Coxeter Matroids

281 Pages·2003·20.17 MB·English

by  Alexandre V. Borovik

Progress in Mathematics
Volume 216
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
Alexandre V. Borovik
I.M. Gelfand
Neil White
Coxeter Matroids
with illustrations by
Anna Borovik
Birkhauser
Boston - Basel- Berlin
Alexandre V. Borovik 1M. Gelfand
UMIST Rutgers University
Department of Mathematics Department of Mathematics
Manchester, MOO 1Q D Piscataway, NJ 08854-8019
United Kingdom
Neil White
University of Florida
Department of Mathematics
Gainesville, FL 32611-8105
Library of Congress Cataloging-in-Publication Data
Borovik, Alexandre.
Coxeter matroids 1 Alexandre V. Borovik, I.M. Gelfand, Neil White.
p. CID. - (Progress in mathematics; 216)
Includes bibliographical references and index.
ISBN-13:978-1-4612-7400-1 e-ISBN·13:978-1-4612-2066-4
001:10.1007/978·1-4612-2066-4
1. Matroids. 2. Gel'fand, I.M. (Izrail' Moiseevich) II. White, Neil, 1945-III. Title. IV.
Progress in mathematics (Boston, Mass.); v. 216
QAI66.6.B67 2003
511'.6-dc21 2003045247
CIP
AMS Subject Classifications: Primary: 05B35, 20F55, 52B15; Secondary: 06CIO, 20E42, 52B4O
Printed on acid-free paper
@2003 Birkhliuser Boston Birkhiiuser
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ISBN-13:978-1-4612-7400-1 SPIN 10468315
Reformatted from authors' files by TEXniques, Inc., Cambridge, MA.
Illustrations by Anna Borovik.
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This book is dedicated to a great mathematician, H.S.M. Coxeter, who
passed away March 31, 2003, as the book was in press. He was 96
years old. His contributions to our understanding of symmetry and
geometry pervade our subject.
Introduction
The subject of combinatorics is devoted to the study of structures on a finite set;
many of the most interesting of these structures arise from elimination of continuous
parameters in problems from other mathematical disciplines.
For example, graphs appear in real life optimization problems as, say, sets of cities
(vertices of the graph) connected by roads (edges of the graph) of certain length. In
combinatorics we look at the structure left after we ignore the lengths of the roads
(which are continuous parameters in the original problem), as well as all topographical
considerations, and so on. The combinatorial structure of the graph determines many
important features of the original parametric problem. If we work on an optimal
delivery problem, for example, it does matter whether our graph is connected or
disconnected.
A matroid is a combinatorial concept which arises from the elimination of con
tinuous parameters from one of the most fundamental notions of mathematics: that
of linear dependence of vectors.
Indeed, let E be a finite set of vectors in a vector space ]Rn. Vectors at, ... , ak
are linearly dependent if there exist real numbers ct, ... , Ck, not all of them zero,
+ ... +
such that ct at Ckak = O. In this context, the coefficients ct, ... , Ck are
continuous parameters; what properties of the set E remain after we decide never to
mention them? The solution was suggested by Hassler Whitney in 1936. He noticed
that the set of linearly independent subsets of E has some very distinctive properties.
In particular, if B is the set of maximal linearly independent subsets of E, then, by a
well-known result from linear algebra, it satisfies the following Exchange Property:
Forall A, B E Band a E A ,B thereexistsb E B,A, such that A ,{a}U{b}
lies in B.
Whitney introduced the term matroid for a finite structure consisting of a set E
with a distinguished collection B of subsets satisfying the Exchange Property. The
origin of the word "matroid" is in "matrix": this is what is left of a matrix if we are
interested only in the pattern of linear dependences of its column vectors.
Matroids naturally arise in many areas of mathematics, including combinatorics
itself. For example, when we take the set E of edges of a connected graph together
viii Introduction
with the collection B of its maximal trees, they happen to form a matroid. Moreover,
the validity of the Exchange Property is almost self-evident and can be established by
a simple combinatorial argument. However, there are deeper reasons why a matroid
arises: it can be shown that the edges of a graph can be represented by vectors in such
a way that linearly dependent sets of edges are exactly those containing closed cycles.
The cohomological nature of the last observation should be apparent to every reader
familiar with algebraic topology.
The work of three generations of mathematicians confirmed that matroids, indeed,
capture the essence of linear dependence. Since linear dependence is a ubiquitous and
really basic concept of mathematics, it is not surprising that the concept of matroid
has proved to be one of the most pervasive and versatile in modern combinatorics.
There are dozens of books on the subject, of which we mention only [77,93,98].
Now let us take a step further in our discussion of structures on a finite set. We
already see that, in mathematics, even such a simple object as a finite set should be
endowed with some extra structure. However, the most fundamental structure on a
finite set--even in the absence of any other structures-is provided by its symmetric
group acting on it. The symmetric group already lurks between the lines of the Ex
change Property in the form of transpositions (a, b) responsible for the exchange of
elements. It is time to reveal that one of the aims of this book is to develop the theory
of matroids in terms of the symmetric groups and expose its hidden symmetries. This
is done in Chapters 1 and 2.
The rest of the book is devoted to further a generalization and development of
this approach. The symmetric group Symn is the simplest example of a finite Coxeter
group (or, equivalently, a finite reflection group). It can be interpreted geometrically
as the group of symmetries of the regular (n - I)-dimensional simplex in IRn with
the vertices
(1,0, ... ,0), (0, 1,0, ... ,0), ... , (0, ... ,0, 1).
As shown in Chapters 3 and 4, we can replace the symmetric group with the group of
symmetries of another Platonic solid in IRn, the n-cube [-1, l]n. (This group is called
the hyperoctahedral group.) Then we get a very natural generalization of matroids,
called symplectic matroids. We will usually refer to matroids (in Whitney's classical
sense) as ordinary matroids, to distinguish them from the more general symplectic
matroids, and later from even more general Coxeter matroids. Symplectic matroids
are related to the geometry of vector spaces endowed with bilinear forms, although in
a more intricate way than ordinary matroids to ordinary vector spaces. Some special
classes of symplectic matroids have already been studied under the names ~-matroids
[30], metroids [35], symmetric matroids [30], or 2-matroids [34]. Our approach al
lows us to develop a very rich, coherent and beautiful theory of symplectic matroids.
Furthermore, Symn is naturally embedded in the group of symmetries of the n-cube,
because we can make Symn permute the coordinate axes without changing their ori
entation; this action obviously preserves the n-cube [ -1, l]n. Thus ordinary matroids
can be also understood as symplectic matroids, the latter becoming the most natural
generalizations of the former.
Introduction ix
Finally, after recapping the theory of finite reflection groups in Chapter 5, we
develop the theory of Coxeter matroids in its full generality in Chapter 6. These
combinatorial objects, which were introduced by Gelfand and Serganova [61], are
related to finite Coxeter groups in the same way as classical matroids are to the
symmetric group. Interestingly, every further level of abstraction allows us to deduce
new concrete results on previously introduced less abstract objects. In particular, in
Chapter 6 the reader will find more results on symplectic matroids. We also have new
results on ordinary matroids themselves.
One of the important tools of the theory is the geometric interpretation of
matroids-ordinary, symplectic, Coxeter-as convex polytopes with certain sym
metry properties; this interpretation is provided by the Gelfand-Serganova theorem.
To help the novice reader develop the necessary geometric intuition, we prove this
crucial theorem three times, for classical matroids, symplectic matroids and in the
most general situation. We hope that it pays dividends, because eventually the geo
metric thread in our narrative leads to a surprisingly simple (although cryptomorphic)
definition of a Coxeter matroid:
Let t::.. be a convex polytope. For every edge of t::.., take the hyperplane that cuts
the edge at its midpoint and is perpendicular to the edge, and imagine this hyperplane
to be a semitransparent mirror. Now mirrors multiply by reflecting in other mirrors,
as in a kaleidoscope. If we end up with only finitely many mirrors, we call t::.. a
Coxeter matroid polytope, which, in view of the Gelfand-Serganova interpretation,
is equivalent to a Coxeter matroid.
Essentially, Coxeter matroids are n-dimensional kaleidoscopes which generate
only finitely many mirror images. A mathematical theory rarely comes to a more
intuitive reinterpretation of its basic concept.
In the final Chapter 7 we revisit the origins of the theory: if the most natural
examples of matroids come from finite collections of vectors in vector spaces (we
call such matroids representable), what is the analogous concept of representation in
the general case of Coxeter matroids? The answer is given in terms of buildings, the
geometric objects introduced by Tits as generalizations of projective spaces. Indeed,
the classical representation of matroids turns out to be a special case of representation
in buildings. We further develop the concept of representation and eventually end up
with its purely combinatorial version, when every ordinary matroid is represented in
what we call a combinatorial flag variety, that is, a certain simplicial complex made
of all matroids on the set of n elements.
This book is intended for graduate students and research mathematicians in com
binatorics or in algebra. It can serve as a textbook, an introductory survey, and a
reference book.
We tried to make the book accessible and as self-contained as possible. However,
in some instances we refer to known results about ordinary matroids, mostly in the
situations when we wish to establish the correspondence between our theory and the
more traditional treatment of matroids. We also refer to some standard facts about
root systems and Coxeter groups, although we develop in some detail those aspects
of the theory of Coxeter groups which form the language of the theory of Coxeter
x Introduction
matroids (Chapter 5). The last two chapters, 6 and 7, present the reader with a steeper
learning curve than presented in the rest of the book.
Every chapter contains a substantial list of exercises. Stars • mark those exercises
that are considerably more difficult. Quite often these are results from research papers,
in which case we give appropriate references.
'l\vo stars •• mark exercises that require some background knowledge from other
mathematical disciplined (say algebra or topology) which is not covered in the book.
Preface for the expert reader
This book is devoted to the following class of combinatorial objects.
Let W be a finite Coxeter group, P a parabolic subgroup in W and ~ the induced
=
strong Bruhat order on the factor set W P W / P. Let ~w denote the w-shifted order,
A ~w B if and only if w-1 A ~ w-1 B. Let M be a subset of Wp• We say that the
set M S;;; W P is a a Coxeter matroid if it satisfies the Maximality Property:
for any w E W, there is a unique A E M such that, for all B E M,
B ~w A.
In the special case when W = An-l is the symmetric group Symn and P is a
maximal parabolic subgroup, Coxeter matroids are exactly (ordinary) matroids in the
classical meaning of this word. Moreover, the Maximality Property becomes the well
known Gale characterization of matroids which has its origin in discrete optimization
theory [55].
At first glance, the definition of Coxeter matroids appears to be dry and abstract;
but, as this book demonstrates, it is very flexible and efficient in proofs, even in the
classical context of ordinary matroids.
The Gelfand-Serganova Theorem translates the definition into geometric terms,
associating with every Coxetermatroid a certain convex polytope. The class of Coxeter
matroid polytopes arising from Coxeter matroids can be characterized by the following
elementary property.
Let A be a convex polytope. For every edge [a, P] of A, take the hyperplane that
cuts the midpoint of the segment [a, P] in its midpoint and is perpendicular to [a, P].
Let W be the group generated by the reflections in all such hyperplanes. Then W is
a finite group if and only if A is a Coxeter matroid polytope.
Most interesting examples of Coxeter matroids (and, in particular, all examples of
ordinary and symplectic matroids in this book which are represented by a matrix of
some kind) come from torus orbits on flag varieties of semisimple algebraic groups.
Here we give only a brief sketch of the corresponding construction; it is fairly obvious
modulo standard results about moment maps [4,63] and semisimple algebraic groups.