AN ERLANGER PROGRAM
FOR COMBINATORIAL GEOMETRIES
by
Joseph Pee Sin Kung
B. Sc. (Hons),
University of New South Wales
(1974)
Submitted in partial fulfillment
of the requirements for the
degree of
Doctor of Philosophy
at the
Massachusetts Institute of Technology
April 11,
0
1978
Joseph Pee Sin Kung 1978
Signature redacted
Signature of Author................
Depertment of Mat1regnatics, April 11,
. . . . . . . . . . . . .
.
redacted
.
. -Signature
y
1978
-7 rh n , Thesis Supervisor
*
Accepted by ..
Signature redactedC
...........
Chairman, Department Committee
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MASSACHUSETTS INSiTUTE
OF TECHNOLOGY
AUG 2 8 1978
LIBRARIES
-2-
AN ERLANGER PROGRAM
FOR COMBINATORIAL
GEOMETRIES
by
Joseph Pee Sin Kung
Submitted to the Department of Mathematics
on April 11,
1978 in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy.
ABSTRACT
The
study of combinatorial synthetic geometries or
matroids may be regarded as the study of the geometric
properties of finite sets of points in space which are
invariant under the action of the general
linear group.
Classical invariant theory suggests three other analogs
of the notion of a matroid. The first,
that of a bimatroid,
is an abstraction of the notion of a bilinear pairing
between two vector spaces. This is studied in detail-highlights include a Minty-style cryptomorphism in terms
of bond-circuit and circuit-bond pairs and a four term
Tutte decomposition theory. The second,
that of an
orthogonal matroid, turns out to be a special case of the
notion of a bimatroid.
The third is the notion of a
Pfaffian structure; our study yields generalizations
3
-
-
of many known results about the matching matroid of a
graph,
and symplectic analogs of many notions in
matroid theory.
Along the way, we discuss several related areas
of matroid theory: core extraction, weak maps,
alternating basis exchange,
matroid joins,
the alpha
function and Radon transforms.
Thesis Supervisor:
Gian-Carlo Rota,
Philosophy
Professor of Applied Mathematics and
-
- 4
To my parents
The bygers of wrath are
wLser than the horses
Of LnsbructLon.
William Blake
5
-
-
CONTENTS
Abstract
2
TabLe of contents
5
Preface
7
Chapter 1: COMBINATORIAL SYNTHETIC GEOMETRIES
1.1 The Bowdoin program
9
1.2 Core extraction
14
1.3 The co-core
23
1.4 Alternating basis exchange
26
1.5 Partitions and intersections
33
1.6 The alpha function
38
Chapter 2: FINITE RADON TRANSFORMS
2.1 The classical Radon transform
43
2.2 The copoint Radon transform
45
2.3 The bond Radon transform
49
2.4 The p-Radon transforms
50
2.5 The maximal chain theorem
54
Chapter 3:
BIMATROIDS
3.1 Matrices and bilinear pairing
57
-
- 6
3.2 Basic definitions and examples
61
3.3 Some elementary exchange-augmentation
properties
70
3.4 The quotient diagram over the comatroid
72
3.5 Perpendicularity
78
3.6 Embedding covectors in the matroid
82
3.7 The bipainting cryptomorphism
83
3.8 Relative bonds and circuits
86
3.9 Reconstructing non-singular minors
91
Chapter 4:
THE TUTTE DECOMPOSITION THEORY FOR BIMATROIDS
4.1 Contractions, deletions and sums
96
4.2 The Tutte decomposition
102
4.3 The rank generating polynomial
105
4.4 Tutte invariants
109
Chapter 5:
PFAFFIAN STRUCTURES
5.1 Symplectic invariant theory
112
5.2 Pfaffian structures
114
5.3 Lagrangian flats and bimatroids
119
5.4 The one-factor Pfaffian structure of a graph 121
5.5 Tutte decomposition
Chapter 6:
123
SKETCHES
6.1 Bimatroids
127
6.2 Pfaffian structures
128
6.3 Orthogonal matroids
130
6.4 Non-commutativity
131
7
-
-
Preface
The first problem that I worked on in the theory of
combinatorial geometries concerned the representation
properties of the linear ring.
the
Later,
I was led to consider
still unsolved problem of finding all possible basis
exchange properties valid in any geometry.
I now realise,
These problems,
are part of the basic question:
What Ls a
combLnatorLaL geometry?
The only answer I know appeals to classical projective
invariant theory and is the
subject of section 1.1.
of view suggests that there
is,
Erlanger Program for the
This point
as in classical geometry,
an
theory'of combinatorial geometries.
This program was first outlined by Gian-Carlo Rota in his
Bowdoin lectures
17113,
and I have accordingly christened it
the Bowdoin Program.
Not all the results in this thesis are in the Bowdoin
Program proper.
However,
they are all in its spirit,
and
illustrates my personal philosophy of combinatorial geometries.
Nearly all the results here are new,
or are unpublished
extensions of my previous work [773,[78i3
and [78 2 J.
The
exception is Chapter 2;
-
- 8
the main results there are
This paper is the second paper I have written,
Schumann's PopLLLons,
in
[7821.
and like
is perhaps- the most humorous work
in my output.
The predominant influence
in this thesis
Gian-Carlo Rota. As his research student,
appropriated many of his ideas;
is that of
I have shamelessly
my debt to him is so vast
that the acknowledged borrowings seem only to repudiate
the unacknowledged,
Greene
because unconscious,
borrowings.
Curtis
(whose work laid much of the foundations of this
thesis),
Jay Sulzberger and Tom Zaslavsky have been very
generous with their advice,
entkusiasm and support.
To all of
them,. I offermy thanks and the following insubstantial
essay.
Cambridge,
Spring 1978
9
-
-
..
. .......
Chapter I
COMBINATORIAL SYNTHETIC GEOMETRIES
1 .1
The BowdoLn
Program
A major undercurrent in the development of geometry
since the nineteenth century has been the separation of the
idea of geometry from the idea of space.
Perhaps the first
geometer to articulate this consciously was Riemann
concept of a multiply extended quantity evolved,
the
twentieth century,
manifold and a scheme.
; his
in
into the related concepts of a
There is another aspect of this
separation which is implicit in the axiomatic approach to
geometry.
The basic properties of space are axiomatized and
space is,
quite simply,
left out of the picture altogether.
Our terminology differs from Riemann's; to him, geometry is
the study of space. However, the distinction between the
concept of space and the basic properties of space is central
to his argument in his famous lecture Uber dLe hypothesen,
weLche der GeomebrLe zu Grunde LLegen.
- ,- ,, - - __ . - - -- --- --
..- ,
-1-1- - I1-, 1- ,
- --
-
10
I _- - - - - - "I.- - 11-1. - - -1-
'M- -
-
-
__-'_"'
W&W -bwbw
-
- ___ -1-_1-,;-'
The most extreme example of this is perhaps the theory of
combinatorial synthetic geometries or matroids.
The theory of combinatorial geometries is concerned
with the abstract properties--that is,
which can be
those properties
stated in purely set-theoretic term3--of
linear dependence. It is intuitively clear that these
invariant under the
abstract geometric properties are
projective
case,
linear group. Although not historically the
the axiomatization of a combinatorial geometry
proceeds very naturally from the two fundamental theorems
2
of invariant theory.
The first fundamental theorem states that in a
vector space V of dimension d over an infinite field,
the
only relative invariants under the general'linear group
are sums and products of the brackets:
[xl
where u1 ,
...
...
xd] =
det <x iu
>
,ud is a basis of dual vectors,
is the value at x.
of u..
Thus,
and (x ju.)
combinatorially,
the
geometric property to focus upon is whether a bracket is
2A development of classical projective invariant theory from
a combinatorial point of view can be found in Desarm6nien,
Kung and Rota [781.
non-zero
or not:
11
that is,
-
-
whether a set of d vectors is
a basis.
The second fundamental theorem tells us what all
the
relations between the brackets are.
can all be
These relations
derived from the following syzygies:
[xE ...
d-T
x(d-11
...
Yd[l
_Id-i
1=1
(yix2''' d3
In combinatorial terms,
l'''
i-1x1yi+1''..dl'
this becomes the basis exchange
axiom:
If B1,
B2 are two bases and x E B,
y E B2
such that
B
1s
then there exists
x u y and B 2 . y ux are both bases.
In the framework,
it is obvious that there are,
analogy to classical geometry,
in
three other combinatorial
structures corresponding to the geometries invariant under
the
general linear group acting on both the vector space
and its dual,
the orthogonal group,
and the symplectic group.
This observation was first made by Gian-Carlo Rota at the
1971 NSF Advanced Seminar in Combinatorics held at Bowdoin
College,
Maine.
In imitation of Felix Klein's Erlanger
12
-
-
Programm, we call the projected systematic development
of these three combinatorial structures the Bowdoin Program
for Combinatorial Synthetic Geometries. This thesis may
be regarded as laying the foundations for this program.
We have concentrated on two notions:
that of a
bimatroid (the analog of a bilinear pairing)
and that of
a Pfaffian structure (the analog of a symplectic space).
The notion of an orthogonal matroid can quite naturally
be
subsumed under the notion of a bimatroid.
Combinatorial
geometries
branches of combinatorics.
(in my opinion)
in nearly
all
Its ubiquity has the fortunate
consequence of its having many names.
Recently, however,
the word matroid has become standard.
In this thesis, we shall use
synonym for
arise
the word "matroid" as a
"combinatorial- pregeometry";
in particular,
a geometry is a simple matroid.
The time has passed when a paper in matroid theory
has to recapitulate the basic concepts in the
There are now two excellent texts,
introduction.
Crapo and Rota [70] and
the encyclopedic Welsh [76). As the lesser known concepts
are
described as fully as space permits,
the reader with
a basic
13
-
-
knowledge of the standard cryptomorphisms (that of
basis exchange, circuit elimination, rank, independent set
augmentation and closure)
should not have
and the theory of strong maps
to refer outside this thesis. Even the
given a bit of
theory of strong maps is not essential;
imagination,
all that the reader need to know is that a
strong map is the analog of a projection from a point in
space to a subspace in general position.
We use the standard set-theoretic notation:
S \T is the difference
particular,
elements in S not in T. However,
in
set consisting of all the
as is usual
in matroid
theory,
sets often appear without attendant curly brackets;
thus,
...
x
x
denotes the set {x1 , ...
,x
}.
14
-
-
1.2 Core extractLon
In this section, we essay another translation of
the Laplace expansion into a coordinate-free language.
This translation provides a cryptomorphism which
distinguishes matroids in the class of complexes.
complex on S is specified
Let S be a finite set. A
by a non-empty order ideal J
lattice of subsets of
that is, J is a collection of subsets of S satisfying
==-
I C J and J E j
I EJ
.
S;
on the
The
subsets in the order ideal are called independent sets.
The
subsets not in J
(which form an order filter)
dependent sets.
called
A = fv
Consider a set
:
=
0.
This implies,
..
v).
by the Laplace
the following dependence relation between the
expansion,
vectors
vi
determinant
j 4 n} of n linearly
suppose that vi = (vi,
Choosing a coordinate system,
the
1 <
n-dimensional Euclidean space.
say,
dependent vectors in,
Then,
are
:
1
.2
1
v
'
j-1
j+1
2
2
J-1
j+1
..
'
2
= 0.
.v2
''''.
1
n
vn
vn
n
vn
p
1___-
- __
---
I _-__--"--
- 15
-_
-,
.-
-
--.-
-
..' _-,
- --
_--
-
, - - - .-
-
-
- --
' 1
- -
A~
_---
In fact, this equation, if non-trivial, is a dependence
relation between the vectors vi which have non-zero
coefficients, that is to say (if we choose a coordinate
system in general position) those vectors vi for which
the set A N vi is linearly independent. This is the
crucial observation.
Definition 1.2.1: Suppose A is a dependent set of a complex
on the finite set S. The core of A is the set
(x E A: A \ x is independent}.
Some simple properties of the core which follow
immediately from the definition are:
Lemma 1.2.2: Let D and E be dependent sets of a complex
on the finite set S. Then,
a. if core D is dependent, core(core D) = D.
b. if D Q E, then core E C core D.
c. core D is the intersection of all the dependent sets
contained in D.
In our new terminology, our observation amounts to:
if D is a linearly dependent set of vectors, then core D
is either empty or linearly dependent.
OwAak--.-
,
-
- _--_'-_-
-
_ - -. ______";_.11_-._1_-_--
Theorem 1.2.3:
16
-
-
Let C be a complex on S.
Then,
the
independent
sets of C form the independent sets of a matroid on S if and
core D is either empty or
only if for all dependent sets D,
dependent.
Proof:
(if) We shall prove the circuit elimination property.
Let C 1and C2 be distinct circuits,
sets. Consider the dependent set C 1 U
x j
C
C2
Ir
hence is
U C2
then C
.
dependent. Thus,
minimality,
C1
must be empty.
C 2 . If x E
core C
U C2 C
for all x E
C
A C
C 2 but
C1 U
or C2,
x contains either C
C 2 is independent,
That is,
minimal dependent
i.e.
.
and
by
But,
and hence, core C1 U C 2
C1
1 U2
C29
- x is
dependent.
(only if)
If A
is dependent,
A must contain a
circuit. If A contain exactly one circuit,
If,
equals that circuit.
(or more)
circuits,
that core A
on the
then core A
other hand, A
contains two
the circuit elimination property implies
0
is empty.
It is worth noting the following detail in the proof:
Proposition 1.2.4:
If A
is a dependent set in the matroid
G on the set S,
then core A equals the unique circuit
contained in A,
if such is indeed the case,
and is empty
otherwise. Moreover,
the core of A, A
17
-
-
for any two elements x and y in
x and A Ny have the same closure.
The.-preceding results provide a new cryptomorphism
called core extraction for the
particularly useful in the
theory of matroids.
It is
study of extensions of matroids
and strong maps. We refer the reader to Kung ['77]
for
these applications. Another application is to the study
of weak maps.
Let G be a matroid on the
on the same set.
H
Then,
of G whenever a set A
set S,
and H
a complex
is said to be a specialization
is independent in H.implies that it
is also independent in G.
If H
is also a matroid, we say
that there is a weak map between G and H.
The following
theorem characterizes those specializations of G that
give rise to weak maps.
Theorem 1.2.5:
Let G be a matroid on S,
specialization of G.
Then,
and let H be a
H is a matroid if and only if
the following two conditions holds:
a. The modularity condition:
If I and J are G-independent
sets which become dependent in H and there exists x E I u J
such that Iu Jx x is
H-independent,
then In
J is
also
18
-
-
dependent in H.
b.
Let I and J be G-independent
The weak closure condition:
sets with the same closure in G.
independent in H.
Then,
Suppose that I remains
for all x C S,
I u x is dependent in H
Proof:
(=*)
=
rH(I)
IIuJI
+ rH(J)
J U x is also dependent in H.
observe that
To prove a,
rH(IuJ)
=
-
and
1,
rH(In J) 4 III
-
+
IJI
-
2 -
rH(I AJ).
By the semi-modular inequality,
II n i
rH(IrnJ)
that is to say,
-
1
I nJ is dependent in H.
observe that as I remains independent,
x E I
To prove b,
first
-H
J c_ I . Hence, as
by assumption,
-
Hence,
rH(J u x) 4
III
=
II.
That is,
J u x is dependent in
H.
(
) We first show that condition b is equivalent to
the following apparently stronger condition:
19
-
-
b'. Let I and J be G-independent sets with the same closure
in G.
Then,
Suppose that I remains independent in H.
for all
subsets K c S,
I u K is
H-dependent ==> J u K is
also H-dependent.
Suppose that IuK is dependent in H.
subset of K such that IuK'
Let K' G
is independent in H
subset exists as I remains independent),
Now,
I uK'
K be a maximal
(such a
and let xE K\ K'.
as I and J
is also independent in G. Moreover,
is independent in G and has
have the same G-closure,
JuK'
the same G-closure as Iu
K'. Applying condition b, we obtain,
as
in H,
I uK'u x is
This implies,
dependent
in turn,
JUK'U x is
dependent
in H.
that JuK is dependent in H.
We shall now prove that H is a matroid by checking
the core extraction property. Let D be dependent
in H.
We
have two cases:
1. D
be
is G-independent.
Suppose that coreHD /
all the elements not in the core of D
core HD =
As x
(D\x
1
is not in coreHD, D
)O
x
...
6.
Let x,..
,x
in H. Then,
((D \x ).
is dependent in H.Applying the
modular condition recursively, we can conclude that coreHD
20
-
-
is dependent.
2. D is G-dependent. As coreHD S.coreGD, we are done if
core GD = 0.
Hence, we can assume that Z = core GD is non-
We shall show that there are three possibilities
empty.
for core HD:
Z,
core HD =
or
coreH(Z \y),
for some y E
Z.
This claim is proved by checking four subcases:
a. Assume that for all x E
D'x is
dependent
contained in Z,
Z, ZN x is dependent in H.
in H for every x E
Z.
As coreHD
Then,
is
this implies that coreHD = 0.
We can now assume that there exists an element x in
Z
such that
b.
independent
in H.
Assume now that D\ x is dependent in H.
every y E Z,
remains
N = D
D \y
Z \ x is
Z
y and Z
Observe that-for
x have-the same G-closure.
independent, the fact that D \x =
As Z \x
(Z\ x) uN (where
Z) is dependent implies that, for every y in Z,
= (Z,\ y) u N is also dependent in H. Therefore,
is empty.
coreHD
21
-
-
We can now assume that for all x in Z, Z \x is
is also H-independent.
H-independent implies that D \x
c.
Then,
by the assumption,
dependent,
d.
Z \x is H-independent.
Suppose that for every xE Z,
coreHD equals..Z,
and hence H-dependent
which is a G-
set.
There remains the possibility that there exists y
such that Z-%y is dependent.
its H-core, by part 1,
As ZN y is independent in G,
is a H-dependent set. Thus, we are
done if coreHD = coreH(Z Ny).
As coreH D!
coreH(Zs y),
reverse containment.
we only need to prove the
Suppose t is contained in coreH Z\y.
We have to prove that D s t is H-independent. By the
it suffices to prove that Z st is H-independent.
assumption,
Suppose it is not. Then, Zht and Z Ny are both G-independent
sets which are dependent in H. Moreover, there exists x
such that Z-,x is H-independent. By the modular condition,
we can conclude
core HZ \ y;
that Z - t,y is
H-dependent.
But t
in
is
this contradiction establishes the reverse
containment,
and also the theorem.
0
After such an exhaustive and exhausting proof, we owe
the reader some matroidal insights. We shall compare the
-
22
-
L
theorem with some of the results in the theory of strong
maps. We assume that the reader is familiar with this theory;
details may be found in Crapo and Rota [70),
[76), chapter 17 and Kung (773,
chapter 9, Welsh
sections 4 and 5.
Observe first that the modular condition must necessarily
hold whenever we "specialize" any independent set I in G;
indeed,
the modular condition is equivalent to the core
extraction property, applied to the matroid H restricted to
I. If the weak map decreases rank by at most one, then the
modular condition reduces to the familiar modular condition
in the theory of elementary strong maps:
I, J are modular? independent sets and I and J both
become dependent implies that I n J also become
dependent.
When G -+H is a strong map,
the weak closure condition
is strengthened to the strong closure condition:
if I and J are independent sets in G with the same
G-closure,
then I and J have the same rank in H.
Alternatively, we can describe a strong map G--*H by saying
that it is a weak map for which any circuit in G is a union
23
-
-
of circuits in H. Rephrasing this in the context of our
proof, we see that of the three possibilities for the
core in H enumerated in the second part of the proof,
only the first two cases,
coreHD = 0 or coreG D
can occur for a strong map.
Incidentally,
this also
characterizes strong maps among weak maps.
Thus,
the difficulties encountered in the theory
reside in the possibility
of weak maps (see Lucas (753)
of creating relative isthmi:
y
that is,
in those elements
contained in a circuit Z such that
core HZ = coreHZ \y.
Note that the element y is now an isthmus in the matroid
restricted
to Z.
1.3 The co-core
As the reader would expect,
there is an orthogonal
version of the core extraction property. A matroid G on
the
set S can be spedified by a-non-empty order filter
2
24
-
-
of subtets of S. The subsets in j are called spanning
sets,
while the subsets not in A
are said to be non-
spanning.
Definition 1.3.1:
complement of A
A
is the
Theorem 1.3.2:
on S
is non-spanning. Then,
the co-core of
set
{x G
Then,J4
Let A be a subset of S such that the
A:
Let
x
u
J
Ac is spanning}
be a non-empty order filter of S.
is the collection of spanning sets of a matroid
if and only if for all non-spanning sets,
co-core A
is either non-spanning or empty.
Proposition 1.3.3:
The co-core of a non-spanning set A is
the unique bond contained in A if such is indeed the case
and empty otherwise.
In the language
of closure operators,
(Ac c
#.
Thus,
if Ac
is a copoint of G,
oteris
co-core A
otherwise.
co-core extraction bears some resemblence to the
interior operation in topology.
and
-
.1- --
-
.1 -
-
25
I-
-
- -1. 1--,
- .1--l.
-
-
- -Ira. --
1--
4--
-
1-
The core and co-core are related by the following
proposition:
Proposition 1.3.4:
set S. Suppose that x E B and y i B.
4=.
Then,
x 6 core(B u y)
y E
co-core((B \ x)C)
The
proof consists of the following equivalences:
.
on the
a basis of the matroid G
Let B be
x E core(B u y)
iff
Bu y\ x is
independent
iff
Bu y xx is
a basis
iff
y
iff
y is in the unique bond contained in (B
is not in the copoint spanned by B \x
c
\x)
The special case of this proposition for a graphic
matroid can be found in Biggs E743,
prop.5.5,
p.32.
The co-core extraction cryptomorphism also yields
a new proof of Crapo's theorem describing the erections
of matroids (see Crapo E70],
rather unperspicuous,
theorem 2).
and is omitted.
This proof is
1 .4
26
-
-
basLs exchange
ALternatbLng
Let X be a vector space of dimension d over a field
F.
If x
we set,
,
,xd is an ordered d-tuple of vectors in X,
...
as always,
[x 1 ...xd]
where
(x.
system.
.)
=
det (xij)
are the co-ordinates of x.
in some co-ordinate
Consider the function t on an ordered v-tuple of
vectors x1 , ...
,x,
defined by
=
.L(x I,...,x0)
x-.-.xkak+l ..adxlk+l''xk+bl+l..bd'''xk+l+..+m+l*.xpcm+l..cd
where
d are fixed vectors. The function t is
''0 'c
ak+1'
not alternating, but we can construct an alternating
function t*
in the following fashion.
We first recall the notion of a shuffle.
I=
{S 1 ,
shuffle
*..
Let
,S } be a partition of the finite set S. A
relative
to the partition
i
is a permutation ei
S such that there exists an i and x E S.1
In other words,
a shuffle
such that crx 4 Si.
is a permutation of S that actually
moves an element from one block of
function t* defined by
of
ir to another. The
t*(xi,...,xv)
where the sum is over all
the partition
v}},
t(x0 ,1 ,. ..,x ,)
=
,
27
-
-
shuffles of (1,...,v} relative
to
{J1,...,k},{k+1,...,k+l},...,tk+l+...+m+1,...,
is an alternating multilinear v-form on the vector
space X.
we observe that the value of t* on any ordered
Now,
set of 0
In the language
linearly dependent vectors is zero.
of matroid theory,
this becomes the following basis exchange
property:
Theorem 1.4.1
(Alternating basis exchange):
be bases of a matroid G on the
Let B 1 ,
,Bm
with each basis
finite set S,
into two blocks:
partitioned
X.i U Y.
B.
such that the sets Y
,
14 i<m
Suppose that
are pairwise disjoint.
uYm is dependent in G. Then, there exists a
Y = Y 1 U1 U
shuffle o' relative to the partition (Y
for all i,
X
uoY
1
,
,Ym}
..
Magnanti
such that
is also a basis.
Proof: We first construct the exchange graph (see
is
...
1753 or Lawler
a way of encoding
all
[763,
Chapter 8);
the possible basis
the
Greene and
exchange graph
exchanges.
The exchange graph is the directed graph (without loops)
28
-
-
defined on the vertex set Y as follows.
as the sets Y
ordered pair;
in a unique basis B
exchange
graph
if
. The
Bi
be an
Let (a,b)
are pairwise disjoint, b is
pair (a,b)
b u a is
is an arc in the
This
a basis.
is
situation
b
aEb
.
indicated by
B.
Evidently,
a-B.--b whenever one of the
equivalent
conditions holds:
(i)
following two
b is in the fundamental circuit of a relative to B.;
is contained in the bond S NB - b.
(ii), a
Now,
Y contains a circuit C.
as Y is dependent,
the
Since a bond is the complement of a closed set,
intersection of the circuit C with any bond D cannot be
a single element set:
that a E B.,
that is,
consider the bond D
1
assuredly contained in Cn Da,
observation,
b is
in
ICn DI
a
1.
For a
= S ,B.Na.
by our
another point b E C, with b E B.,
the bond Da;
that
is
to say,
a.
b-.
C such
As a is
1
there must be,
e
such that
Thus,
from
J
any point in C,
C
we can always retreat to another point in
through an arc of the exchange graph.
As C
is finite,
this
implies that the exchange graph contains a directed cycle.
29
-
-
Choose a directed cycle of minimum length:
The
d Bk-a,
...
c
b
(*)Ba
permutation V on Y defined by sending a to b,
and keeping the points not in the cycle fixed,
a shuffle,
is in fact
since consecutive arcs in the cycle cannot be
same basis.
labelled by the
a basis;
b to c,.
Moreover,
for all
i,
this is proved in Greene and Magnanti
X
u O-Y
[763,
is
p.533,
but the proof is worth repeating.
p
Let xp B
p
B
We claim that for all p,
=B
y
For p = 1,
is a basis.
be all the arcs in the cycle
in order of appearance from left
say,
labelled B1 , arranged,
to right in (*).
n,
p
...
ypu x 1 ... xp
this follows from the
definition.
Suppose that this holds up to q-1. Note that the fundamental
circuit of x
since,
that x
relative to B.
if it were not,
q
y,
r
minimum size.
is contained in B.%
there would be an r,
y1 ..
y q1U x
14 r< q-1,
such
contradicting the fact that the cycle is of
Moreover,
the fundamental circuits of x
q
relatie
toiBand1Bare
the same. Hence, we can replace
;
y
the basis B.
inq to obtain
B i,q-1
q by x q in
Setting p = n,
follow by induction.
B.
i,n
= X.uo'Y.
is a basis.
1
1
30
-
-
.
Our claim now
we conclude that
The proof of the theorem is now
complete.
A
special case of the theorem is obtained if we
postulate
r(G) + 1;
IYI >
that
this is first proved by
in (74], and may be regarded as a generalization of
Greene
exchange
the basis
axiom.
The roles played by circuits and bonds are symmetrical
orthogonal duality provides us with
Thus,
in the proof.
another metamorphosis:
same initial hypotheses,
Corollary 1.4.2: Under the
that X = X 1 u
...
exists a shuffle a'
such that,
A
is a non-spanning set.
uX
Then,
suppose
there
of Y relative to the partition {Y 1 ,...,Ym
for all i, X
is also a basis.
u cYo
further analysis of the proof yields a third
metamorphosis:
Theorem 1.4.3:
Gi,
...
Let B1,...
,
Bm be bases of the matroids
Gm on the finite set S,
with each basis partitioned
into two blocks:
B
such that the sets Y
X
U Y
,
14i4 m
are pairwise disjoint.
Suppose that
uY
contains a
subset Y'
such that Y'
is a
union of circuits in each of the matroids G . Then,
exists a
shuffle of relative to the partition
such that for all i, X 1 U aY.
is a basis
Y
there
,...,Yma
in the matroid G
.
...
Y = Y1u
31
-
-
The proof is an easy modification of the proof of
Theorem 1.4.1. The orthogonally dual version is left as
an exercise.
The
last version of the alternating basis exchange
property has an interesting interpretation in terms of
matrices.
Theorem 1.4.4:
Let M be a matrix with rows indexed by T
and columns indexed by S.
be
Let U, V G T,
and let GIU and GIV
the matroids on S defined by the matrices MIU and MIV
(where the matrix MIU is the matrix consisting of the rows
of M indexed by U).
and suppose that A I
A
n
Let B
B1,
and B2 be bases of GIU and GIV,
A2 C B2 are subsets such that
A2 = 0 and there exists no shuffle a, of A1 U A2 relative
to {A ,A2}
bases.
such that B1. A 1 u
Then,
AI
the subset A 1 u A 2
and B2N A2u oA 2 are both
is an independent set in
the matroid GIU uV defined by the matrix MIU uV.
Proof:
32
-
-
We note that GIU and GIV are projections (or,
matroidal terminology,
quotients or strong map images)
of GIU uV. Under a projection,
onto a union of circuits.
Thus,
any circuit is projected
if there exists no
shuffle o' of the form specified, AI u
a circuit in GIU UV:
in GIU uV.
in
A
2
that is to say, AI u
cannot contain
A2
is independent
0
1-1-1- 11.
- -
1 11
. -
-
.
I .
, 1 11 1-1-
-
,
33
.
11
-
.
. I
-
.
I
I
I
.-
1. 1.
1 -
ffi
, -.
1 11 -
-
-
11W
---
".
1.5 PortLtLons and LntersectLons
The exchange graph introduced in the previous section
was first invented to deal with the problem of matroid
partition:
Let G1,
G2,
...
Gk be matroids on the finite
,
set S and let T be a subset of S.
Then,
T is
an independent set partition relative to G ,
1
there is an ordered partition T 1 ,
T
in G
is independent
,Gk if
of T such that
theory of matroid partition
The major result in the
is the
,Tk
..
.
for all i,
...
said to have
following theorem of Edmonds and Fulkerson E653:
Theorem 1.5.1:
function r
Let G.,
, and let T
14i.
; S.
k be matroids on S with rank
Then,
T has an independent
set partition if and only if for all subsets A C; T,
i=1
(A) > ||
The concept of independent set partition leads to
the definition of the join of two matroids. Let G and H be
two matroids on S; consider the collection of subsets I 9 S
such that I has an independent set partition relative to G
and H. This collection is in fact the collection of
independent sets of another matroid GvH,
called the join
--
,
-,-I.- -
-
-
11-1-.1--1-.---,-
34
-
-
p.121,
(see Welsh C763,
of G and H
or section 4.1 of this
thesis). The join operation has been studied mostly from
the
point of view of graph theory (Tutte [611,
Nash-Williams
[661) or matching theory (see Welsh [763). We outline here
a more
geometric
approach.
We first set up a co-ordinate picture of the join.
Let M
and N be two matrices,
same set S,
with columns indexed by the
The
defining matroids M and N on S.
join MvN
is co-ordinatized by the following matrix:
M
MvN
NX
obtained by the construction:
Let Ixa',
x':
a
a ES} be an
algebraically independent set of transcendentals over the
ground field k.
The matrix Mx is obtained from M by multiply-
ing the a-th column by xa;
similarly.
These
the matrix N
is obtained
two matrices are then stacked on top of
each other as shown above.
It is easy to check that the matrix
Mv N does in fact co-ordinatize MvN,
as the
transcendentals
we have introduced prevents any "accidental" linear
dependence.
35
-
-
The fact that deleting a row from a matrix M
corresponds to a projection of the associated matroid M
is the geometric motivation for the following result:
Proposition 1.5.2:
Let G and H be matroids on S. Then,
GvH-+rG,
GvH-+H
are both strong maps.
Proof:
L.
S, AK
Suppose that x E
a Gv H-independent
Now,
G v H:
set I 9 A such that
that is,
x u I is
there exists
dependent.
I is independent in the join implies that there is a
I1,I2 of I such that I
partition
in G and H, respectively.
and I2 are independent
Moreover, x uI
the join implies that both x u I
and x uI
is dependent in
2
are dependent.
a G-independent set I
C A
such
We have found,
therefore,
that x u I
dependent in G. This implies that x E I
is
x E TH.
Similarly,
1
There is an orthogonal version of the
the meet,
.
A C
Recall that K--L is a strong map if for any subset
join,
called
defined by
GAH =
where G* is
(G*vH*)*,
the orthogonal dual of G. Using the fact that
a strong map G-+ H dualises to a strong map H*-+ G*
(see
ku --
-
36
--
,
.1 1
1
-
...
.........
Welsh (76], p.321), we obtain
Proposition 1.5.3:
strong.
The maps G--GA H and H-+ GAH are
the collection of flats
In particular,
of GAH
is contained in the intersection of the collections of
flats of G and of H.
This proposition is the foundation of an approach to the
meet operation via the theory of strong maps.
if H
For example,
is a matroid of nullity one, with the unique circuit C,
then GAH is the elementary quotient
of G whose modular filter
is the principal filter generated by C. We hope to develop
this approach further elsewhere.
The concept of matroid intersection is orthogonal
to that of matroid partition when we are concerned with two
matroids.
The major result here is due to Edmonds [703.
Theorem 1.5.4: Let G and H
there
is a subset I C
be matroids on the
set S. Then,
S of cardinality i which is independent
in both matroids if and only if for all partitions of S into
two blocks S 1 and S2'
+ rH(S 2
)
rG(S1
Such a subset is called a matroid intersection of G and H.
1-
-
---
-
.
.
I
I-
1 1,
-
. -
37
-
-
1
1
. . ,
-
-
-
1.
.
. . I.
I
I
-
I -
1-4k.,
- . ,
- I Vbg!4
-
-
A neat way to look at matroid intersection is the
following:
Let M
and N be matroids co-ordinatizable over
the same field by the matrices M
and N.
Let MX
and N
be the transcendentalized matrices defined as earlier.
Then,
the maximum size of a set independent in both
matroids is the rank of the product matrix
S(x') t
Mvi.(N
)
-
where t denotes matrix transpose.
This idea
can be
rephrased combinatorially.
Another way is through the concept of perspectivity.
and r(U)
= r(V)
Suppose that we want to find a common independent
set of
Let U
and V be flats of G. Let r(G)
G/U and G/V.
= d,
= r.
This is equivalent to finding a common modular
complement for the flats U and V. In the terminology of
lattice theory,
flats are
case
this is the problem of deciding when two
perspective. Note that this apparently special
is in fact general,
for,
by Higgs'
theorem for strong maps (Higgs C683),
the
representation
any two matroids of
same rank can be represented as contractions of a
suitable extension of the
free matroid on S.
Note also that perspectivity,
not transitive.
as defined here,
is
Intuitively,-this failure of transitivity
-
-
, -
-
1
38
-
-
is related to the difficulty (if not impossibility)
of
finding a polynomially bounded algorithm for the three
matroid intersection problem.
1.6 The aLpha functLon
We sketch here a theory of the alpha function of a
matroid.
For details,
we refer the reader to Kung [
78
3.
The alpha function was first introduced by Mason in his
pioneering paper [72] on matroids induced from directed
graphs.
Let G be a matroid on the finite set S.
any subset A C
S,
Then,
for
the alpha function is defined recursively
by
o(#)
=
0
o'(A)
=
n(A)
-
Z
o(F)
FCA
where n(A)
is the nullity of the subset A, and the summation
is over all the flats in G strictly contained in A. For a
flat F of G,
39
dF) = E
-
-
(E,F)n(E)
EC: F
the
summation being over all flats contained in F. Thus,
we can regard the alpha function as the first difference
of the nullity.
Indeed,
the alpha function measures the
change in nullity at a set which is not predictable from
the matroid structure lying below A.
We shall compute,
as an example,
the alpha function
of the graphic matroid of the complete graph Kn
(which is
also the lattice of partitions of an n-element set). If F
is a flat which is disconnected (in the matroid sense),
then
((F) = 0 as the nullity of F is predictable from the nullity
of its components.
of K
Hence,
= 0 unless the flat F consists
o(F)
for some m(.n together with isolated vertices.
subgraph is called a complete
subgraph;
Such a
its order is the
number of non-isolated vertices. We claim that for a flat F
o(F)
As F is a flat,
m,
(-1)m+1 if F is complete of order
otherwise
{0
it suffices to check that
E]
EGF
((E)
= n(F).
and
But,
40
-
-
after breaking up F into its connected components,
It is also easy to
this becomes the binomial identity.
show that
a(A)
for
=
any set
|AI
-
n -
A 9 S,
#connected components
(-,)order C + 1
E
C:C is a complete
subgraph
The classic result in the theory of the alpha
function (for which we
Theorem 1.6.1
give a new proof in [781
(Mason's alpha criterion):
) is:
A matroid G on
the finite set S is the orthogonal dual of a transversal
matroid if and only if for all subsets A
of S,
o((A) > 0
This theorem can be generalized to an alpha
criterion for truncations of transversal matroids;
problem was attempted by Brualdi and Dinolt [75].
this
We first
describe some technical results about lifts and truncations
which are
due to Cheung and Crapo [733.
The operation of truncation is orthogonally dual to
the operation of lift (towards the free matroid).
G--H is an elementary strong map.
Then,
Suppose
G is said to be
the lift of H,
and H
41
-
-
the anti-lift of G,
filter associated with the
if the modular
strong map is
D = the modular filter generated by all
the dependent
flats in G.
We emphasize that D need not be the collection of all
dependent flats,
If
a flat
F is
h(F)
and if
define a function h
A
h(A)
of G,
flats by:
then
(-1) #ways of expressing F as a join of
the generators of D
= T.
is
on the
a subset in S,
h(F)
=
.
We
although it certainly contains them.
all maximal flats F
contained in A
Recall
that the generators of a modular filter are
minimal elements. The function h
its
is called the anti-lift
function of the matroid G.
Theorem 1.6.2:
A matroid is the orthogonal dual of the
truncation of a transversal matroid if and only if for
all subsets A
42
-
-
h(A).
oc(A) >
in S,
The proof is omitted;
Kung
178 1
,
for readers familiar with
the following hints should be sufficient;
use
and the fact that the flats in the collar of D
Lemma 2,
and hence, have alpha function zero.
are all independent,
One
can iterate this criterion;
recall that the Higgs
factorization of a matroid G of nullity n
is the
sequence
of strong maps
.
o(A)
where h.
For any subset A
=
dual of the
h
0
(A),
A matroid G on S is the orthogonal
subsets A
ot(A) .
= 0 G,
n
in the matroid G,
j-th truncation of a transversal matroid if
and only if for all
1
-Y
is the lift of G.
is the anti-lift function for the matroid G.
Proposition 1.6.4:
where h
and G.i+
.
G nis the free matroid,
n
Proposition 1.6.3:
G
--r G i= Gt
+
where
-+
*
Gn--G
in S,
h
(A)
is the anti-lift function of the matroid G!,
1
and G+
+1
=
anti-lift G.i
1
where
43
-
-
......
.
......
....
Chapter 2
FINITE RADON TRANSFORMS
2.1
The cLassLcaL Radon
Let f:
bronsform
R n -+R be a real-valued function in n
variables which satisfies certain integrability conditions
irrelevant in this context.
Its Radon transform is the
function
Rf:
(R n)* -+R,
x*
f-+
f,
the integration being over the hyperplane H perpendicular
to the dual vector x*.
This definition can be formally
in the case of a finite
imitated
collection
P of subsets of S,
Definition 2.1.1:
function
Rf:
P -k,
S,
equipped with a
as follows:
The Radon transform with respect to the
collection P of a function f:
-+ Tf(p)
set
T
p
S-+k (k a field)
is the
44
-
-
The general concept of a finite Radon transform is due
to Ethan Bolker who first presented it at the M.I.T.
Combinatorial Theory Seminar during spring term,
In this chapter,
we discuss some
theory of matroids
1976.
applications to the
1
We shall be concerned only with geometries or
simple matroids. We first recall the following variant of
the closed set cryptomorphism due to Crapo (see C713,
A
geometry G on the finite set S
operator Ai-* A
on its subsets.
p.1):
is specified by a closure
The closed sets or flats in
S satisfy two conditions:
single element set
(a)
A
(b)
(the
u,...
is closed (and is called a point);
partition property) Let t be a flat in S and
,um be all the flats covering t. Then, the sets
u. \t are the blocks of a partition of S\ t.
1Some of our results are implicit in Dowling and Wilson
[753. A synthesis of our approach with their Mbius
function technique should be very rewarding.
45
-
-
2.2 The copoLnt Radon bronsform
Let f:
set S
S-+,k be a function from the finite
to a field k pf characteristic zero.
geometry on S.
Suppose G is a
The Radon transform of f is
of G by
defined on the flats
Rf(t)
the function
=
E t
f(p)
The restriction of Rf to the flats of rank i is called
the
rank i Radon transform.
Less pedantically, we also
speak of the copoint and coline Radon transform.
The mass
of the function f is defined by
mass(f)
= Rf(S)
=
EpES
f(p)
Over a field of characteristic zero,
the
copoint Radon transform is invertible.
Theorem 2.2.1:
Let G be a geometry on S of rank greater
than or equal to two,
and f
a function S-a-k,
field of characteristic zero.
Radon transform of f,
determined.
Then,
where
k is
given the copoint
the function f itself is uniquely
a
46
We shall exhibit an algorithm for inverting the
Proof:
copoint Radon transform.
since
-
-
the case r =
Lemma 2.2.2
i-1 flat,
We can assume that rank G = r g 3,
2 is trivial.
Let t be a rank
(the truncation equation):
with i < r,
and ul,
,um be all the rank i
...
flats covering t. Then,
Rf(t)
Proof:
=
- 1
m
Consider the sum
property,
the sum,
while
exactly
m times.
Hence,
_
Rf(u1 ))
Rf(u
u
)
,Rf
Algorithm 2.2.3:
of f is given.
each point
f(p)
+
(m
Suppose that the
Let M be
in t contributes
-
p
tfp)
pE
1)
thought of as
an indeterminate,
coline Radon transform in terms of the
truncation equation.
the rank 1 Radon transform,
obtained in the form
0
copoint Radon transform
representing the as yet unknown mass of f.
by the
mass(f)].
By the partition
Rf(u.).
mi
-
each point in S not in t contributes exactly
once to
~i-
[=
Compute the
indeterminate M
Iterate this procedure till
which is the
function f,
is
f(p)
(*)
= a
Sp
47
+ b M,
-
-
a,
p
b
p
E k.
By definition,
>ii
M =
5
a
+
(
pES b
It is clear by induction that b
and bp 4 0; this
implies
that
1
)M.
is a rational number,
-
p E S b9
p
0. Hence,
PCS a
M =
1
Zp
E S bp
Substitute this numerical value for M in (*).
This concludes
3
the algorithm, and also the proof of the theorem.
The k-vector
Map(S,k)
of all
functions
standard basis consisting of delta functions
S--ok
6
,
has a
space
p E S,
defined by
6 (q) = 1 if
Let C be the
p = q,
and O otherwise.
set of copoints of G.
The copoint Radon transform
is a linear operator from Map(S,k) to Map(C,k). With respect
to
the standard basis on both vector spaces,
the matrix
of the copoint Radon transform is just the point-copoint
incidence matrix,
M
cp
i.e. the matrix (Mcp) with
= 1 if pec, and 0 otherwise.
-
- 48
We have thus proved the following results,
due to
Basterfield and Kelly C683 and Greene (70).
Corollary 2.2.4:
Let G be a geometry on S of rank > 2.
Then,
(a)
the number of points is less than or equal to the
number of copoints;
(b)
over a field of characteristic zero,
the
rank of the
point-copoint incidence matrix equals the number of points;
(c)
there exists
an injection
cK:S-+ C such that
for
all
p,
p 6 a(p).
It
should be possible to use the
geometric idea
contained in the partition property to invert the classical
Radon transform (and its modern variants).
The basic idea
here is to compute the inverse Radon transform by going
down the subspaces one rank at a time by an analog of the
truncation equation involving integrals over the
Grassmannian.
49
-
-
2.3 The bond Radon transform
The complementary Radon transform of a function
f:S-+ok is the function defined on the flats of G by
pt
Cf(t)
=
f
The restriction of Cf to the copoints is called the bond
(since a bond is the
Radon transform
copoint).
set complement of a
The theory of the complementary Radon transform
can be developed in a fashion similar to that of the
Radon transform; we
(The complementary truncation equation):
Let t be a rank i-1 flat,
the rank i flats covering t.
Cf(t)
Theorem 2.3.2:
= m
-1
< r, and u1 ,
with i
Then,
Cf(u
Let G be a geometry on S of rank
k a field of characteristic zero.
Then,
)
a function
f:S-,k is determined uniquely by its bond Radon
transform.
,um all
...
)
Lemma 2.3.1
shall only state the main results.
2,
and
Corollary 2.3.3:
50
-
-
a geometry on S of rank'> 2.
Let G be
Then,
the rank of
(a)
over a field of characteristic zero,
the
point-bond incidence matrix equals the number of points;
(b)
there exists an injection f:S-+.C such that for all p,
p 0 P(p).
Part
(b)
is a result of Curtis Greene [701.
2.4 The p-Radon Transforms
Suppose now that the field k is of positive
characteristic p.
Our theory may break down entirely as
the truncation equations
end of the algorithms)
(as well as the
computation at the
may no longer be valid.
over fields of characteristic p,
For example,
the geometry of 2p points
in general position in p-space has non-invertible bond and
copoint Radon transforms.
The following inequalities are all the results known
at present.
Let G be a geometry on S of rank d >
2.
Let
I p(G)
51
-
-
= rank of the bond Radon transform,
and
Pp(G) = rank of the copoint Radon transform,
both computed over a field of characteristic p.
can derive,
given the mass of the function,
Since we
the bond from
4 1
|-p - Pp
All three cases can occur;
three-point line,
three
b.
= b
bi+1..bd,
022 for the
2 for the free geometry on
by considering the copoints
points. Moreover,
.. b
over GF(2),
P2
while
-
the copoint transform and conversely,
where b,...bd
is a basis of G,
we
see that
d
7p
The p-Radon transform is of importance in the theory
of coding and packing. In particular, the p-Radon transforms
of a projective geometry over GF(pk) have been extensively
studied (see,
for example, Jamison [773 and Smith (683).
Indications are that the
study of the p-Radon transform for
arbitrary geometries is very difficult.
One of the reasons for the failure of the inversion
algorithm over characteristic p is the divisibility by p
of the upper covering numbers:
the covering numbers of a
52
-
-
a flat t in a geometry are defined by:
m*(t) = number of flats in G covering t
m,(t) = number of flats in G covered by t
has proved that in a finite modular
Dilworth
(in C54J)
lattice,
the numbers (for which we propose the name Dilworth
)
numbers of the second kind2
Bk
Ck
t with m*(t)
= k)
=
#(flats
=
#
=
#(flats t with m*(t) = k)
=
# flats covering k flats
flats covered by k flats,
are equal. However,
and
the obvious generalization,
Bk >, Ck
(the direction of the inequality is dictated by the fact
that #copoints = B 1 >
by the
#points = Ck) is false as is shown
following counterexample:
b
C -o-o----od
e
For ,the Dilworth numbers of the first kind,
and Rota [701.
see Harper
53
-
-
k
0
1
2
3
4
5
6
Bk
1
6
1
4
0
1
0
Ck
1
5
4
2
0
0
1
This is also a counterexample to the conjecture that the
sequences Bk or Ck are unimodal.
We have the following result,
however.
Let
B u
LG(u) =
be the generating polynomial of the numbers B .
GX H (u)
A
=
Then,
BG(u)BH(u)
similar result holds for the "lower"
Dilworth numbers.
The proof relies on the
fact that the lattice of flats
of the direct sum G X H
is the direct product
of the lattices of flats of G and of H;
(as lattices)
details are easy
and omitted.
Finally,
it should be
interesting to
investigate the
structure of G from the viewpoint of its Dilworth numbers;
for
example,
BI = C
Is
there
Greene [70) has proved that
=- G is modular
a generalization
=4 Bk = Ck for all k.
of this?
54
-
-
2.5 The maxLmaL chaLn theorem
Consider the lattice
S of rank d
>
of flats of a geometry G on
2. A maximal chain in this lattice is a
chain
u1
where u.
m(G)
2
U d-1
is a flat of rank i.
Let
= the maximum size of a family of pairwise
disjoint maximal chains in the lattice
of flats
of G.
As each maximal chain contains a point,
m(G)
it is evident that
is at most the number of points in S.
That this upper
bound can be attained was first proved by Mason [731,
using an extension of the matching technique of Greene
(70).
We shall present another proof.
A
preliminary observation is in order.
set of all rank i flats in G.
T: Map(F.,k)-+Map(F.
-i-i ,k)
-1
standard basis,
Consider the
defined,
Let F
be the
linear operator
with respect to the
by the matrix (Ttu), where,
rank i-1 flat covered by m rank i flats,
if t is a
r1/m-1
tu
L0
if
55
-
-
t C u, and
otherwise.
The complementary truncation equation implies that the
following diagram (where C.
is the complementary Radon
transform restricted to the
flats of rank i)
Map(F
,k)
T
C.
C
Map(S,k)
Note that if E 1
T
,J
commutes:
Map(F._ 1 ,k).
is a subset of
,
the maps C
and
are still well-defined when we replace the codomain by
k). Moreover,
Map(E.
the above diagram, with the right
hand corner replaced by Map(E ,,k)
Theorem 2.5.1:
remains commutative.
In a geometry G on S of rank )
2,
there
exists a family of pairwise disjoint maximal chains of
size
ISI.
Proof: We
Assume
shall construct the required family inductively.
that we already have
of the form
a family of n
disjoint chains
uk,1
u
c
= (u
Let E.
-i -1
part of the
56
-
-
:
k$, i1
16 kn}.
1
,k)
Map(F
Map(S,k)
possible
C
indexed by E.,
the
E.
-i-i
0 iff t C
columns E.
14 k 4 n.
also assume,
the diagram
,k)
Map(E
=
T o
C.
,'1
k)
By the induction
is of rank n. This implies that it
and T,
Q
F
is
such that both of
restricted to the rows or columns
are of rank n.
u,
as
that the operator
C.
C
,
is of rank n. Now,
to choose an n-subset E
the matrices C.
Ttu
C.1-
that is to say,
hypothesis,
We shall
induction hypothesis,
C _1 : Map(S,k)--Map(E
commutes,
uk, i-1
. . .
As the matrix T satisfies
the non-singularity of T restricted to
implies that we can find a matching from
to E.. We can now use this matching to extend the
-1
family of disjoint chains up another level.
the induction step and also the proof.
This completes
0
57
-
-
Chapber 3
BIMATROIDS
Where necessLby enforced a
passage,
vanLty
suppLed a grobbo.
Samuel Johnson
3.1
MatrLces and bLLLnear
pLrLngs
Much as the notion of a matroid is the combinatorial
abstraction of a set of points in a vector space,
the
notion of a bimatroid is the combinatorial abstraction of
a matrix. For our purposes, it is best to regard a matrix
as the table of values of a bilinear
two vector spaces.
More precisely,
pairing between
let X and U be two
vector spaces of dimension d over the same field k. Suppose
that there is a bilinear pairing between X and U.
Then,
picking a finite set S from X and a finite set T from U,
we can form the matrix with rows and columns indexed by S
and T and the xu-th entry defined to be the value of the
58
-
-
bilinear pairing on the pair of vectors x and u. More
simply, we can consider the case when U is the dual vector
space V*. In this case, we can use the fundamental theorems
of invariant theory for the general linear group acting on
both a vector space and its dual as a guide to a convincing
axiomatization for bimatroids.
The first fundamental
theorem states that the only
relative invariants are the determinants
(x 1 .. x nlu
where <x
..
un
= det[(xilu
3'y
lu >is the value of the bilinear pairing between
the vector x.
and the covector u .. Thus,
the essential
feature to focus upon is whether a minor is singular or
non-singular.
In addition, we need the analog of the
exchange property.
Here,
the second fundamental theorem
admits several interpretations.
For example,
relations satisfied by the brackets <x 1
may be
the
though all the
..x nlu
.. un)
derived from the vanishing of all d+1X d+1 minors,
direct combinatorial translation of this fact yields
exchange properties which are too weak. Our final choice
is a combinatorial version of a weak form of the generalized
Laplace expansion (see Desarmenien, Kung and Rota [78],
p.69):
59
-
-
un
. .
i
'
n u
x y . . x
yi :-ym
..
j=1
x _lyxi+l
y
j=1
y2
Our choice is motivated,
..
'''
m
firstly,
ym..
V''
1
fn u1
..
u
m
/
n
''
j-1 j+1 '
m
by the fact that this
identity is central to the straightening algorithm approach
to invariant theory (see Doubilet,
Rota and Stein [74]),
and secondly, by the naturalness of the results obtainable
from this axiomatization.
Remark:
The idea of abstracting the non-singularity
properties of the minors of a matrix must be as old as
the idea of a matroid.
There are two previous attempts
at doing this.
The chronologically later axiomatization is the
notion of a tabloid, due to Hocquenghem [pre]. A tabloid
is,
quite simply,
a birank function r(A,B)
defined on
60
-
-
pairs of subsets of two sets S and T such that the
sections
r(A,-) and r(*,B) are rank functions of matroids on S and
T for arbitrary subsets A
and B.
This is perhaps the most
general possible axiomatization.
The earlier axiomatization is the notion of a
quotient bundle,
due to Alan Cheung and Henry Crapo [733.
The study of quotient bundles is the study of the
independence property of two
distinguished)
abstract
(which are
sets S and T
of vectors in the same vector space. Viewing
a matrix as a bilinear pairing between V
and V*, we think
of the dual vectors as being represented as vectors in V,
and we are careful to distinguish between vectors and dual
vectors.
This axiomatization has several advantages,
the
main one being that the dual vectors are already part of
the space; however, we can no longer consider subspaces
as being rigid with respect to each other,
as
there are
possibly many ways of realising a given quotient bundle
as
a bilinear
pairing.
The theories of bimatroids and quotient bundles
share many common themes,
although these themes are usually
inversions of each other. We shall draw attention to these
similarities in the course of our exposition.
61
-
-
3.2 BasLc defLnLtLons and exampLes
Let
u
S and T be two finite sets.
Suppose x C S and
same cardinality n.
g T are two subsets of the
pair x,u is said to be an nxn minor. Frequently,
often written as x x 2
A
n,
thus,
u 1 u 2 .. .u
the
we display
the minor x,u is
.
a minor by listing its elements:
Then,
bimatroid B on the sets S and T is specified by a
partition of the minors into two
(disjoint) classes:
class of regular (or non-singular)
singular minors.
the
minors and the class of
The statement that a minor x,u is regular
is abbreviated by
x
we shall also use
u.
Similarly,
u;
the locution x is independent relative to
the statement that the minor x,u is singular
is written
We
impose three conditions on the class of regular minors.
The first is that the empty minor is regular:
Secondly,
that is,
we require that the regular minors satisfy the
62
-
-
following exchange-augmentation property:
If
x 1 x 2 '0 'Xn
u1u2.u
Yl---YM
then,
*6*
1
m
that both x,u and y,v are regular
(meaning, of course,
minors),
V
at least one of the following
y,
for y 1 C
holds:
such that
There exists x
(a)
x 1..x
- yxi+1'-
x iy2
'
u
n
1"
un
,
v 19..1vm
YM
or
There exists v. such that
x y
x 1..
y2'''Y
We call
If (a)
while if (b)
u v
....
1...v
m
the element y1
operation.
u1
j-vj+1'''e
nj
m
'
(b)
the focus of the exchange-augmentation
holds,
we say that we can exchange,
holds, we say that we can augment.
Finally,
we also require the regular minors to satisfy the analogous
exchange-augmentation property with the roles of x, X and
u,
v
reversed.
Our axiomatization is
symmetric with respect to S and
T.
is a bimatroid on S and T,
if B
Thus,
63
-
-
its transpose
defined between T and S by setting u I x if and only
B t,
if x
I u,
A
on the
is also a bimatroid.
bimatroid between the sets S and T defines matroids
sets S and T.
Lemma 3.2.1:
u in T,
u,
x
in S such that there exists
The subsets x
form the independent sets of a matroid
on S.
Proof:
We shall prove that the maximal
satisfy the basis exchange property.
y = y1...ym
u,
Now,
.. .xn and
v. We may assume
I u and X
choose u, v such that they have
of maximum size.
element v
Let x = x
be maximal sets for which there exists subsets
v in T such that x
n > m.
independent sets
in v
u.
Suppose that v
Then,
that
an intersection
there is an
is not in u. Consider the
such that v
exchange-augmentation operation focussed on v:
..
xu
y...ym
We cannot augment,
In particular,
u
v2vm
as x is maximal; hence, we can exchange.
for some ui,
x1
... Xn
u
... u
ivui+ 1 ...sun
ui u v increases
Replacing u by u \
intersection,
64
-
-
the
size of the
contradicting our initial choice.
Thus,
we have
xi
...1
xn
ul...umum+1...un
yl...ym
U,u... um*
Now consider the exchange-augmentation operation focussed
on um+1.
Hence,
Neither exchange nor augmentation is possible.
we must have m
X
yl
xn
...
yn
=
That is,
n.
ui ...
u
...
un
ul
we have
Focussing now on y1 , we cannot augment;
able
to exchange y 1
into x.
hence,
we must be
This proves the basis exchange
property.
A neater proof may be obtained by using Edmonds'
optimization cryptomorphism
(see Edmonds
[703,
p.69);
details are left to the specialist.
0
The matroid on S obtained in this fashion is called
the associated matroid on S,
G,
and is denoted G , or simply
when no confusion is possible.
replace T by any subset T' 5
credibility of the proof.
It is clear that we can
T without damaging the
The matroid on S obtained in
65
-
-
this way is called the associated matroid on S restricted
to
T'
GIT'. We say that a set x in S is
and is denoted
independent relative to T',
independent
x is
T',
in symbols x
whenever
in the matroid GIT'.
By the symmetry in our axiomatization,
everything
on T
is called the associated comatroid,
and is denoted H
,
we have done applies to T as well. The associated matroid
or simply H. The associated matroid on T restricted to S'
is denoted by S'jH.
The elements in S are called vectors
and those in T are called covectors.
The rank r(A,B)
More notation is necessary.
pair of subsets A
and B is defined to be the maximum size
of a non-singular minor x,u With x C A and u S B.
it
of a
is the rank of B in AIH,
or A
in GIB. We
Alternately,
shall use the
r(A,B)
We
= rB(A)
= rA(B)
.
following notation interchangeably:
shall use a similar notation for the other matroidal
concept:
for example,
-B
A
In the sequel, we
is the closure of A in GIB.
shall state and derive results
about the matroid G and its restrictions, leaving the
analogous results for H
However,
and its restrictions unstated.
because of the symmetry in our axiomatization,
we can,
66
-
-
and will, freely use both versions without comment.
Finally, note this technical detail:
Proposition 3.2.2:
If x is a basis of the matroid G and
u is a basis of the comatroid H,
Proof: As x and u are bases,
x
then x I u.
there exists v and y such that
I v and y I u. By the argument in the first part of the
proof of the previous lemma, we can exchange elements of u
for elements of v until we obtain x I u.
The classic example of a bimatroid is a
co-ordinatizable bimatroid. A matrix M over a field k
specifies a bimatroid on the index sets of the columns and
rows
'as follows:
x ... xn
ul...un iff the minor <x1 .. .xnIu .. .un>
0.
A special case of a co-ordinatizable bimatroid is a
transversal bimatroid. Let S and T be two finite sets and
let R be a relation between them. We specify a bimatroid
between S and T by: x ...xn
I ui...un if and only if there
exists a permutation or on 1...n such that x Ru01
for all i.
The transversal bimatroid is co-ordinatizable over a
67
-
-
suitably large extension field of the rationals by the
indexed by S and
free matrix F-R whose rows and columns are
T,
and whose entries are
<x.u.)if x.Ru. and 0 otherwise,
where the non-zero entries
<x 1 u > form an algebraically
independent set over the rationals.
Our next example shows that any two matroids may
be related via a bimatroid.
and T.
Let G and H be matroids on S
We define a bimatroid on S and T by setting x
if and only if x is independent in G and u
in T.
Iu
is independent
The proof that this is in fact a bimatroid is an
easy but tedious case by case analysis;
omitted.
the details are
Note that if G and H are not of the
same rank,
the associated matroids may be truncations of the original
matroids.
Such a bimatroid is said to be constructed by
placing G and H in general position in the same space.
We
can also construct a bimatroid from any given
matroid G of rank d on the finite set S.
of G.
Choose a basis B
We define a bimatroid between the sets S and B by
specifying the regular minors as follows:
for any minor
x,
x
68
-
-
b,
b if and only if x u (B\ b)
is a basis in G.
The proof that this is a bimatroid is particularly instructive
as to the relation between the exchange property for matroids
and the exchange-augmentation property for bimatroids. First
observe that as B is a basis, d
and y I b'.
.
Now, suppose that x I b
That is,
x1... xnbn+1
...
bd
n
1
' mbm+1. .. bd
I
d
are both bases, where bb
b 1
0
*
b' =
=d~B \b'.
b.
M+ib*i
d = B\ b and bm
Consider y,. By the basis exchange property for the matroid
G, at least one of the following must occur:
(a)
such that
There exists x.
x1...x
i-ylxi+1'''*n b n+1...b d
x y2
..
m+1 b.
**
' 'mb
Translating this into bimatroid notation, we obtain
x1 ...x i-ylxi+1
x y2
(b)
..
'
m
There exists b. such that
,J
.xn
'
b
.,, -- -
-
x1
-xnyibn+1..
.bib
that is,
Note that b. i B \b':
X 1.Xn
1
b. E b'. Rewriting this
-b u b
m -b'b.
..
focus on an element in B.
indeed defined a bimatroid between S and B:
this bimatroid is called the
to
j+... bd
similar argument works when we
Thus, we have
----------
we have
in bimatroid notation,
A
---
y* bb'+1 l b
y2
y2''
69
,-,.--,
-,,-%
- ,-,I- -. ,- ,
-- 2,- ,.III II -II .I- -..- --,..I ,,-- -I,,,-
-
, - ,.II ,lik-- -z 1- 1.. ,I- - - ,-
-
1 -. . --II. II-- ---. I. -, .
.
I'll- - -.111
1
bond
bimatroid of G relative
the basis B.
The
question arises of whether it is possible to
define a bond bimatroid between the elements in S and the
bonds of the matroid G,
in analogy to Whitney's construction
of the circuit matrix (Whitney C353,p.526;
is orthogonally dual
his construction
to ours). I can see no way to do this
and two facts argue against its possibility.
The first is
that the bond matrix of a matrix is dependent on the
matrix,
and not simply on the
induced matroid structure.
The second is that this is related to the problem of
finding an adjoint for an arbitrary geometric lattice.
This problem has been shown to be intractable in general
by Alan Cheung [743.
3.3 Some
70
-
-
properbLes
eLementary exchange-ougmentatlon
One of the basic properties of determinants is the
multiple Laplace expansion:
(x
+
=
... 1x kxk+1'''
.x+k<u
0x
..
n u
...
un
.ui )(xk+
1 l''
1
n
k+1
the sum being over all partitions of u 1 .. .u .
enquire
n
(We need not
The bimatroid analog is the
into the signs here.)
following result (compare Greene and Magnanti [753,
3.2,
>,
...ui
k
theorem
p.535):
Let
Proposition 3.3.1(the multiple exchange property):
for all k,
Then,
x,u be a non-singular minor.
a partition of u into blocks u
..
, u
.u.
1
k
there exists
, such
... u1
k+1
n
x
...
xk+1
xk
x''
n
u 1.. .u k
u
...u
k+1
Remark:
basis,
.
that
n
For the circuit bimatroid of G with respect to a
this result reduces to
the multiple exchange
property for matroids first proved by Greene in [73).
Proof:
Our result is equivalent to the statement:
there
-
-
71
exists a matroid partition of u relative to the matroids
Xl...xkIH and xk+l''XnI H. We shall use the matroid partition
It suffices to prove that for all
p.533).
Magnanti [751,
subsets w = w1 .
.w
C u,
r x 1000 k (w)
+ r
r
(x1 .
~
k+1'''n
the left hand side equals,
|wj.
>
(w)
xy...k ~
But,
or Greene and
section 1.5 of this thesis,
theorem (see
by symmetry,
.xk) + rw (Xk+1'''
n
Consider the following exchange-augmentation operation
focussed on-w':
n
where wp
w
... w w
+
..
w
xi
... x
...
wn is the set complement of w
p+l*
is possible,
p+l
n
n
and hence,
.
w ..wp%+ .w
n lp
we must have
p
Continue to do this until none of the w'.
3
have a subset x'
No exchange
such that
an x
'xn
xi-1ii+1
in u.
of x such that x'
appears. We
w. But now,
thus
as x'
is
independent relative to w,
rw(x 1
' k)
+ rw (x k+'*xn)
r w(xi ...xkn x')
x1 ... xkn
lxI
+
+ rxw
k+1.. xn
lxk+l...x nC'x
n x')
I.1'I
=1
I.
72
-
-
E
This proves the equality and also the proposition.
An epecially important case of this proposition is
Corollary 3.3.2:
minor.
Then,
all
x.
i,
Let x 1 . . .xn,u1 . . .u
be a non-singular
there exists a permutation 0
such that for
I u T.
Finally, we have an analog of core extraction (see
section 1.2 of this thesis):
Proposition 3.3.3:
core
(d)
=
If d is dependent in G,
(x G
d:
d --,x
I
and u c T,
then
u}
is either dependent or empty.
Proof:
Note that d is also dependent in Glu and use Theorem
1.2.1.
3.4 The quotlent dLogram
We first observe
over
the
comobroLd
the important fact that GIT'
only on the closure of T'.
depends
73
-
-
Proposition 3.4.1:
Let U be a set of covectors.
GIU = GIU,
is the
where U
closure of U
contains a basis of U,
Proof: As U
Then,
in the comatroid H.
this is essentially
0
Proposition 3.2.1.
Corollary 3.4.2:
tt for some x E c
c - x
Proof:
=4
for all
Note that c s x and c\y have
Now,
H.
If c is a circuit in G,
then,
y E c,
c %y
I t.
the same closure.
consider the lattice of flats of the comatroid
To each flat U,
we associate the matroid GIU.
association is called the
This
quotient diagram over the comatroid.
Our terminology is justified by the next proposition.
Proposition 3.4.3:
Let U S V be
flats in the
GIV--GIU is a strong map. Moreover,
GIUv u -GIU
Proof:
comatroid. Then,
for u i U,
the map
is a non-trivial elementary strong map.
Recall that H-+L is a strong map (or,
L is a quotient
of H) whenever a set is closed in L implies that it is also
closed in H. Dually,
in H becomes
this can be restated as:
a union of circuit
every circuit
in L.
As the composition of two strong maps is again a
74
-
-
it suffices to consider the case
U vu -- oGU
G
.
strong map,
Suppose c is a circuit in GIUv u.
Then c \c
is independent
in GIUv u.
Extend c \c0 to a basis so that we have
(*)
c
u.
n e n+1''er
for
a basis
is
where u 1 ... ur-
SupDose first that c-%c
ur-1 u
U.
remains independent in GIU.
Then,
by the previous corollary,
well.
Thus,
this is true for csc1
as
c remains a circuit in GIU.
We can now assume that c \c
Partition (*)
er. As c Nc 0
relative to c 1 .. .cno e n+1
dependent relative to U,
c
is dependent in GIU for all i.
.. 0c nIu
Partition again,
is
we must have
1.u
n-iu
this time relative to u 1.. .u.
,
There
u.
exists c. such that
c
Thus,
hence,
c\ c0 c.
c0
..c
cj+1..C
u 1..u
is independent in GIU.
n-1
But c \c.
c.,
is contained in the core of c
is dependent;
that is,
the
c.
unique circuit of GIU contained in c \-c
.
also contained in a circuit in c. Thus,
c is a union of
Similarly,
is
75
-
-
circuits in GIU.
Finally,
observe that r(Uv u)
-
r(U) = 1;
hence,
the strong map GIU vu--GIU is a non-trivial elementary
0
strong map.
We now prove a useful lemma which answers the question:
Let A
be a flat in G. What is the closure of A
Lemma 3.4.4:
The closure of A
in GIU?
in GIU is the maximum flat B
= rB(U).
such that rA(U)
Proof: As a flat is a maximal set of a given rank, we need
only check that the maximum flat adumbrated above
fact exist:
that is, we need to check that given any two
flats B and B'
= rA(U),
Observe
does in
oontaining A and satisfying rB(U)
their join also satisfies rB VB'(U)
= rB'(U)
= rA(U).
first that rBAB (U) = rA(U). By the semimodular
inequality in the matroid AIH,
rBv B'(U)
But clearly,
rB
( rB(U) + rB'(U) B'(U)
rB /B'(U)
= rA(U)
) rA(U). This proves the equality.
0
76
-
-
The remainder of this section concerns the finer
structure of the quotient diagram; it is inspired by
results in the theory of quotient bundles and may be omitted
by the
casual reader.
Let V S U be flats in the comatroid. We set
M(UV)
= the collection of flats A
rU(A)
-
= r(U) -
r (A)
As GIU--GIV is a strong map,
(see Cheung and Crapo E731,
in GIU for which
r(V).
M(U,V)
is a modular filter
section 4).
The following
is a weak version of the main "exchange" property for
quotient bundles:
Proposition 3.4.5:
and V be a modular pair of flats
Let U
in the comatroid. Then,
M(U vV,U) n M(Uv V,V)
Proof:
= M(U vV,U AV).
Let U and V be a modular pair of flats, with bases
chosen as follows:
U vV = wuuuv
U=
wuu
UA V=w
V=
wuv
77
As the rank of any flat A
-
-
relative to a flat in H depends
only on the closure, we can restrict attention to the
bases.
Now, A E M(UVV,U AV)
of A in GIUv V,
implies that if a
then any non-singular r x r
... ar is a basis
minor must be
of the form
a1...ar
w
where w' c
in u and v,
be
lul
+
I w'uu u
v
(if not, we can delete the covectors contained
but the decrease in rank of a
lvi).
... ar would not
0
The proposition is now obvious.
For a quotient bundle,
the analogous equality holds
for an arbitrary pair of flats.
in general for a bimatroid.
This is,
I think,
false
78
-
-
3.5 PerpendLcuLorLby
Since a relevant vector or covector may not be
present in a bimatroid,
the notion of perpendicularity
is not as useful in bimatroid theory as in the theory of
bilinear pairings.
Indeed,
only the most elementary
properties of perpendicularity carry over to bimatroids.
We
shall give a list of counterexamples at
the end of
section.
this
Let A
Definition 3.5.1:
be a
The set A
subset of S.
is
defined to be
Cu F T:
a I u for all a E A}.
is in fact a flat of the comatroid,
The set A
called the
flat perpendicular to A, and depends only on the closure
of A.
Proposition 3.5.2:
(a)
Suppose x 0 x 1
14i !n.
Then,
x0
''*Xn is a circuit in G,
and xi
I u.
The set AL is a flat in the comatroid.
(c)
A- = A
.
(b)
u for
(a)
Proof:
79
-
-
By Proposition 3.4.1,
u is a loop in x .. .xn IH
implies that u is also a loop in x
1
xn IH.
Parts (b) and
(c) now follow immediately.
closed,
Since ALL is
A1 L. This containment
A 9
very
is
often strict.
Another property which carries over is
Proposition 3.5.3:
The rank of AL
is less than or equal to
the corank of A.
Proof:
Let corank A = c and suppose that rank A
there exists a collection uc-1ucuc+1..
covectors
such that
for all
a
4 ui.
a
...ad-c contained in A.
Moreover,
i,
.
>
c. Then,
ud of independent
c-14 i 4d,
and all
there exists an independent
aE A,
set of vectors
Extending both collections to
bases, we have
a1...ad-cad-c+1...da
However,
ui...uc-2uc-luc...d"
we cannot partition this into non-singular 1X 1
minors as,
in any such partition,
minor of the form a ,u
there is at
least one
. This contradicts the multiple exchange
property and establishes the proposition.
0
Again,
80
-
-
the inequality is often strict. A consequence
of the previous proposition is
Corollary 3.5.4:
If rank A
=
corank A,
The converse is not true in general.
of flats of the form A
of Galois connections
then A
A
A characterization
can be obtained from the
theory
1
Briefly, we have a Galois connection between the
lattice of flats of G and the lattice of flats of H given by
This defines a closure operator on the lattices of flats:
A+A,
The
U--+U
jj-closed sets form a lattice, and the flats in G which
are jj-closed are precisely those flats A for which A = U,
for some flat U in H.
The lattice of
iL-closed sets inherits almost none
of the properties of the geometric lattices of flats, as
can be seen in the following three examples.
These examples
are all transversal bimatroids; we symbolize a
d by a plus
sign + and a I d by a blank space
1
Detailed expositions may be found in Birkhoff [673,
5.8, and Ore (623, Chapter 11.
section
-
81
LottLce of iL-cLosed sets
Example I:
d
e
abc
f
bc
+
a +
-
... ..
. ......
...............
+
b
b
00
+
+
C
d
e
a+
+
+
b
+
+
Example II:
I
abc
f
bc
C
+
c
Example III:
e
a +
+
bc
+
b
abc
f
+
d
b
C
+
c
The conjectures about perpendicularity disposed of includes:
(a) T
(b)
= A
(almost every example);
= corank A
rank A
(c) A = A
1
==)
rank A
(Let A = a in example
= corank A
(Again, let A
These examples also show that the lattice of
sets need not be
ranked
(I),
self-dual
semimodular (I),
(III),
I);
i-closed
distributive
or atomic
(II
= a in I).
(I),
and III).
82
-
-
3.6 EmbeddLng covectors Ln the mobroLd
Given a matrix,
it is always possible to consider the
row vector-s as vectors in the column space such that the
matrix entries are the values of a suitable bilinear form
on the column space. However,
is rather unsatisfactory,
as the theory of perpendicularity
we cannot expect the
same result
for bimatroids. We do have the following construction,
however,
which is the reverse of the construction of the
bond bimatroid of a matroid G relative to a basis B
(see
section 3.2).
Let B-_ be a bimatroid on S and T;
a basis of the comatroid.
let b = b
...
1**
b
d
be
We specify the matroid G+ on
S u b by:
x u b'
is a basis in G+ if and only if x
b \
b'
in B.
That this is a matroid can be proved by reversing the proof
in section 3.2. Note that when restricted to S,
G +b'
= Gjb\
b'.
The matroid G+ so obtained is called the matroid augmented
by the basis b.
3.7
The
83
-
-
bLpaLntLng cryptomorphLsm
In this section,
we present the bimatroid analog of
Minty's painting cryptomorphism for matroids (Minty [661).
This
cryptomorphism defines a matroid G on S by specifying
two collections C and J
of non-empty subsets of S,
circuits and bonds respectively.
called
The circuits and bonds
satisfies:
c and bond d;
1 for any circuit
(a)
|c n d l
(b)
for any element x E S,
and any partition of S Nx into
either there exists a circuit c
two blocks r and b,
consisting of x and elements in r,
or there exists a bond
d consisting of x and elements in b.
(c)
so are the bonds.
the circuits are incomparable subsets;
We now describe the bipainting cryptomorphism for
bimatroids.
Let S and T be two
disjoint. A bimatroid B between S and T
collections
t and
'
assumed to be
finite sets,
of elements from 2
is specified by two
x
2
T
called the
circuit-bond pairs and the bond-circuit pairs respectively.
These two collections satisfy the
following axioms:
(a)
84
-
-
bond pair,
and (d',c') be
a bond-circuit pair.
I (c n d') u (d A C') I
(b)
Let (cd) be a circuit-
the Minty intersection property:
the bipainting property:
Then,
;
1
Let x E S be chosen. A
painting
of S and T with basepoint x is a partition of S - x into two
labelled blocks (which may be empty)
a and w,
called the green and white covectors.
T
S
red
green
X blue
Figure
3.1
whLte
PcLntLng
a bLmatroLd
For any painting of S and T with basepoint x,
there exists a circuit-bond pair (c,d)
x
called the
and a partition of T into two labelled
red and blue vectors,
blocks
r and b,
and only red vectors otherwise,
covectors,
either:
such that c contains
and d contains only white
85
-
-
or,there exists a bond-circuit pair (d,c)
such that d
contains x and only blue-vectors otherwise,
and c contains
only green covectors.
As usual,
we also require the transposed version of this
axiom to hold;
(c)
lexicographic incomparability:
be two circuit-bond pairs.
c c c'
cannot hold;
Then,
Let
(c,d)
and (c',d')
the conjunction
and d c d'
similarly,
bond-circuit pairs are also
lexicographically incomparable.
(d)
The pair (0,T)
(S,5)
is never a circuit-bond pair;
similarly,
is never a bond-circuit pair.
We
shall prove
to the earlier one
next two sections.
the equivalence
of this cryptomorphism
in terms of non-singular minors in the
86
-
-
3.8 ReLotbve bonds and cLrcuLts
Let B be a bimatroid on the sets S and T specified
by non-singular minors. Let G and H be the matroid and
comatroid associated with B.
A circuit-bond pair is a pair (c,d) of subsets, with
c c S and d g T such that
(a) d is either a bond in the restricted comatroid cIH,
or
is empty; in the former case, the complement dc is a copoint
in c H,
(b)
and
c is a circuit in the matroid Gldc
A bond-circuit pair is defined in a transposed manner. Note
that if (c,d) is a circuit-bond pair, r(c,d C)
Lemma 3.8.1:
Let x = x 0 X 1 '''Xn,
= IC
-
1.
u = u 1 ...un be a pair of
subsets contained in S and T such that r(x,u) = n. Suppose
that c is the unique circuit of Glu contained in x. Then,
the closures of u in xlH and in cIH are the same.
Proof: As c g x, xlH- oc[H is
-x
-C
Now,
relabel
a strong map (Lemma 3.4.3).
Hence, !- g U-.
x so that
x O1-''xk are all the elements
in c;
v E
_
87
-
-
in particular, the rank of u in cIH is k. Suppose that
that is, r (u uv) = r(u), or equivalently, for every
k-subset u' of u, c f u'u V.
-x
suppose that v $ u-;
x 0x1...x n I
vu ...
u
that is,
.
Now,
Consider the partition of x into c and x \c:
by the multiple
exchange property (Proposition 3.3.1), we should be able to
find a corresponding partition on u uv such that the blocks
pair up into non-singular minors. However, as v E u-,
c
u' u v, while, as c is a circuit in Glu, c I u" for any
k+1 subset of u. This contradiction establishes our
0
contention.
Corollary 3.8.2 (the refining lemma): Let x, u and c be as
in the previous lemma. Then,
(c,d) where
-x
d = the complement of uis a circuit-bond pair.
Proof: We need only note that
-x
_u-
is a copoint in xIH, and
remains a copoint in c1H. Moreover, c is a circuit in
Gx
= Gidc.
We shall now check that the circuit-bond pairs and
88
the bond-circuit pairs,
-
-
defined in the above manner,
satisfy the four axioms in the bipainting cryptomorphism.
Proposition 3.8.3:
The circuit-bond and bond-circuit pairs
satisfy the Minty intersection property.
Proof: Let (c,d) be a circuit-bond pair, and (d',c') be a
I (c n d')
u (d n
1
I') I
.
bond-circuit pair. We have to show that
Let c = c 0 c41...cn. As d is a bond or empty,, there
exists covectors u ...
u n such that
Cl ...cn I u 1 ...un
(1)
and u1 ...
un
is the copoint complementary to d in cIH.
Similarly, we can find xl...xm such that
x 1 ... xm
where c'
-
= c'c' .. c', and x ...
O1
m
1
Suppose that c n d'
c'
I c
x
is complementary to d'.
m
4 g. That is, there exists an
element, c 0 say, in c which is also in d'. But as d' is
(2)
c x0 0...
xm
M
I
d'
c'c'...c'
0 1
m
.
non-empty and a bond in GIc', c 0 E
implies that
89
-
-
Consider the exchange-augmentation operation for the
non-singular minors (1)
occurs,
and (2)
then there exists c
Ic n d'I >
focussed on co. If exchange
which is in d': that is,
2.
If not, there exists c!1 such that
c~c
that is,
c! E
...c
I u 1 ... u c!
d. In this case,
Idric'
> 1.
In either case,
I (c n d') u (d n c')
The case when drnc'
I
2.
d but crnd'
0
=
d is handled in a
similar fashion.
0
Proposition 3.8.4: The bipainting property holds.
that x E rZ. Then,
where all the c
there exists a circuit c = xc1 ...c
,
Proof: Using-the same notation as in Section 3.7, suppose,
are painted red, in the matroid GI.. Now,
-c
Z-
90
-
-
is either a copoint in cIH or empty: in either case, its
complement d is painted white. Thus,
(c,d) is a circuit-
bond pair painted as required.
If x 0 rZ, then there exists a bond d in the matroid
GIL consisting of x and blue vectors only, extracted by
choosing a copoint in GIL containing r. Let e1 ...ek be a
basis for that chosen copoint, and let g ...
gk+l be a
maximal independent set in L in the comatroid; the sets
gi...gk+1
and e1 *...ek satisfy the hypotheses in the refining
lemma (Gorollary 3.8.2), and so we can construct a bondcircuit
pair painted as required.
Proposition 3.8.5:
0
Bond-circuit pairs are lexicographically
incomparable; so are circuit-bond pairs.
Proof:
Suppose that (c,d) and (c',d') are two circuit-bond
pairs, and that d c d'. Then,
every circuit in Gid
c
By Lemma 3.4.3,
is a union of circuits in Gid'c. As
circuits are incomparable, c
is c = c'; but,
d''C ,c.
#
c'. The only possibility left
if this is true, d and d'
are both bonds in
the same matroid cIH, and cannot be comparable.
0
It remains to show that ( ,T) is not a circuit-bond
pair,
91
-
-
and (S,S) is not a bond-circuit pair. But this follows
from the fact that the empty minor is always independent.
3.9 ReconstructLng non-sLnguLor MLnors
Now, suppose that the bimatroid B is specified by its
collections of bond-circuit and circuit-bond pairs;
these
two collections are assumed to satisfy the four axioms given
in Section 3.6. We shall reconstruct the non-singular minors
of B from this information.
Let x % S and u S T be two subsets.
is
We say that x
dependent relative to u if there exists a circuit-
bond pair
(c,d)
such that
c (
x and u c- d
c
This circuit-bond pair is said to be contained
in x,u.
Similarly, we say that u is dependent relative to x if
there exists a bond-circuit pair (d,c) such that c g u
and x S d
c
A pair of subsets x,u is a non-singular minor if
x is independent relative to u and u is independent relative
to x.
That is,
92
-
-
if there does not exists either a circuit-
bond pair (c,d) such that c s x and u c dc or a bond-circuit
pair
(d',c') such that
c' c
u and x c d' c
First, we note that, by Axiom (d),
,
is a non-
singular minor.
Lemma 3.9.1: If x,u is a non-singular minor, and u E u,
then u %u
is not a non-singular minor; moreover,
and (c 2 2
)are two circuit-bond pairs contained in x,u su,
then,
if (cl,dl)
.l = c2
Proof: Consider the transposed painting:
X
X
C
C
U
U\-"U
U
green
bLuie
whLte
Figure 3.2
red
UntLtLed
As x,u is a non-singular minor, there cannot be a bondcircuit pair (d,c), with c containing u and only red
covectors, and d containing only white vectors. Hence,
there must exist a circuit-bond pair (c,d) such that c
contains only green vectors
(i.e.
c 9 x),
and d contains
93
-
-
u and only blue covectors (i.e. u %,u
dependent relative
Now,
; dc). Hence, x is
to u - u.
suppose (cl, d) and (c2'1 2 ) are two circuit-bond
pairs cont Lined in x,us..u. If c,
1
c2'
let y be a vector in
2 2 not in c.
X
X
C
U
blue
red
C
U '- U
whLte
d
green
U
d2
C,1-2
d
Figure
3.3
UnLqueness
of cLrcuLt
Consider the painting with base point y, the rest of x
painted red,
a
C
painted blue, u
C
painted white and u painted
green. Now, we cannot have a circuit-bond pair (c,d) with c
containing y and red vectors, and d containing only white
vectors, for it would be contained in x,u. Hence, there exists
a bond-circuit pair (d,c) such that d contains y and blue
vectors, and c
; u. Now, as (c 2 9
2
) is a circuit-bond pair,
and c 2 n d = y, 1 2 n c must contain the only covector it can
contain,
-
94
-
namely u. But then,
n d) u (d. n c) = u,
(c
contradicting Axiom (a).
0
Corollary 3.9.2: If x,u is a non-singular minor, and u E u,
then there exists x E x such that x xx,u Nu is again a
non-singular minor.
Proof: By the previous result,
every circuit-bond pair in
x,u-,u is of the form (c,d), where c is a fixed circuit;
thus,
by removing any point x E c, we have a pair x-\x,u\u
which contains no circuit-bond pair.
Now, suppose (d',c') is a bond-circuit pair contained
in
x - x,u\u:
that
is,
d' c
(x\x)C ,
and c' c
u-.u.
As
x,u is non-singular, x E d'. Now, let (c,d) be any circuitbond pair in x,u \ u. Then, as c c _x,
d
.E
(u - u) c
(c n d') u(c'n d) = x,
0
contradicting Axiom (a).
Proposition 3.9.3: If x,u is a non-singular minor, then
lxI
=
lul.
Proof: If x,u is non-singular, we can, by the previous
corollary, remove an element from each so that x -x,u
.
u
95
-
-
is a non-singular minor. By an inductive argument,
it
suffices to show that x,O and 9,u are not non-singular
minors; but if they were non-singular minors,
Lemma 3.9.1
would imply that 0,0 is not a non-singular minor.
0
It remains to check that the exchange-augmentation
property holds: but this is a simple consequence of the
Minty intersection property and can safely be left to the
reader. This finishes the proof of the equivalence of our
two cryptomorphisms.
To conclude, we present a criterion for a bimatroid
to be binary,
that is, co-ordinatizable over GF(2).
Proposition 3.9.4: A bimatroid is binary if and only if
for all circuit-bond pairs (c,d) and all bond-circuit pairs
(d',c'),
1(c nd') u(dnc')l
is
even.
There is a theory of binary bimatroids, paralleling the
theory of binary matroids; but time and space conspire to
prevent its presentation here.
-
96
-
....
. ....
....
Chapter 4
THE TUTTE DECOMPOSITION THEORY FOR BIMATROIDS
4.1
ContrctLons,
deLebLons and sums
The natural notation when we
are dealing with the
Tutte decomposition theory is to write a bimatroid B on
S and T as GIH, where G and H are its matroid and comatroid.
Although this is ambiguous in principle,
it is always clear
in practice what the bimatroid structure between G and H
is.
The operation of deletion is simply that of removing
vectors or covectors.
More precisely,
If a E S,
the
deletion
of a from GIH results'in the bimatroid G \aIH on the sets
S\ a and T.
The non-singular minors in the deletion are
those non-singular minors x,u in G H such that x
C
S - a.
___-.-_-." . - I-.
,- -
..
---
.
"I. ---
1-1111 _'1_.1-1_1--'1!'._-
'-_ -"_'1'_'1 -
-
Contraction
is
non-singular minor.
97
I
.
1
1. I, 11--l-1-1
:- -_
1_".1___11"' 1- -
-
I
.
-
I
-
-1
-
1
.1. .1-
-
-
-1- 11
less
obvious.
Let
a,b be a 1 x 1
The contraction G/aIH/b of the bimatroid
GIH with respect to a,b is the bimatroid between S \a and
T \ b with the non-singular minors specified by:
x
I u in G/aIH/b if
and only if
xua
I uUb in GIH.
It is an easy computation to show that the contraction is
a bimatroid;
note that we require a,b to be non-singular
because the empty minor is always non-singular.
What is the contraction of a matrix? Let M be a
matrix and suppose that its ba-th entry is non-zero. Then,
by first performing row operations and then column operations
we can reduce M to
(or the other way around),
0
0
M
= b
-red0
o o
..
o
1
0
..
the form:
I
0
The contraction of the matrix M relative to a,b is the
with the b-th row and a-th column deleted. To
matrix M
-red
see this construction is equivalent to the earlier one,
,
1.
1
.
I I -
L14"
98
-
-
simply observe that the row and-column operations in the
reduction preserves the non-singularity of a minor containing a,b.
The picture for contraction in terms of the quotient
diagram is given by:
Proposition 4.1.1:
The quotient diagram of G/aIH/b is the
diagram over the interval
of H,
[b,1)
of the lattice of flats
with the quotient at the flat U
(which contains b)
(GIU)/a.
Proof:
First,
we check that H/b is indeed the comatroid
of the contraction G/aIH/b.
equivalences:
b
... b d-
This follows from the
is a basis in H/b iff b 1 .. .bd-1 b
is a basis in H iff there exists x S S,
But by Proposition 3.2.2,
contain a.
x
I b 1 ... b d-1b
x can always be chosen to
The same argument works for the second part of
0
the proposition.
Contractions and deletions evidently commutes.
A bimatroid which can be obtained from GIH by a sequence
of contractions and deletions is called a Tutte minor of
G IH.
The
last Tutte operation is the direct sum. Let B
be a bimatroid between S and T,
and B'
a bimatroid between
S'
and T'.
-
99
-
sum B 9 B'
The direct
bimatroid on S u S'
is
(or GeG'IHeH')
the
and T U T' with non-singular minors
specified by:
x I u if and only if xn S = unT and xfnS' = uflT', and
x
S
I u nT and xnS'
In co-ordinate terms,
matrices M
I unT'.
this corresponds to putting two
together as shown:
and M'
.
M
M'
0
Proposition 4.1.2:
0
The quotient diagram of the
sum is the quotient diagram on the
direct
lattice of flats of
HeH'
(which is the product lattice of the lattices of flats
of H
and H') with the quotient on U u V
in H and V
a flat in H')
the matroid GIU
(where U
is a flat
s G'IV.
The proof is obvious.
A bimatroid which can be expressed as a direct sum of
non-empty bimatroids is said to be disconnected;
the connected
bimatroids which are its summands are called its connected
components.
At -
.1 1
.. .
I
1 1.
1 111
'.
I . -. _-_.I.-
I
-
.-
1 .---
1 -__----I-_,
- 1-1.. -1 -
I
. I . I- I
- 1-1-1
11-1-
-
.- _- 1.
- 100-
A cryptomorphic version of connectedness is
given by
Proposition 4.1.3: A bimatroid B between the sets S
and T is disconnected if and only if there exists a
and S2
partition of S into non-empty blocks S
a partition of T into non-empty blocks T
and
and T 2 , such
that for all circuit-bond pairs (c,d),
either c G S,,
T 2 q d or c cS 2
'T
d,
and for all bond-circuit pair (d,c),
either c G T 1 , S2
d or c = T 2 9 S 1 Q d.
The proof is easy, once one recalls the method of
reconstructing the non-singular minors from the bondcircuit, circuit-bond pairs described in Section 3.9.
There is another notion of "summing" two bimatroids
which corresponds to the join operation (see Section 1.5)
in matroid theory. Let B be a bimatroid on S and T, and
B' a bimatroid on S and T'. The join B vB'
bimatroid defined on S and T uT'
by:
is the
101
-
-
x I u if and only if there is a partition of x
into x
and x
into possibly empty blocks of the
appropriate sizes such that x I
It
is
easy to prove that this
T and
u
2
u
T.
in fact defines a bimatroid,
and is the most natural proof I know. Proposition 1.5.2
can also be derived from general bimatroid theory.
This approach to joins is complementary to the
approach via strong maps outlined in Section 1.5 and
will be fully developed elsewhere.
i
102
-
-
4.2 The Tubbe decomposLbtlon
Let us begin with two typical examples. Suppose we
wish tu count all the non-singular minors (of whatever
size)
of a bimatroid GIH.
Then,
decomposition for matroids 1,
in analogy with the Tutte
we look for a recursion
relating the number of non-singular minors
of GIH with the
number of non-singular minors of appropriate contractions and
deletions.
Let iGIH be the number of non-singular minors of the
bimatroid GIH.
T)
iGIH
it satisfies the recursion:
Then,
"GsaIH
+
GIH\b
GiaIHNb + iG/aIH/b
where a,b is a "typical" pair of elements from S and T.
The proof consists of dividing the non-singular minors in
GIH into four kinds:
containing a only,
both a and b.
those
containing neither a nor b,
those containing b and those containing
These four kinds of minors are
counted with
multiplicities given by the following table:
1
those
see Tutte (473 and Brylawski [723 for an account.
I-L. 11.1 1 1 1. I
- I.
..-.-- I.. - . I'll '-, , "II-- 1-1
"-, - -. I
---- - "'
II I; - . .11.-
-
103
GNaIHNb
GJ Hb
G.aIH
II
.I. I.I-III I . I-
I
G/aIH/b
1
1
-1
0
a..l...
0
1
0
0
... lb..
1
0
0
0
a..lb.,
0
0
0
1
...
...
Everything works out! Moreover,
(T 2 )
GeG'IHoH'
apply the
We
of sets:
r(A,B)
that is,
= d,
this number;
Now,
we have the recursion:
'GIH G'IH'
same technique to counting spanning pairs
pairs of subsets A,B in S,T such that
the rank of the bimatroid GIH.
again,
similar
essentially
.I. .- "At.-
-
-, '.--I ,.:
we have the
Let sGIH denote
two recursions and
proofs.
we should analyse more carefully what we mean by
a "typical"element;
that is,
we should point out the
exceptions.
First, note that for contraction to be defined at all,
we must have alb. However,
loop of G H:
suppose that a,b is an absolute
that is, a is a loop in the matroid G and b is
a loop in the comatroid H.
recursion (T 2 ) to write
Then, we can take advantage of
GIH
104
-
-
a!b G-aIHNb
I
where a!b is the unique bimatroid on a,b of rank zero,
which we shall call the absolute loop.
Now,
suppose
a,b is an absolute isthmus;
every d xd minor contains a and b.
that is,
Note that this happens
if and only if a and b are both isthmi in G and H.
Then,
sGNaIlH counts none of the spanning pairs of GIH; rather,
it counts some of the pairs of rank d-1. However,
the
absolute isthmus need not be a connected component; for
example, consider the bimatroid given by the matrix:
1
1
We cannot use
1
(T 2 ) as in the theory of matroids.
The same difficulty crops up when a is an isthmus
in G, but b is not an isthmus in H. What we need in both
cases is a modification of (T1 ) which makes appropriate
terms in the recursion zero.
Before we consider this new phenomenon,
one definition. A lx1
let us make
minor in GIH is said to be
_
.----
'1_1-- -
- ,
-11-1111111-11-11
. " 1'_"'_r__'_""1_--
1
_
_-_---'1-'_
-
_ '. -
". -,
. I . 1
105
1
A
-
-
-
,
,, .
-1-1
I
.
.
I
'I
I
I
L, -
- -1 -
. - - .-
- I - -
I
-
- --
unexceptionable2 if it is non-singular,
and neither a nor
b is an isthmus in G or H. In particular,
an unexceptionable
minor is neither an absolute loop nor an absolute isthmus.
4.3 The
generatLn.g
rank
poLynomLaL
The most natural way to develop the Tutte decomposition
theory for bimatroids is through the
rank generating
polynomial.
Definition 4.3.1:- Let GIH be a bimatroid on S and T.
The
rank generating polynomial of GIH is defined by
2
LGIH(xq,IJ)
=
A
A CS,B
r(S,T)-r(A,B) XAI -r(AB)P IBI-r(A,B)
CT
Note that the rank generating polynomial behaves well under
transposition:
LGIH(
'X,k)
The analogs of recursions
2
rHIG(XIA
(T 1 ) and
(T 2 ) are given in the
Our terminology is borrowed from Jane Austen;
nothing else is.
hopefully,
I
. 1 -1
I w
L W
106
-
-
proposition:
Proposition 4.3.2(the Tutte decomposition for the rank
generating polynomial):
bimatroid GIH.
Let a,b be a 1xi minor of the
Then,
T . If a,b is an unexceptionable minor,
If a
is a loop in G,
=
rG
Similarly,
a1H
;
=
(1
+
g)rGIH*b
but a
is not a! loop in G,
;
if a,b is an absolute loop,
rG
T3
2)r
if b is a loop in H,
GlH
Finally,
(1 +
but b is not a loop in H,
=
(1
+ 2)(1 +
If a is an isthmus in G,
-GIH
Similarly,
-GalH
-GIH
)rGralHb
but b is not an isthmus in H,
b
-
xrGaIHb + r G/aIH/b
if b is an isthmus in H,
but a is not an isthmus
in G,
rGIH
Finally,
GaIH + x GIHb
'
T2
- G-aIHNb + rG/alH/b
rG-aIH + -GIH-b
-GIH
-GxaG..HNb
if a,b is an absolute isthmus,
+
G/ajH/b
EGIH
T4
4
107
-
-
+ xrGIHb
.G,aIH
~ XrGs aIH %b
+ rG/aH/b
The rank generating polynomial is multiplicative with
that is,
respect to direct sum:
LGoG'IHsH'
rGIHEGIIH'
constitute the Tutte decomposition
The four recursions T.
for a bimatroid.
The proof consists of straight-forward computations,
akin to those presented in 4.2.
The only things to note are
that we may have to put in factors of x to compensate for
a decrease
the
in rank where appropriate,
and that the rank in
contraction is given by
= rGIH(Au a,B u b)
rG/aIH/b(AB)
-
1
Instead of boring the reader with details, we
example.
shall do an
Let us consider the bimatroid
a
c
*
b
which is the
transversal bimatroid between a and bc with
alb and aic.
Then,
108
-
-
r(x, 2 ,9) = Xp2 + 2x/4 + xA + x + u + 2.
The computation via Tutte decomposition proceeds as follows
(everything but the subscripts is suppressed):
where
- x()+
x(*
,
S
I symbolizes the phantom column:
between 0 and a single element set b.
Now, note
transpose the phantom row.
polynomial
the unique bimatroid
Similarly, we call the
that the rank generating
of the phantom column is given by
.LI(x,
,P)
= 1 + 1,
while the rank generating polynomial of the absolute isthmus
is given by
r,(x,.,g)
=
1 +
+ X/4 + x
A
Thus,
=
x(1+ )
2
+ 1 + xA + x/
+ 1 + i
+ x -
,
r(x,2,4)
which checks with the earlier computation.
We end with the observation:
Lemma 4.3.3:
GIH
(
- 2 r(S,T)(1+A ) 1S (1+j,)ITI
x(1 + tc)
109
-
-
4.4 Tubbe LnvorLants
A
function t defined on bimatroids taking values
in the complex numbers (or any ring)
satisfying the Tutte
decomposition for a certain value x is said to be a
Tutte invariant with parameter x.
the rank generating
By evaluating
can obtain a lot of Tutte
is
invariants;
polynomial,
we
a preliminary list
as follows:
(1,0,0) = #non-singular minors in GiH
r
-GIH
r Gi
(0,91,1)
GIH(1,1,0)
GjH(1,1,1)
H (0,0,0)
Now,
= #spanning pairs
= #pairs A,u:
in Gill
u is independent in AIH
lSl+ITI
=
2
-
#non-singular d X d minors
recall that if A is a flat in the matroid G,
function can be computed by the following formula
p.352):
PG(OA)
Thus,
(-1) IEI
=
E: E=A
the MH*bius
(Rota C64J,
(-1)r(ST)
Gk
S)
GIH (0,-19,-1) = PLG(0,S)pH(0,T)
.
=
110
-
-
These formulas are specializations of the following:
LGIH
A!;S,
B S
BET
xr
E
(xy,
S,T)-r(A,B) y
IAI -r(A,B)
B 9T
r(S,B)-r(A ,B)
B) F.
A 9S
x r(ST-rS
IAI-r(A,B)
rH(T)-rH1l(B)
x
GIB X y)
BET
corankH F
(#bases of F)x
F is
rIF xy)
a flat
in H
where
rGIF(x,y)
is the rank generating polynomial of the
restricted matroid GIB (see Crapo [69J,. p.213).
A
111
-
-
similar computation yields:
Fr
=corank
F is a flat
in H
All these Tutte invariants are evaluations of the
rank generating polynomial:
in fact,
all Tutte
invariants
are such evaluations.
Theorem 4.4.1:
x.
Then,
t be a Tutte invariant with parameter
Let
t is an evaluation of the rank generating
polynomial.
Proof:
The function t is determined by its parameter x,
and its values at the two indecomposables,
and column.
the phantom row
Note that the absolute isthmus *
indecomposable
(as in matroid theory);
* = x(
I
) + x(
~ ) - x
( '
We find this a surprising result;
is no longer
indeed,
) + '
.
0
the reader is urged
to compare
this with the analogous result in matroid theory
(Brylawski
[723,
Theorem 3.6).
Combinatorial applications (for example, to the
bimatroid analog of the critical problem) will appear
elsewhere.
I
112
-
-
Chapter 5
PFAFFIAN STRUCTURES
Leb
be be
the fLnoLe of seem
Wallace Stevens
5.1
SympLectLc
LnvorLant
theory
We shall simply sketch the
symplectic group;
invariant theory for the
the combinatorial
interpretations are
similar to those for bilinear pairings.
that of de Concini and Procesi
Let V be a vector space
field k,
[763.
of even dimension 2d over a
not of characteristic two,
symplectic bilinear form (,)
Our account follows
equipped with a
that is,
,> is a bilinear
form satisfying
(x,x)
=
0
The group of automorphisms which preserve this bilinear form
is the symplectic group Sp(V).
113
-
-
The first fundamental theorem of invariant theory
for the symplectic group states that all the invariants
are generated by the Pfaffians
Ex 1 ... x
=
/
det
(x,x.>
It is a classic result that the Pfaffian is in fact a
polynomial. As in the bimatroid case,
the second fundamental
theorem is less explicit for our purposes; we choose the
following version which yields a natural combinatorial
interpretation:
The
relations among Pfaffians can all be derived from
the syzygies of the form
+(-1) Ly 2xm 1'i-1x1i+1
m
11x 1 .. XmEY1'''yn
.
n
n
m
+ E
(-1)fEx2*x j-xj+1'
j=2
.x 1m i'y. y
114
-
-
5.2 PfaffLan structures
Let S be a finite set. A
Pfaffian structure L on S
is specified by a collection of subsets of S called
composite
sets;
the subsets not in this collection are
called prime sets.
The composite sets satisfy the following
three axioms:
P :
The empty set is composite.
P 2 : Any one element set is prime.
P3
1
m **n are composite, then, at.least one of
X1 y1''
the following holds:
(a)
exchange:
there exists y
such that yix 2 ... xm and
yl..y 1 _1 xlyi+'' n are both composite,
(b)
augmentation:
there exists x
or
such that x 2 .. x i-1xi+'
m
and xI x y...yn are both composite.
As
in the theory of bimatroids,
exchange-augmentation operation.
composite
we call x 1
Note
the focus of the
that a
subset of a
set need not be composite.
As we have hinted in section 6.1,
the paradigm of a
Pfaffian structure is a finite set of vectors in a symplectic
space,
with x 1 .. .xm composite if and only if Ex 1 .. .xm
0.
. _-I'II-._I-'_-';'--' -'-.__"'._-_-_.._.' -
- ."'-.'-____'_ ____
-
-
More
,- I .
__ - . -.111.
115
-
I. I
.-- I.. I I
.
' I
gw
N_
examples will be given in subsequent sections.
The Pfaffian of an odd sized skew symretric matrix
This is reflected by
is zero.
5.2.1:
Lemma
Let L be a Pfaffian structure on S.
X 1 ** .xm
is composite, m
Proof:
Observe that x 1 . ..
Thus,
if
Then,
is even.
xm and the empty set are composite.
by an exchange-augmentation operation focussed on x
we have
,
- .. - -1 _-
(after relabelling)
x1x2
both composite;
and x 3 *
.xm
continue this. But a single element set is
necessarily prime:
therefore,
m must be
Now consider the maximal composite
cannot be augmented
exchange.
0
even.
sets. These
by definition; hence,
sets
we can always
But the exchange property is just the basis
exchange property for a matroid.
Lemma 5.2.2:
We have
thus proved
The maximal composite sets of a Pfaffian
structure L on S are the bases of a matroid on S.
This matroid is called the ambient matroid of L, and is
denoted by L also.
The ambient matroid is the
underlying vector space of a symplectic
form,
analog of the
and necessarily
116
-
-
has even rank.
Lemma 5.2.3:
Let x1 ... xm be an independent set in the ambient
matroid. Then,
there exists xm+1 ...xk such that x ... xmxm+1'
1
is composite. That is,
the independent sets in the ambient
matroid are precisely those sets which are subsets of
composite sets.
Every matroid of even rank is the ambient matroid of
some Pfaffian structure.
Indeed,
let G be a matroid on S of
even rank. The generic Pfaffian structure L
-G
on S on the
--
ambient matroid G is specified by
x 1 ..
xm is composite if and only if x 1 *..xm is independent
is even.
and m
The only non-trivial axiom to check is P 3 . Let x 1 .. .xm and
y1 ... y n be independent sets of even cardinality.
that we wish to exchange or augment,
firstly that x
1
E
yy...yn;
focussing on x . Assume
relabel so that xly...yk is the
unique circuit contained in xy.. .yn.
r(y1 .
.
x)
a.yk2 '
> r(x.
..xm)
= m,
Then,
Now suppose that x 1
as
there exists a y 1
circuit such that yix 2 '' .xm is independent:
we can exchange.
Suppose
in the
that is to say,
is not in y 1 ... y , and
-
117-
x2 '
that we can find yi
Xm.
y . We are left with the case:
for all i.
Then, x
x. 0
= n+1 (
IX y1 ... ynI
y.. .yn and yi E
m-1;
This implies that n
can be exchanged for
but m is even.
x 2 ,, 'm
Thus,
m. By the independent set augmentation
This concludes the proof
axiom for matroids, we can augment.
.
of axiom P
3
A useful property to note is
Proposition 5.2.4: Let y = y...y
element of S. Then,
be composite,
there exists yi
and x an
such that y1 ... yi
is composite if and only if there exists y. E
xyi+
'''
1
y such that
xy. is composite.
Proof:(=-) Suppose y 1 .. .y
xyi+1''
n is composite.
By
expanding into two element composite sets as in Lemma 5.2.1,
there must be y. such that xy.
(4-=)
The-sets xy
is composite.
and yl...y
are both composite; but,
if we focus on x, no augmentation can take place.
Hence,
El
we can exchange.
Another useful property is that compositeness depends
only on closure.
To be more precise,
n
Proposition 5.2.5:
118
-
-
Let x1 .. .x
and y 1 ...y
be independent
sets in the ambient matroid with the same closure.
Then,
x 1 .. .Xn is composite if and only if y.. .yn is composite.
Proof:
Assume that x 1 ..
is independent,
xn
is composite.
Then,
as y
...
y
we can extend y to a basis of the ambient
matroid
--
~
'
y1---yny,
we can consider the
As a basis is necessarily composite,
composite sets
y
.. Ynyn+1'..
focussing on y1 , where y1
dependent
(and hence,
we cannot augment.
x.,the
2d '
x 1 .' xn
i x 1 ..
x
.nAs x
.. xnYl is
cannot be extended to a composite set)
Thus,
we can exchange to obtain,
for some
composite set
x 1 0. .xiylxi+
1
Iterating this,
.. xn
we conclude that y
...
yn is also composite.
The proposition now follows by symmetry.
ri
-
5.3
LagrangLan
119-
fLabs and
bLmatroLds
subset x 1 .. .xn in S is said to be
A
is prime for every pair x
if x x
a Lagrangian set
every
for example,
and x
single element set is Lagrangian.
The closure of a Lagrangian set is Lagrangian.
Lemma 5.3.1:
Let x = x 1 ... x
Proof:
x 1 .. .x . Now,
1
x'...x'
r
l'et x.
1
be a Lagrangian set,
be an arbitrary element.
is a basis for x contained in x.
xxi
xi...
and let x be in
Consider the sets
xr
focussing upon x. As x E x, we cannot augment;
exchange
since x.x'.
Suppose
is prime. Hence,
xx.
nor can we
must be prime for
J1
any element x
01
in x.
Lemma 5.3.2: The rank of a Lagrangian flat in the ambient
matroid of a Pfaffian structure of rank 2d is at most d.
The proof is by decomposition into d composite pairs.
Now suppose that we have a bimatroid B between S and
T.
We can consider it as a Pfaffian structure on the disjoint
union S 6 T by specifying
that
X = X ...
x 2n
120
-
-
is composite if and only if we can partition
x into two blocks x 1 , x 2 of size n
S
S,
T and x 1x2 is a non-singular minor in B.
-
-2
-
x
such that x,
The only possible doubt about this construction is whether
P3 holds. But the exchange-augmentation operation focussed
for the pair of composite
xi...x
sets
xl'-X
... , y... ymYJ'''Ym
'
on x 1
(the primed elements are in T)
is the same as the following
exchange-augmentation operation in the bimatroid:
X1
~ ~ *.XXI.X
.
focussed again on x
1
The Pfaffian structures arising in this manner are
just those Pfaffian structures with a partition into
Lagrangian flats. That is,
flats SI and S 2
is the
set S.
there is a pair of Lagrangian
such that r(S 1 ) = r(S 2 ) = d,
The elements in Z = Si n
and S
U S2
S2 are loops in the
ambient matroid. The bimatroid structure on S' .Z
and T which
induces the Pfaffian structure can now be constructed by
reversing the earlier construction.
We conclude with an obvious observation:
bimatroid B is co-ordinatized by the matrix M,
if the
then our
121
-
-
construction results in the Pfaffian structure co-ordinatized
by the skew-symmetric matrix
0:
-Mt
5.4 The
0
one-factor PfaffLan
Let
A
M
r
matching
structure of a
graph
be a simple undirected graph on the vertex set S.
of P is a subset F of edges of T such that no
two edges are incident. A subset T S S is said to be composite
if there exists a matching F of t such that T is precisely
the set of all vertices incident on an edge in F; F is called
a one-factor of the restricted graph lIT.
The first result in the theory of Pfaffian structures
is the following observation of Tutte
[47 2
Proposition 5.4.1: The composite sets of a graph ' on S
forms the composite sets of a Pfaffian structure on S.
--
-
--
--
I -- 1 1 -. 1 - - I .
. __ ' -
-1.1-
- 1-1 I I __ -11 - - -
-
__-
- -
-1
__ I - -, - 'b ___- _-- I - - - , - 1, - . , -
-
122
, II '
-
I . , .I~ .,
Linearly order the vertices.
Proof:
.-- .
-
,
- . --
, --
N- - afilih&6&
-
- -.
I - lel.
1-1
-___'_
-
j
free incidence matrix T P=
(t.
.)
The skew-symmetric
of the graph ' is the
square matrix with rows and columns indexed by S,
and the
uv-th entry given by
Xuv if u and v are adjacent,
-Xuv
0
where
if u and v are adjacent,
and u < v,
and u
and
> v,
otherwise,
the non-zero terms Xuv are an algebraically independent
The proposition
set of indeterminates over the rationals.
and T has even
now follows from the fact that if T C S,
cardinality,
the Pfaffian of the square submatrix indexed
by T is given by
Sv
the
v2 v3v4
v2n-1
v
2
n
summation being over all partitions of T
into two-
El
element blocks.
An application of Lemma 5.2.2 yields the following
result due to Edmonds and Fulkerson
Proposition 5.4.2:
composite
on S.
[653:
The subsets of S which are subsets of
sets of P forms the
independent sets of a matroid
_
_
5.5 Tubbe
123
-
-
decomposLblon
In this section,
we outline the Tutte
theory for Pfaffian structures.
of rank 2d.
structure on S,
A
decomposition
Let L be a Pfaffian
The Pfaffian rank of a subset
of S is defined by
p(A)
= maximum size of a composite
set contained in A.
The Pfaffian rank is an increasing set function,
and increases
in steps of two.
Definition 5.5.1:
The
(Pfaffian) rank generating polynomial
of a Pfaffian structure L on S is given by
.L xX
=
xp(S)-p(A)AIAI-p(A)
AGS
Interesting evaluations of the rank generating polynomial
include:
EL ( 1,1)
L (1,0)
=
2 S1
= #sets
of Pfaffian rank 2d
= #spanning sets in the ambient matroid
L (0,0)
= #composite
=
sets of size 2d
#bases in the ambient matroid
r
124
sets in L
= #composite
(0,1)
-
-
The concept of deletion from a Pfaffian structure is
obvious;
the analog of contraction is the following construction.
Let ab be a composite set in L. The contraction of L by ab,
written L/ab,
is the Pfaffian structure on S ,ab with composite
sets defined by
in the contraction if and-only if
cl...cn is composite
c 1 . ..c ab is composite
in L.
The ambient matroid of L/ab is the contraction (as a matroid)
L/ab;
its lattice
of flats is isomorphic to the interval
and the composite flats are those flats in the interval
fab,13
which are composite. The Pfaffian rank in the contraction is
given by
PL/ab(A) = p L(A uab)
If L
-1
S1
and L
-2
and S 2 ,
-
2
are Pfaffian structures on the disjoint sets
the direct sum
1 sL 2
is the Pfaffian structure on
S Iu S2 with the composite sets specified by: A
if and only if AnS1
A
is composite
and A nS2 are both composite.
pair of elements ab in S is said to be unexceptionable
whenever ab is composite,
in the ambient matroid,
a is neither a loop nor- an.-isthmus
and b is neither a loop nor an'isthmus
125
-
-
in the ambient matroid.
Proposition 5.5.2
(the Tutte decomposition for the Pfaffian
Let L be a Pfaffian structure
rank generating polynomial):
Then,
on S.
T 1 . If ab is an unexceptionable pair,
L
T2
+ rLb
-a
+ rL/ab
-ELxab
If a is a loop in the ambient matroid,
=
r
-L
T3 .
(1
+ A)r
-L sa
Let ab be composite.
matroid,
but b is not,
2
L- a + r Lsb
EL Similarly,
-1 -2
-
x rLab + r L/ab
2
2
+ x
La
L 1L
then,
Lb
x
rL ab +
rL/ab
= EL EL
-
-
.
If a is an isthmus in the ambient
if a and b are both isthmuses in the ambient matroid,
2
L
T
then
These four recursions constitute the Tutte
decomposition
for a Pfaffian structure.
A
function t from Pfaffian structures to the complex
numbers
(or any ring)
parameter x
126
-
-
is said to be Tutte
invariant with
if it satisfies the Tutte decomposition.
Theorem 5.5.3:
A Tutte invariant t with parameter x is an
evaluation of the Pfaffian rank generating polynomial.
Proof:
The only indecomposable in the Tutte
is the
loop
L,
decomposition
the unique Pfaffian structure on a one
element set whose ambient matroid is of rank zero. Hence,
the value of any Tutte invariant is determined by its
parameter and its value on the
E (x,1)
it
loop. But,
= 1 + 2.
is now easy to show that t is
A
A
rL (x,1), where A
just the evaluation
is the value of t on the loop.
0
Let B be a bimatroid on the disjoint sets S and T. If
L is the Pfaffian structure constructed from B with S and
T a partition into Lagrangian flats,
r
(x,k)
= r B(x
21.
then
127
-
-
Chapter
6
SKETCHES
As too many poets have
completed, only abandoned;
said, a poem is never
this is even truer of
mathematical theses. We sketch here some of the work we
have not had time to work out in detail. We hope to present
full accounts in the future.
6.1
BLmotroLds
(a)
Strong maps: A strong map between two bimatroids B 1
and
B2 is a system of strong maps between the quotient diagrams
of B
and B2 which commutes with the restriction maps. Such
maps are generalizations of contractions, and the analog of
Edmonds' theorem holds: every strong map is an extension
followed by a contraction.
(b)
128
-
-
Excluded Tutte minors:
There
is
an analogous
theory of excluded Tutte minors for representation problems.
(c) Bimatroid multiplication:
The matrix product of two
bimatroids can easily be defined using the Cauchy-Binet
theorem. The problem here is whether such an object is still
a bimatroid. For example,
transversal,
if one of the bimatroids is
then the product is still a bimatroid; this
generalizes matroid induction (see Lindstrom [72) and also
Hocquenghem (pre]). This concept should unify many matroidal
constructions.
(d)
Orientable bimatroids:
This is the combinatorial theory
of matrices over formally real fields, and is very unwieldy.
Unless there are applications, it seems doubtful if a full
development would be worthwhile.
6.2 PfoffLan structures
(a) Core extraction: There is a core extraction property
for Pfaffian structures:
Proposition 6.2.1:
Let b a 1 ...an,
i
=
l,...,k be prime.
129
-
-
Then,
core b 1 ... bk (a1 ...an)
=bl...bk
a : a1 ... a
a
i+1 ... an is composite}
is either prime or empty.
The proof is a rephrasing of the exchange-augmentation
property.
In this form, it is too weak to be crypto-isomorphic;
such a crypto-isomorphism would be very useful, say in
the single-element extension theory.
(b)
The one-factor Pfaffian structure: The ambient matroid
of such a Pfaffian structure is, by a theorem of Edmonds
and Fulkerson 165J, a transversal matroid. Their proof relies
on choosing a basis (or one-factor), and then constructing
a relation;
it
would be useful to have a "canonical"
construction. A deeper study of the Pfaffian structure is
of the utmost importance, both for the theory of Pfaffian
structures and of graphs. One example should suffice: a
Lagrangian set is simply a stable set of vertices (i.e. a
set of vertices no two of which are adjacent). The coloring
problem becomes a problem of packing with Lagrangian sets!
(c)
Symplectic
130
-
-
optimization: There are two major results
in this area:
"Greedy" Algorithm,
the symplectic analog of the
and a duality theory
for Pfaffian intersection and partition?
6.3
OrthogonaL mabroLds
The theory of orthogonal matroids is the combinatorial
abstraction of an inner product on a vector space. It can
be regarded as a special case of the theory of bimatroids:
an orthogonal matroid M on the finite set S is a bimatroid
between the set S and the set S, such that the associated
matroid and comatroid are the same, and, for any minor
I y if and only if y
I x.
This theory offers another example of a Pfaffian
structure. Let M be an orthogonal matroid on S such that
x I x for every odd cardinality set x. Then, the sets
x
for which y I y are the composite sets of a Pfaffian
structure on S. The proof is an easy computation.
The basis exchange properties of orthogonal bases
131
-
-
is particularly interesting. By allowing extensions,
there
is an analog of Witt's contiguity theorem for orthogonal
bases.
The graphic orthogonal matroids are also of great
interest: here, instead of spanning trees,
are "sesquilinear graphs"
the "bases"
(see Harary [623).
6.4 Non-commutatLvlty
Reading through P.M. Cohn's "Skew Field Constructions"
[773, we find,
at the very end, another analog of a matroid
suggested by algebraic dependence over a skew field.
Roughly speaking,
the bases here satisfy the exchange
property, but their subsets need not be independent. This
phenomenon also arises in Pfaffian structures. It would
be of interest to develop a general theory of dependence
structures without an abstract elimination procedure.
132
-
-
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BLography
of the author
Joseph Pee Sin Kung
(
1952,
In 1967,
in Hong Kong.
) was born on April 22,
he went to Australia and
received part of his high school education there.
1970 to 197J,
he
From
studied mathematics at the University
of New South Wales, where he graduated in 1974 with
the
degree of Bachelor of Science
(Honours) and was awarded
the University iedal for pure mathematics.
From Fall,
1974 till the present, he has been a graduate student
in applied mathematics at M.I.T..