Table Of ContentRelations, Matroids and Codes
Thomas Britz
Ph.D. thesis
Department of Mathemati
al S
ien
es
University of Aarhus
O
tober 2002
To Rikke, my al
hemist...
Prefa
e
Thisthesisdo
umentsmostofmyresear
hduringthePh.D.programme,
andto explainthepolytone nature of thework, a briefhistoryof theproje
t
is in order.
In April 1998, Gian-Carlo Rota agreed to supervise my thesis. Shortly
after
ommen
ing the Ph.D. programme, under the formal supervision of
J(cid:28)rgen Brandt, I moved to Boston and was enrolled as Visiting Student
at M.I.T. for two terms. There, I followed
ourses and seminars and
on-
du
ted resear
h under Gian-Carlo Rota's supervision. Rota en
ouraged me
to (cid:12)rst a
umulate a broad and solid foundation of ba
kground knowledge
and introdu
ed me to several interesting topi
s and a number of
olleagues,
someofthelatterwithwhomIhave sin
e
ollaborated,resultinginmaterial
presented in this thesis.
Following Gian-Carlo Rota's sudden death in April 1999, I de
ided to
remain at M.I.T. until the end of the term, while
ontinuing the
ollabora-
tions. Meanwhile, I made enquiriesto experts inthe (cid:12)eld and was fortunate
to (cid:12)nd Peter Cameron willing to
ontinue Rota's supervision.
During the summer of 1999, I spent two weeks at Queen Mary and
West(cid:12)eld College inLondon, visitingPeter Cameron inorder for usto agree
upon the terms of the supervision and of the thesis topi
. During this visit,
Peter Cameron proposed the
urrent thesis topi
, namely the
onne
tion
between representable matroids and linear
odes. Sin
e then, I have visited
Queen Mary twi
e for longer periods, namely the autumns terms of 2000
and 2001.
Thisthesis isdividedinto three parts. Thethirdpartpresentsa number
of arti
les (Chapters 13-19); four of these arti
les have previouslybeen pub-
lished, in slightly di(cid:11)erent forms, in various journals, and Chapters 11, 16,
18, and 19 are ea
h to be re-editted and submitted. The arti
les presented
in two of these
hapters were written in
ollaboration with other authors.
Chapter 13 was written together with Sergey Fomin, while Chapter 14 was
written together with Matteo Mainetti and Luigi Pezzoli. Furthermore,
Chapter 18 would probably not have
ome into existen
e had it not been
for a number of inspiring dis
ussions with Carrie G. Rutherford, as well as
her knowledge of the program GAP whi
h she kindly shared with me.
PartIIIalso
ontainsChapter20 whi
hbrie(cid:13)yre
apitulatesmanyofthe
problems and future dire
tions of my resear
h mentioned in Parts I and II.
These (cid:12)rst two parts
on
ern the two main areas of resear
h, namely that
iii
iv PREFACE
of binary relations, and that of the relationships between linear
odes and
representable matroids. Most of the polished results in these two areas
appear in the
hapters of Part III. The role of Parts I and II is thus mainly
to introdu
e and dis
uss. Some new results do, however, appear in these
(cid:12)rst two parts, as Chapters 2, 5, 7, 10, 11, and 12.
It seems that there is still a bit of Prefa
e left, whi
h provides me with
an ex
ellent opportunity to thank
ertain people.
First all, mu
h gratitude is dueto J(cid:28)rgen Brandt, Gian-CarloRota, and
Peter Cameron for their willingness to spend time and e(cid:11)ort in supervising
me. Likewise, I very gratefully thank Sergey Fomin for willingly daring to
ollaborate with an unknown and green Ph. D. student on a proje
t whi
h
from the beginning guaranteed a fair workload. Along with these mentors,
I thank my parents and Ivan Tafteberg for tea
hing and inspiring me.
My gratitude also befalls a number of people, near and far, with whom
I have
orresponded in
onne
tion with this thesis; foremost are Matteo
Mainetti, Ri
hard Brualdi, Carsten Thomassen, Ildiko Sain, Carrie Ruther-
ford, Luigi Pezzoli, Joseph Kung, Alan Sokal, James Oxley, and Alexander
Barg.
Among the many friends to whom I am grateful, I would espe
ially
like to thank Henrik Kragh S(cid:28)rensen for years of fertile dis
ussions on all
mannersof subje
ts, as well as Fran
is Poulin, Adrian Vetta, Peter Cli(cid:11)ord,
Colva Roney-Dougal, and Matt Ollis, for their warm-hearted re
eption and
support during my ventures abroad.
And of
ourse my Rikke, not only for her forebearen
e and support but
also for everything that she has taught me.
Thomas Britz
July 2002
Contents
Prefa
e iii
Part I. Binary Relations 1
Chapter 1. Binary relations 3
1. Binary relations 3
2. Preorders, posets, and equivalen
e relations 5
3. Hasse diagrams 7
4. Latti
es 9
Chapter 2. Homomorphisms on binary relations 13
Chapter 3. Finite posets 17
Chapter 4. Operations on equivalen
e relations 21
Chapter 5. Free matri
es 23
1. Free matri
es 23
2. The produ
t of two free matri
es 26
3. Compound relations 28
Part II. Codes and Matroids 33
Chapter 6. Matroids 35
1. Some basi
de(cid:12)nitions in matroid theory 35
2. Operations on matroids 37
3. Geometri
al representations of matroids 38
4. Matroid polynomials 39
5. Representable matroids 41
Chapter 7. The basis problem 45
Chapter 8. Codes 49
1. The basi
de(cid:12)nitions in
oding theory 49
2. The Ma
Williams identity 50
3. Generalised weights 50
4. Codes and t-designs 51
5. Quaternary
odes 52
6. Code
hains 53
v
vi CONTENTS
Chapter 9. Codes and matroids 55
1. Representable matroids and linear
odes 55
2. Supports of linear
odes and matroid polynomials 55
3. Covering radii and the Tutte polynomial 57
4. Chains of
odes and matroids 58
Chapter 10. Matroid representability over rings 59
1. Finite lo
al rings 61
Chapter 11. Matroid
hains 63
1. Some
lasses of matroid
hains 66
2. Representations of matroid
hains 73
3. Tutte polynomials for matroid
hains 78
Chapter 12. Equivalen
es of
odes and representable matroids 83
Part III. Arti
les and A List of Tasks 87
Chapter 13. Finite posets and Ferrers shapes 89
1. Introdu
tion 89
2. Main Theorems 90
3. The Robinson-S
hensted
orresponden
e 94
4. The S
hu(cid:127)tzenberger involution 99
5. Saturation and orthogonality 102
6. Nilpotent matri
es 104
7. Network (cid:13)ow preliminaries 107
8. Frank's network. Proof of Theorem 13.1 108
9. Three proofs of Theorem 13.2 111
10. Proof of Theorem 13.3 117
11. Proof of Theorem 13.4 117
12. Proof of Theorem 13.5 120
Chapter 14. Operations on the Family of Equivalen
e Relations 121
1. Introdu
tion 121
2. Preliminaries 122
2.1. Equivalen
e relations 122
2.2. Finite types 123
3. R(cid:0)T Compatible Relations 125
4. Some appli
ations 128
5. Generalising (cid:3) to n-ary operations 132
Chapter 15. The inverse of a non-singular free matrix 137
1. Introdu
tion 137
2. Preliminaries 138
(cid:0)1
3. When is M free? 138
4. Proof of Theorem 15.3 139
A
knowledgements 140
CONTENTS vii
Chapter 16. Inverting a free matrix 141
1. Introdu
tion 141
2. Free matri
es and digraphs 142
3. The digraphs of a free matrix and its inverse 143
4. An appli
ation of Theorem 16.4 144
5. The sign patterns of a free matrix and its inverse 145
A
knowledgements 146
Chapter 17. Ma
Williams identities and matroid polynomials 147
1. Introdu
tion 147
2. Support enumerators of a linear
ode 148
3. The ve
tor matroid of a linear
ode 152
4. An alternative proof of Theorem 17.8 159
A
knowledgements 162
Chapter 18. Covering radii are not matroid invariants 163
1. Introdu
tion 163
2. Two pairs of
odes 164
Chapter 19. The supports of a linear
ode 167
1. Introdu
tion 167
2. The ve
tor matroid of a linear
ode 167
3. A generalisation of the Criti
al Theorem 169
4. The rth higher matroid 171
5. The matroids and various enumerators of a
ode 174
A
knowledgement 177
Chapter 20. And now what? 179
Summary 181
Dansk Resum(cid:19)e 183
Appendix A: Some
o-
hromati
planar graphs 185
Bibliography 221
Index 229
viii CONTENTS
