Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

The chromatic number of the q-Kneser graph for large q

We obtain a new weak Hilton-Milner type result for intersecting families ofk-spaces inF2kq, which improves several known results. In particular the chromaticnumber of theq-Kneser graphqKn:kwas previously known forn >2k(except forn= 2k+1 andq= 2) ork 5, so that the only remaining open cases are (n, k) = (2k, k) withq∈{2,3,4}and (n, k) = (2k+ 1, k) withq= 2

Cometric Association Schemes

The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes

Intriguing sets of partial quadrangles

The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in particular, the well-studied objects known as \textit{generalised quadrangles} are each partial quadrangles. An \textit{intriguing set} of a generalised quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalised quadrangles by Bamberg, Law and Penttila to partial quadrangles, which surprisingly gives insight into the structure of hemisystems and other intriguing sets of generalised quadrangles

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch

Cameron-Liebler kk-sets in AG(n,q)\text{AG}(n,q)

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\text{AG}(n, q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\text{PG}(n, q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\text{AG}(n, q)$ correspond to certain equitable bipartitions of the Association scheme of $k$-spaces in $\text{AG}(n, q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\text{AG}(n, q)$. We define Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ and $\text{PG}(n, q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets. This paper focuses on $\text{AG}(n, q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\text{AG}(3, q)$ was already treated separately

Cameron-Liebler sets of k-spaces in PG(n,q)

Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F. Ihringer. We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also present a classification result

On the incidence map of incidence structures

By using elementary linear algebra methods we exploit properties of the incidence map of certain incidence structures with finite block sizes. We give new and simple proofs of theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained also for diagrams geometries. By mean of an extension of Block’s Lemma on the number of orbits of an automorphism group of an incidence structure, we give informations on the number of orbits of: a permutation group (of possible infinite degree) on subsets of finite size; a collineation group of a projective and affine space (of possible infinite dimension) over a finite field on subspaces of finite dimension; a group of isometries of a classical polar space (of possible infinite rank) over a finite field on totally isotropic subspaces (or singular in case of orthogonal spaces) of finite dimension. Furthermore, when the structure is finite and the associated incidence matrix has full rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that certain families of incidence structures have no sharply transitive sets of automorphisms acting on blocks