Although many excellent papers and some specialised monographs on different aspects of design theory had been published, prior to 1985 (when the first edition of this book appeared) there was no comprehensive monograph on the field. Thus it was our plan to cover the main concepts and ideas of modern design theory without being encyclopedic, but including a deeper study of several representative topics. As it turned out, these aims (which required a rather long book) seem to have been met by the first edition which has been quoted extensively in the research literature.

A first draft of this book was obtained by merging different manuscripts of the authors. The first edition then grew from an iterative process of rewriting in which all the authors contributed their ideas to each of the parts. For reasons of time constraints and shifting research interests, the major part of the revision now presented has been done by the second author, drawing on his previous update Jungnickel (1989a) and surveys Jungnickel (1990a, 1992a); in particular, this holds for Chapter VI. Of course, there has still been considerable help from his co-authors (with Chapter XIII being the first author's contribution), but nevertheless he is willing to accept most of the blame for the mistakes introduced during the process of revision.

Algebraic combinatorics involves the use of techniques from algebra, topology and geometry in the solution of combinatorial problems, or the use of combinatorial methods to attack problems in these areas. Problems amenable to the methods of algebraic combinatorics arise in these or other areas of mathematics, or from diverse parts of applied mathematics. Because of this interplay with many fields of mathematics, algebraic combinatorics is an area in which a wide variety of ideas and methods come together.

During 1996-97 MSRI held a full academic year program on Combinatorics, with special emphasis on algebraic combinatorics and its connections with other branches of mathematics, such as algebraic geometry, topology, commutative algebra, representation theory, and convex geometry. Different periods of the year were devoted to research in enumeration, extremal questions, geometric combinatorics and representation theory.

The rich combinatorial problems arising from the study of these various areas are the subject of this book, which represents work done or presented at seminars during the program. It contains contributions on matroid bundles, combinatorial representation theory, lattice points in polyhedra, bilinear forms, combinatorial differential topology and geometry, Macdonald polynomials and geometry, enumeration of matchings, the generalized Baues problem, and Littlewood- Richardson semigroups. These expository articles, written by some of the most respected researchers in the field, present the state-of-the-art to graduate students and researchers in combinatorics as well as algebra, geometry, and topology.

1. The Varchenko B Matrices

Let A = ﹛H1,…, Hl﹜ be an arrangement of hyperplanes in ℝn and let r(A) = ﹛R1,…, Rm﹜ denote the set of regions in the complement of the union of .A. Let L(A) denote the collection of intersections of hyperplanes in A. Included in L(A) is ℝ n, which we think of as the intersection of the empty set of hyperplanes. We order the elements of L(A) by reverse inclusion thus making it into a poset. It is well known that this poset is a meet semilattice and is a geometric lattice if the arrangement is central. We will abbreviate L(A) to L when the arrangement is clear.

This article is based on my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. I thank the organizers (Sergey Fomin, Curtis Greene, Phil Hanlon and Sheila Sundaram) for bringing together a group of outstanding combinatorialists and for giving me a chance to bring to their attention some of the problems that I find very exciting and beautiful. In preparing the note for this volume (October 1998), I made a few small changes in the original version [Zelevinsky 1997], and added in the end a brief (and undoubtedly incomplete) account of some exciting progress achieved since April 1997. I am grateful to the referee for helpful suggestions.

Theorem 1. LRr is a finitely generated subsemigroup of the additive semigroup Pr3 ⊂ ℤ3r. This is a special case of a much more general result well known to the experts in invariant theory. A short proof (valid for any reductive group instead of GLr(ℂ)) can be found in [Elashvili 1992]; A. Elashvili attributes this proof to M. Brion and F. Knop. The semigroup property also follows at once from “polyhedral” expressions for that will be discussed later (see Theorem 5 and below).

Introduction

A variety of questions in combinatorics lead one to the task of analyzing a simplicial complex, or a more general cell complex. For example, a standard approach to investigating the structure of a partially ordered set is to instead study the topology of the associated order complex. However, there are few general techniques to aid in this investigation. On the other hand, the subjects of differential topology and differential geometry are devoted to precisely this sort of problem, except that the topological spaces in question are smooth manifolds, rather than combinatorial complexes. These are classical subjects, and numerous very general and powerful techniques have been developed and studied over the recent decades.

A smooth manifold is, loosely speaking, a topological space on which one has a well-defined notion of a derivative. One can then use calculus to study the space. I have recently found ways of adapting some techniques from differential topology and differential geometry to the study of combinatorial spaces. Perhaps surprisingly, many of the standard ingredients of differential topology and differential geometry have combinatorial analogues.

1. Introduction

This article is an explication of some remarkable connections between the two-parameter symmetric polynomials discovered by Macdonald [1988] and the geometry of certain algebraic varieties, notably the Hilbert scheme Hilbn( ℂ2) of points in the plane and the variety Cn of pairs of commuting n x n matrices (“commuting variety”, for short). The conjectures on diagonal harmonics introduced in [Haiman 1994; Garsia and Haiman 1996a] also relate to this geometric setting.

I have sought to give a reasonably self-contained treatment of these topics, by providing an introduction to the theory of Macdonald polynomials, to the “plethystic substitution” notation for symmetric functions (which is invaluable in dealing with them), and to the conjectures and related phenomena that we aim to explain geometrically. The geometric discussion is less self-contained, it being unavoidable to use scheme-theoretic language, constructions such as blowups, and some sheaf cohomological arguments. I do however give geometric descriptions in elementary terms of the various algebraic varieties encountered, and review whatever of their special features we might use, so as to orient the reader not previously familiar with them.

The linchpin of the geometric connections we consider is the so-called “n! conjecture” [Garsia and Haiman 1993; 1996b], which remains unproved at present.

1. Introduction

Matroid bundles are combinatorial objects that mimic real vector bundles. They were first denned in [MacPherson 1993] in connection with combinatorial differential manifolds, or CD manifolds. Matroid bundles generalize the notion of the “combinatorial tangent bundle” of a CD manifold. Since the appearance of McPherson's article, the theory has filled out considerably; in particular, matroid bundles have proved to provide a beautiful combinatorial formulation for characteristic classes.

We will recapitulate many of the ideas introduced by McPherson, both for the sake of a self-contained exposition and to describe them in terms more suited to our present context. However, we refer the reader to [MacPherson 1993] for background not given here. We recommend the same paper, as well as [Mnev and Ziegler 1993] on the combinatorial Grassmannian, for related discussions.

We begin with a key intuitive point of the theory: the notion of an oriented matroid as a combinatorial analog to a vector space. From this we develop matroid bundles as a combinatorial bundle theory with oriented matroids as fibers. Section 2 will describe the category of matroid bundles and its relation to the category of real vector bundles.

1. Introduction

How many perfect matchings does a given graph G have? That is, in how many ways can one choose a subset of the edges of G so that each vertex of G belongs to one and only one chosen edge? (See Figure l(a) for an example of a perfect matching of a graph.) For general graphs G, it is computationally hard to obtain the answer [Valiant 1979], and even when we have the answer, it is not so clear that we are any the wiser for knowing this number. However, for many infinite families of special graphs the number of perfect matchings is given by compellingly simple formulas. Over the past ten years a great many families of this kind have been discovered, and while there is no single unified result that encompasses all of them, many of these families resemble one another, both in terms of the form of the results and in terms of the methods that have been useful in proving them.

Introduction

In January 1997, during the special year in combinatorics at MSRI, at a dessert party at Helene's house, Gil Kalai, in his usual fashion, began asking very pointed questions about exactly what all the combinatorial representation theorists were investigating. After several unsuccessful attempts at giving answers that Gil would find satisfactory, it was decided that some talks should be given in order to explain to other combinatorialists what the specialty is about and what its main questions are.

In the end, Arun gave two talks at MSRI in which he tried to clear up the situation. After the talks several people suggested that it would be helpful if someone would write a survey article containing what had been covered in the two talks and including further interesting details. After some arm twisting it was agreed that Arun and Helene would write such a paper on combinatorial representation theory. What follows is our attempt to define the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status.

1. Introduction

The generalized Baues problem, or GBP for short, is a question arising in the work of Billera and Sturmfels [1992, p. 545] on fiber polytopes; see also [Billera et al. 1994, §3]. The question asks whether certain partially ordered sets whose elements are subdivisions of polytopes, endowed with a certain topology [Björner 1995], have the homotopy type of spheres. Cases are known [Rambau and Ziegler 1996] where this fails to be true, but the general question of when it is true or false remains an exciting subject of current research.

The goal of this survey is to review the motivation for fiber polytopes and the GBP, and discuss recent progress on the GBP and the open questions remaining. Some recommended summary sources on this subject are the introductory chapters in the doctoral theses [Rambau 1996; Richter-Gebert 1992], Lecture 9 in [Ziegler 1995], and the paper [Sturmfels 1991]. The articles [Billera et al. 1990; 1993], though not discussed in the text, are nonetheless also relevant to the GBP.

Before diving into the general setting of fiber polytopes and the GBP, it is worthwhile to ponder three motivating classes of examples.

1. Introduction:

“A Formula for the Number of Lattice Points... “

The first main object of this paper is the integer lattice 𝕫d ⊂ ℝd consisting of the points with integer coordinates. We define the second main object.

We are interested in the set P ⋂ 𝕫d of lattice points belonging to a given rational polyhedron P. For example, we may be interested in finding a “formula” for the number of lattice points in a given rational or integer polytope P. But what does it mean to “find a formula“? We consider a few examples.