Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems

2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12

Lifting Induction Theorems

AbstractWe show that—in some suitable sense—any induction theorem for the character ring of a finite group can be lifted to an induction theorem for the Green ring. (A precise statement see Theorem A herein.) This provides a uniform proof of important induction theorems, as, for example, the ones of Conlon and Dress. Moreover, we prove an analogue of Brauer's induction theorem for the Green ring (cf. Theorem B herein)

Star sets and related aspects of algebraic graph theory

Let μ be an eigenvalue of the graph G with multiplicity k. A star set corresponding to μ in G is a subset of V(G) such that [x] = k and μ is not an eigenvalue of G - X. It is always the case that the vertex set of G can be partitioned into star sets corresponding to the distinct eigenvalues of G. Such a partition is called a star partition. We give some examples of star partitions and investigate the dominating properties of the set V (G) \ X when μ ε {-I, a}. The induced subgraph H = G - X is called a star complement for μ in G. The Reconstruction Theorem states that for a given eigenvalue μ of G, knowledge of a star complement corresponding to μ, together with knowledge of the edge set between X and its complement X, is sufficient to reconstruct G. Pursuant to this we explore the idea that the adjacencies of pairs of vertices in X is determined by the relationship between the H-neighbourhoods of these vertices. We give some new examples of cubic graphs in this context. For a given star complement H the range of possible values for the corresponding eigenvalue μ is constrained by the condition that μ must be a simple eigenvalue of some one-vertex extension of H, and a double eigenvalue of some two-vertex extension of H. We apply the Reconstruction Theorem to the generic form of a two-vertex extension of H, thereby obtaining sufficient information to construct a graph containing H as a star complement for one of the possible eigenvalues. We give examples of graph characterizations arising in the case where the star complement is (to within isolated vertices) a complete bipartite graph

Some Implications on Amorphic Association Schemes

AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition

The Indecomposables of Rank 3 Permutation Modules

Transitive permutation groups of finite order are viewed as linear groups over fields of characteristic p &gt; 0 by having the group permute the basis elemerits of a vector space M. The decomposition of M into the direct sum of invariant subspaces is investigated, and criteria given for whether M is decomposable, and if it is, how many direct summands occur, in the special case the group has rank 3, i.e., it has 3 orbits on ordered pairs of points. In the case that each orbit is self-paired, M decomposes into the maximum possible number of indecomposables, and the group has every p'-element conjugate to its inverse, irreducibility results are obtained for the indecomposables. This last result holds for any rank. It applies in particular to the symmetric and thence to the alternating groups, which enables us to describe certain modular irreducibles of these groups.</p

Well-Quasi-Ordering by the Induced-Minor Relation

Robertson and Seymour proved Wagner\u27s Conjecture, which says that finite graphs are well-quasi-ordered by the minor relation. Their work motivates the question as to whether any class of graphs is well-quasi-ordered by other containment relations. This dissertation is concerned with a special graph containment relation, the induced-minor relation. This dissertation begins with a brief introduction to various graph containment relations and their connections with well-quasi-ordering. In the first chapter, we discuss the results about well-quasi-ordering by graph containment relations and the main problems of this dissertation. The graph theory terminology and preliminary results that will be used are presented in the next chapter. The class of graphs that is considered in this research is the class W of graphs that contain neither W4 (a wheel graph with five vertices) and K5\e (a complete graph on five vertices minus an edge) as an induced minor. Chapter 3 is devoted to studying the structure of this class of graphs. A class of graphs is well-quasi-ordered by a containment relation if it contains no infinite antichain, so infinite antichains are important. We construct in Chapter 4 an infinite antichain of W with respect to the induced minor relation and study its important properties in Chapter 5. These properties are used in determining all well-quasi-ordered subclasses of W to reach the main result of Chapter 6

Restricted permutations, antichains, atomic classes and stack sorting

Involvement is a partial order on all finite permutations, of infinite dimension and having subsets isomorphic to every countable partial order with finite descending chains. It has attracted the attention of some celebrated mathematicians including Paul Erdős and, due to its close links with sorting devices, Donald Knuth. We compare and contrast two presentations of closed classes that depend on the partial order of involvement: Basis or Avoidance Set, and Union of Atomic Classes. We examine how the basis is affected by a comprehensive list of closed class constructions and decompositions. The partial order of involvement contains infinite antichains. We develop the concept of a fundamental antichain. We compare the concept of 'fundamental' with other definitions of minimality for antichains, and compare fundamental permutation antichains with fundamental antichains in graph theory. The justification for investigating fundamental antichains is the nice patterns they produce. We forward the case for classifying the fundamental permutation antichains. Sorting devices have close links with closed classes. We consider two sorting devices, constructed from stacks in series, in detail. We give a comment on an enumerative conjecture by Ira Gessel. We demonstrate, with a remarkable example, that there exist two closed classes, equinumerous, one of which has a single basis element, the other infinitely many basis elements. We present this paper as a comprehensive analysis of the partial order of permutation involvement. We regard the main research contributions offered here to be the examples that demonstrate what is, and what is not, possible; although there are numerous structure results that do not fall under this category. We propose the classification of fundamental permutation antichains as one of the principal problems for closed classes today, and consider this as a problem whose solution will have wide significance for the study of partial orders, and mathematics as a whole

Permutation Groups and Binary Self-Orthogonal Codes

Let G be a permutation group on an n-element set Ω. We study the binary code C(G,Ω) defined as the dual code of the code spanned by the sets of fixed points of involutions of G. We show that any G-invariant self-orthogonal code of length n is contained in C(G,Ω). Many self-orthogonal codes related to sporadic simple groups, including the extended Golay code, are obtained as C(G,Ω). Some new self-dual codes invariant under sporadic almost simple groups are constructed