This book contains the invited papers of the conference “Algebraic combinatorics and extremal problems” held at the Université de Montréal from July 28 to August 2, 1986. This was the first time a conference focusing on these two subjects was held. The main reason for organizing such a meeting was that these apparently mutually distant parts of combinatorics are becoming more and more intertwined. This may be seen, for example, in the paper of P.J. Cameron bringing together - among other things - the representation theory of symmetric groups and the extremal problems of finite structures.

The conference proved quite useful in bringing together some of the best specialists in those two fast developing areas of combinatorics as well as others from related fields. All papers in the proceedings contain new results and most of them provide up-to-date surveys of the respective subfields accessible to non-specialists. There is a list of problems presented at the conference.

The organizers would like to use this opportunity to thank NSERC Canada for the financial support (provided by a conference grant) and the Université de Montréal and its Département de mathématiques et de statistique for hospitality. Finally we would like to thank Antoine Deza, Lucien Haddad and Marc Rosenberg for organizational help.

INTRODUCTION

Four years ago, some members of the present audience were also gathered in Montreal, for the 1982 meeting “Finite Groups: Coming of Age” organized by John McKay of Concordia University. At that time, I presented a survey lecture [Sm1] on the comparatively new and rapidly developing area of “groups and geometries”. By now the area has come to be a more established branch of modern mathematics. It seems appropriate to make today's lecture in effect a sequel to that 1982 lecture – to describe further progress on the main problems then open and indicate important new directions which have opened up since that time.

It will be convenient to follow again the general outline of the earlier lecture, namely:

1) Motivation and applications.

2) Background on geometries and diagrams.

3) “Sporadic” geometries.

4) Properties and characterizations.

Of course, events have rendered parts of this organization a little outdated, but the parallel treatment should help emphasize developments since 1982.

MOTIVATION AND APPLICATIONS

At the earlier meeting, I presented “groups and geometries” as an area whose more widespread development had begun in the latter days (late 70's) of the classification of finite simple groups. In overview, that massive result shows that a non–abelian finite simple group must be one of:

Gemoetric Methods in Group Theory

1. (a) an alternating group;

2. (b) a group of Lie type, defined qver a finite field;

3. (c) one of 26 “sporadic” groups – not contained in the families (a) (b).

INTRODUCTION

The well known theorem of Borsuk [Bo] is the following.

Theorem 1.1 (Borsuk)

For every continuous mapping f :Sn → Rn, there is a point x ϵ Sn such that f (x) = f (−x). In particular, if f is antipodal (i.e. f(x) = −f(−x) for all x ϵ Sn) then there is a point of Sn which maps into the origin.

This theorem and its many generalizations have numerous applications in various branches of mathematics, including Topology, Functional Analysis, Measure Theory, Differential Equations, Approximation Theory, Geometry, Convexity and Combinatorics. An extensive list of these applications, some of which are about fifty years old, appears in [Ste].

Most combinatorial applications of Borsuk's Theorem were found during the last ten years. The best known of these is undoubtfully Lovasz's ingenious proof of the Kneser conjecture. Kneser [Kn] conjectured in 1955 that if n ≥ 2r + t − 1 and all the r-subsets of an n-element set are colored by t colors then there are two disjoint r-sets having the same color. This was proved by Lovász twenty years later in [Lo]. Shortly afterwards, Bárány [Ba] gave a charming short proof. Both proofs apply Borsuk's theorem.

The content of this paper is, thought slightly extended, based on my expository survey talk of the same title at the Montreal meeting: Algebraic Combinatorics and Extremal Set Theory, July 27 – August 2, 1986.

The aim of this paper is to study nice finite subsets in the sphere Sd and other (nice) metric spaces. This kind of study has a long history in mathematics. Its origin is perhaps traced back to the study of regular polyhedrons in R3 (by Platon?). In this paper, however, we restrict the scope of our discussion to the study of finite subsets which are extremal from the viewpoint of Delsarte theory (which we call Algebraic Combinatorics).

This paper consists of the following four sections:

1. §1. Harmonics on Sd and finite sets in the sphere Sd.

2. §2. Combinatorics of finite sets in compact symmetric spaces of rank one.

3. §3. Combinatorics of finite sets in noncompact symmetric spaces of rank one.

4. §4. Rigid t-designs in Sd.

In §1, we give a very brief and sketchy review of the theory of finite sets in Sd (i.e., spherical codes and designs) by Delsarte, Goethals and Seidel [18], which was the starting point of the study of finite sets in topological spaces from the view point of Algebraic Combinatorics.

INTRODUCTION

In Faradẑev et al. (1936) an algorithm for enumeration of intersection arrays of distance–transitive graphs (d.t.g.) having given valency k is described. This algorithm was used in Faradẑev et al. (1936) for classification of the d.t.g.'s of valency k = 5, 6 and 7. This paper is a logical continuation of Faradẑev et al. (1986). Here we supply the algorithm with some new feasibility conditions. The new version of the algorithm enable us to classify the d.t.g.'s of valency up to 13.

All definitions and notations concerning d.t.g.'s can be found in Faradẑev et al. (1936) (see also Bannai & Ito (1984) and Brouwer et al. (1937)).

Let us summarise briefly the history of classification of d.t.g.'s having small valences. Chronologically the first result in the area is the classification of cubic d.t.g.'s (k=3), obtained by Biggs & Smith (1971). Their scheme of classification can be generalised on the case of an arbitrary valency k ≥ 3 and in general consists of the following four steps:

(1) to prove that the order of the vertex stabiliser in the automorphism group of d.t.g. having valency k is bounded by some value f = f(k);

(2) to bound the diameter of a d.t.g. having valency k by some value D = D(k);

(3) to enumerate all feasible intersection arrays for the given valency k;

(4) to construct all graphs with intersection arrays found on the step (3).

INTRODUCTION

It is known that sharply t–transitive groups of permutations of an infinite set exist only for t ≤ 3 (Tits (1952)), while sharply t–transitive sets exist for al l t (Barlotti & Strambach 1984).

Geometric groups and sets of permutations have been proposed as a natural generalisation of sharply t–transitive groups and sets (Cameron & Deza (1979)). Our purpose i s to investigate such objects on infinite sets. Not surprisingly, we give nonexistence results for groups, and constructions for sets.

Let L = {ℓ0, ℓt …, ℓs–1) be a finit e set of natural numbers, with ℓ0 < … < ℓs–1. The permutation group G on the set X is a geometric group of type L if there exist points x1, …, xs ϵ X such that

1. (i) the stabiliser of x1 …, xs is the identity;

2. (ii) for i < s, the stabiliser of x1 …, xi fixes ℓi points and acts transitively on its non–fixed points.

INTRODUCTION

After the definition of matroids in the 30's a lot of work has been done on vector representations of matroids. Much less work has been devoted to algebraic representations of matroids in the proper sense: after the pioneer work of S. MacLane fifty years ago comes the work of A.W. Ingleton et al. fifteen years ago. Not more than five years ago I decided to try to solve some of the open problems in this neglected field. Recently more people have become interested in it. I intend to give a survey of contributions I know about.

First I recall some definitions. For a better introduction I recommend Chapter 11 of the book of Welsh (1976).

Let F be a fixed field and K an extension of F. Elements e1,…,en in K are algebraically dependent over F when there is a non-zero polynomial p(X1,…,Xn) ϵ F[X] such that p(e1…,en) = 0. If E is a finite subset of K then the algebraically independent subsets of E over F give the independent sets of a matroid M(E). Such a matroid is called algebraic. The rank r(A) of a subset A ⊆ E in this matroid is the transcendence degree tr.d.F F(A) of the field F(A) over F.

INTRODUCTION

There is a well–known and intimate connection between the character theory and the ordinary representation theory of a finite group: characters are traces of matrix representations, and determine the representations up to equivalence. For a finite non–empty quasigroup Q, the connections are much more obscure. The character theory is that of the association scheme (Q×Q; C1,…,Cs) determined by the permutation representation of the combinatorial multiplication group on the quasigroup [1, pp. 181·2] [3] [6]. The representation theory is that of free groups determined by the quasigroup – the universal multiplication group and point stabilizers within it [6]. Representations are classified by almost periodic functions on these free groups. The character theory furnishes generalized Laplace operators Δ1,…,Δs acting on the almost periodic functions.