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Coxeter Matroids

Authors: Alexandre V. Borovik, I. M. Gelfand, Neil White

Publisher: Birkhäuser Boston

Book Series : Progress in Mathematics

Included in: Professional Book Archive

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About this book

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry. This largely self-contained text provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group.

Key topics and features:

* Systematic, clearly written exposition with ample references to current research
* Matroids are examined in terms of symmetric and finite reflection groups
* Finite reflection groups and Coxeter groups are developed from scratch
* The Gelfand-Serganova theorem is presented, allowing for a geometric interpretation of matroids and Coxeter matroids as convex polytopes with certain symmetry properties
* Matroid representations in buildings and combinatorial flag varieties are studied in the final chapter
* Many exercises throughout
* Excellent bibliography and index

Accessible to graduate students and research mathematicians alike, "Coxeter Matroids" can be used as an introductory survey, a graduate course text, or a reference volume.

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Table of Contents

Frontmatter

1. Matroids and Flag Matroids

Abstract

The main idea in this chapter is to establish the intimate connection between matroids and the symmetric group Sym _n. This will be seen most clearly in the Maximality Property, which is really just a reformulation of the well-known characterization of matroids in terms of the Greedy Algorithm. It says, briefly, that for every linear ordering of the set of elements of the matroid, there is a unique maximal basis. But linear orderings of a finite set can be interpreted as its permutations. This brings the symmetric group into a pivotal role in matroid theory to an extent that has never been appreciated previously. Coxeter matroids are essentially just the generalization of matroids obtained when the group Sym _n is replaced by an arbitrary finite Coxeter group. Thus this first chapter will not only cast matroid theory in this new light, but also prepare the way for, and provide a prototype for, the more general Coxeter matroids.

Alexandre V. Borovik, I. M. Gelfand, Neil White

2. Matroids and Semimodular Lattices

Abstract

In this chapter we continue the process of explaining the intimate connection between matroids and the symmetric group by first switching to semimodular lattices and seeing how they are related to the symmetric group. We develop a viewpoint of a semimodular lattice as a chamber system with a kind of metric that gives it a structure only slightly weaker than that of a building over Sym_n. This leads to a natural way to represent flag matroids in semimodular lattices, which in turn leads up to our representation of arbitrary Coxeter matroids in buildings in Chapter 7.

Alexandre V. Borovik, I. M. Gelfand, Neil White

3. Symplectic Matroids

Abstract

We have seen how matroids and semimodular lattices are intimately related to the symmetric group. Now we replace the symmetric group by another Coxeter group, namely, BC _n the hyperoctahedral group. The resulting structures are called symplectic matroids, and they are in some sense rather general Coxeter matroids, as they include ordinary matroids and a third type, orthogonal matroids, as special cases. This will also prepare us to tackle Coxeter matroids in full generality in the later chapters.

Alexandre V. Borovik, I. M. Gelfand, Neil White

4. Lagrangian Matroids

Abstract

Lagrangian matroids are much better behaved than symplectic matroids. Indeed, as we shall see in this chapter, Lagrangian matroids are in several ways more closely related to ordinary matroids than are general symplectic matroids.

Alexandre V. Borovik, I. M. Gelfand, Neil White

5. Reflection Groups and Coxeter Groups

Abstract

This chapter is of an auxiliary nature and contains the modicum of the theory of finite reflection groups and Coxeter groups which we need for a systematic development of the theory of Coxeter matroids. A reflection group W is a finite subgroup of the orthogonal group of ℝⁿ generated by some reflections in hyperplanes (mirrors or walls). The mirrors cut ℝⁿ into open polyhedral cones, called chambers. The geometric concepts associated with the resulting chamber system (called the Coxeter complex of W) form the language of the theory of Coxeter matroids. The reader familiar with the theory of reflection groups and Coxeter groups may skip most of the chapter. However, we recommend that this reader look through Sections 5.12 “Residues,” 5.14 “Bruhat order” and 5.15 “Splitting the Bruhat order.”

Alexandre V. Borovik, I. M. Gelfand, Neil White

6. Coxeter Matroids

Abstract

In this chapter we develop the general theory of Coxeter matroids for an arbitrary finite Coxeter group, thus generalizing most of the results from Chapters 1 and 3. The keystone to the whole theory is the Gelfand—Serganova Theorem which interprets Coxeter matroids as Coxeter matroid polytopes (Theorem 6.3.1). As we shall soon show (Theorem 6.4.1), the latter can be defined in a very elementary way:

Let Δ be a convex polytope. For every edge [ α, β] of Δ, take the hyperplane that cuts the segment [α, β] at its midpoint and is perpendicular to [α, β]. Let W be the group generated by the reflections in all such hyperplanes. Then W is a finite group, if and only if Δ is a Coxeter matroid polytope.

Alexandre V. Borovik, I. M. Gelfand, Neil White

7. Buildings

Abstract

We begin this chapter with a return to basics and look at the underlying combinatorics of the Gaussian elimination procedure. This classical routine involves permutation of rows and columns of a matrix. The rules these permutations obey are extremely simple; when axiomatized in group-theoretic terms, they become what are known as axioms for a BN-pair (or a Tits system) and very quickly lead to Coxeter groups appearing on the scene.

Alexandre V. Borovik, I. M. Gelfand, Neil White

Backmatter

Title: Coxeter Matroids
Authors: Alexandre V. Borovik
I. M. Gelfand
Neil White
Copyright Year: 2003
Publisher: Birkhäuser Boston
Electronic ISBN: 978-1-4612-2066-4
Print ISBN: 978-1-4612-7400-1
DOI: https://doi.org/10.1007/978-1-4612-2066-4