresolvethisquestionandwe
give some partial results for higher dimensions. We also prove the 1-dimensional case
of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally,
we completely characterize the Stanley-Reisner ideals of matroid complexes.
KEYWORDS: simplicial complex, matroid, h-vector, arithmetic degree, Stanley-
Reisner ideal
Author’s signature: Erik Stokes
Date: July 22, 2008
THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF
SQUAREFREE STRONGLY STABLE IDEALS
By
Erik Stokes
Director of Dissertation: Uwe Nagel
Director of Graduate Studies: Qiang Ye
Date: July 22, 2008
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Name Date
DISSERTATION
Erik Stokes
The Graduate School
University of Kentucky
2008
THE h-VECTORS OF MATROIDS AND THE ARITHMETIC DEGREE OF
SQUAREFREE STRONGLY STABLE IDEALS
DISSERTATION
A dissertation submitted in partial
fulfillment of the requirements for
the degree of Doctor of Philosophy
in the College of Arts and Sciences
at the University of Kentucky
By
Erik Stokes
Lexington, Kentucky
Director: Dr. Uwe Nagel, Professor of Mathematics
Lexington, Kentucky 2008
Copyright(cid:13)c Erik Stokes 2008
TABLE OF CONTENTS
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Stanley-Reisner Ideals and Commutative Algebra . . . . . . . . . . . 7
Chapter 3 Arithmetic Degree . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 A Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 An Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 4 Matroid Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1 The Set of Dimension 1 Matroid h-vectors . . . . . . . . . . . 48
4.3 A Conjecture of Stanley in Dimension 1 . . . . . . . . . . . . . . . . 55
4.4 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.1 Gorenstein Matroid Complexes . . . . . . . . . . . . . . . . . 62
4.4.2 The Sub-facet complex . . . . . . . . . . . . . . . . . . . . . . 65
4.4.3 The Stanley-Reisner Ideals of Matroid Complexes . . . . . . . 69
Appendix: Computer Code for SAGE . . . . . . . . . . . . . . . . . . . . . . . 74
Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
iii
LIST OF FIGURES
2.1 The 2-simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 A simplicial complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.1 A matroid complex with 6 vertices . . . . . . . . . . . . . . . . . . . . . 30
4.2 A non-matroid complex with 6 vertices . . . . . . . . . . . . . . . . . . . 30
4.3 A non-pure complex whose proper restrictions are all pure . . . . . . . . 31
4.4 S3K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1 3
4.5 S2S2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 1 2
4.6 ∆ and ∆ have the same h-vector but are not isomorphic . . . . . . 43
111 2000
4.7 The edge complex of an octahedron . . . . . . . . . . . . . . . . . . . . . 66
iv