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.

Independence is a unifying concept for linear algebra and graphs. A deep generalization of both through this unification, is contained in the notion of graphoids. Every graph can be seen as an interpretation of a graphoid in a particular ‘coordinate system’, called a 2-complete basis. From this prospect, a graphoid is an essential, coordinate free, geometrical notion for which each associated graph, if it exists, is just a particular view of the same generality. The concept of a graphoid can also be seen as a pair of set systems (dual matroids) whose members are called circuits and cutsets. The set of all circuits (cutsets) together with all their distinct unions we call a circ (cut) space. In this chapter, in the context of circuits and cutsets, we concentrate on two concepts of independence within graphs and graphoids. We first introduce independent collections of circuits and cutsets and then we use this concept to define independent edge subsets, that is, circuit-less and cutset-less subsets. Prom this point on, we take circuits and cutsets as primary notions. This material may be seen as a bridge between traditional graph theory and matroid theory. We also give a brief overview of properties of graphoids and methods in topological analysis of networks.

The graphoidal point of view

The space of all graphs can be divided into disjoint classes such that two graphs belong to the same class if they are 2-isomorphic.

This research monograph is concerned with two dual structures in graphs. These structures, one based on the concept of a circuit and the other on the concept of a cutset are strongly interdependent and constitute a hybrid structure called a graphoid. This approach to graph theory dealing with graphoidal structures we call hybrid graph theory. A large proportion of our material is either new or is interpreted from a fresh viewpoint. Hybrid graph theory has particular relevance to the analysis of (lumped) systems of which we might take electrical networks as the archetype. Electrical network analysis was one of the earliest areas of application of graph theory and it was essentially out of developments in that area that hybrid graph theory evolved. The theory emphasises the duality of the circuit and cutset spaces and is essentially a vertex independent view of graphs. In this view, a circuit or a cutset is a subset of the edges of a graph without reference to the endpoints of the edges. This naturally leads to working in the domain of graphoids which are a generalisation of graphs. In fact, two graphs have the same graphoid if they are 2-isomorphic and this is equivalent to saying that both graphs (within a one-to-one correspondence of edges) have the same set of circuits and cutsets.

Historically, the study of hybrid aspects of graphs owes much to the foundational work of Japanese researchers dating from the late 1960's. Here we omit the names of individual researchers, but they may be readily identified through our bibliographic notes.

First two chapters could be seen as a bridge between traditional graph theory and the graphoidal perspective.

The concept of graph inherently includes two dual structures, one based on circuits and the other based on cutsets. These two structures are strongly interdependent. They constitute the so-called graphoidal structure which is a deep generalization of the concept of graph and comes from matroid theory. The main difference between graph and graphoid is that the latter requires no concept of vertex while the first presumes vertex as a primary notion. After some bridging material, our approach will be entirely vertex-independent. We shall concentrate on the hybrid aspects of graphs that naturally involve both dual structures. Because of this approach, the material of this text is located somewhere between graph theory and the theory of matroids. This enables us to combine the advantages of both an intuitive view of graphs and formal mathematical tools from matroid theory.

Starting with the classical definition of a graph in terms of vertices and edges we define circs and cuts and then circuits and cutsets. In this context, a circuit (respectively, cutset) is a minimal circ (cut) in the sense that no proper subset of it is also a circ (cut). We consider some collective algebraic properties and mutual relationships of circs and cuts. Vertex and edge-separators are introduced and through these we define various kinds of connectivity. The immediate thrust is towards a vertex-independent description of graphs, so that later all theorems and propositions will be vertex-independent.

The last two sections of the chapter are devoted to the notions of multiports and Kirchhoff's laws which are basic concepts in network analysis.

Many properties of pairs of trees of a graph are related to the Hamming distance between them. This is important for several graph-theoretical concepts that have featured in hybrid graph theory. Here the notions of perfect pairs and superperfect pairs of trees have played a part. We define and characterize these notions in this chapter and describe necessary conditions for the unique solvability of affine networks in terms of trees and pair of trees.

The small number of theorems and propositions collected together in the opening paragraphs of Chapter 3 will again be frequently referred to here. Familiarity with the basic concepts of graphs such as circuit and cutset are presumed in this chapter. A maximal circuit-less subset of a graph G is called a tree of G while a maximal cutset-less subset of edges is called a cotree. These terms (circuit, cutset, tree and cotree) will be used here to mean a subset of the edges of a graph. Let F be a subset of E. Then the rank of F, denoted by rank (F), is the cardinality of a maximum circuit-less subset of F and the corank of F, denoted by corank (F), is the cardinality of a maximum cutsetless subset of F. The complement of F is the set difference E\F, denoted by F*. By |F| we denote the number of elements in (that is, the cardinality of) the subset F.

Diameter of a tree

Given a graph G, each its tree t can be classified according to the non-negative integer rank(f).

In this chapter maximal edge subsets that are both circuit-less and cutsetless (which we call basoids) and the related concepts of principal minor and principal partition of a graph are considered. The fact that basoids may have different cardinalities provides a rich structure which is described through several propositions. Transitions from one basoid to another (which provides a basis for augmenting basoids in turn) and the concept of a minor of a graph with respect to a dyad (a maximum cardinality basoid) are also investigated in detail. It is shown that there exists a unique minimal minor with respect to every dyad of a graph G.This edge subset, called the principal minor and its dual called the principal cominor, define a partition of the edge set of Gcalled the principal partition. The hybrid rank of a graph is defined to be the cardinality of a dyad of the graph. This is a natural extension of the definitions of rank and corank of a graph, defined as the cardinalities of a maximum circuit-less subset and a maximum cutset-less subset of the graph, respectively. In the last section of this chapter an application to hybrid topological analysis of networks is considered. An algorithm for finding a maximum cardinality topologically complete set of network variables is also described.

The material of this chapter is general in the sense that it can be easily extended from graphs to matroids. To ensure this generality, the Painting Theorem, the matroidal version of the Orthogonality Theorem as well as the Circuit and the Cutset axioms are widely used to prove propositions.