Association schemes and permutation groups

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

On Maximal Closed Subsets in Association Schemes

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry.

Representations of finite association schemes

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

We present the theory of equiangular tight frames (ETFs) from association schemes as a generalization of harmonic ETFs. This theory has the pleasing consequence of including not only harmonic ETFs from abelian groups but also real ETFs with centroidal symmetry and the ETFs discovered by Renes and Strohmer coming from skew-symmetric conference matrices. Finally we present a theory of nonabelian harmonic ETFs from the perspective of association schemes. An infinite class of ETFs arising as group frames from nonabelian groups is presented.

On the tensor product of permutation modules

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory; cf. [8] and [7]. One of these generalizations is the observation that, similar to groups, association schemes are commutative if they have at most five elements and not necessarily commutative if they have six elements; cf. [6, (4.1)]. (In this article, all association schemes are assumed to have finite valency). While any two noncommutative groups of order 6 are isomorphic to each other, there exist infinitely many isomorphism classes of noncommutative association schemes of order 6 (among them all finite projective planes). The present article is a first attempt to obtain insight into the structure of noncommutative schemes of order 6.

Primitive coherent configurations (PCCs) are edge-colored digraphs that generalize strongly regular graphs (SRGs), a class perceived as difficult for Graph Isomorphism (GI). Moreover, PCCs arise naturally as obstacles to combinatorial divide-and-conquer approaches for general GI. In a natural sense, the isomorphism problem for PCCs is a stepping stone between SRGs and general GI. In his 1981 paper in the Annals of Math., Babai proposed a combinatorial approach to GI testing via an analysis of the standard individualization/refinement (I/R) technique and proved that I/R yields canonical forms of PCCs in time exp(~O(n1/2)). (The tilde hides polylogarithmic factors.) We improve this bound to exp(~O(n1/3)). This is faster than the current best bound, exp(~O(n1/2)), for general GI, and subsumes Spielman's exp(~O(n1/3)) bound for SRGs (STOC'96, only recently improved to exp(~O(n1/5)) by the present authors and their coauthors (FOCS'13)).Our result implies an exp(~O(n1/3)) upper bound on the number of automorphisms of PCCs with certain easily described and recognized exceptions, making the first progress in 33 years on an old conjecture of Babai. The emergence of exceptions illuminates the technical difficulties: we had to separate these cases from the rest. For the analysis we develop a new combinatorial structure theory for PCCs that in particular demonstrates the presence of asymptotically uniform clique geometries among the constituent graphs of PCCs in a certain range of the parameters.A corollary to Babai's 1981 result was an exp(~O(n1/2)) upper bound on the order of primitive but not doubly transitive permutation groups, solving a then 100-year old problem in group theory. An improved bound of exp(~O(n1/3)) (with known exceptions) follows from our combinatorial result. This bound was previously known (Cameron, 1981) only through the Classification of Finite Simple Groups. We note that upper bounds on the order of primitive permutation groups are central to the application of Luks's group theoretic divide-and-conquer methods to GI.

Let G be a permutation group on a finite set f2 of size n. Then G acts naturally on the set P (f2) of all subsets of f2. In this note we shall show that if G is primitive on f2 and A, $ G then in all except finitely many cases G has a regular orbit on P (O). We were led to consider this question by some recent work of Siemons and Wagner [5] and Inglis [4], where the following situation was considered. Given permutation groups G and H on the same finite set f2, write G ~ H if G and H have the same orbits on P (f2). It is shown in [4] and [5] that if G is primitive, A, ~ G and G ~ H then with a few explicitly known exceptions the same primes divide [GI and [HI. It is an immediate consequence of our result that under the same hypotheses with finitely many exceptions in fact G = H. (Indeed, with the weaker assumption that the longest orbits of G and H on P (f2) have the same length, we can conclude that [ G I = I HI with only finitely many exceptions.) Unlike the arguments in [4] and [5], ours require the classification of finite simple groups. We do however get a good deal of information even without the classification and can obtain our conclusion by elementary methods if G is 2-transitive on f2. The problem of determining pairs G and H of groups with G ~ H without the assumption of primitivity seems more difficult. Note, however, that if G ~ H and G is primitive then so is H: for G is primitive if and only if the only orbits of G on P (O) whose members partition f2 are the trivial ones. Similarly, but more easily, if G ~ H and G is transitive then so is H. We start with a preliminary well-known result. Recall that the minimal degree m of a permutation group G is the least number of points moved by a non-identity element of G, while the base size b is the least number of points of which the stabilizer is the identity.

First we define and study the exponentiation of a cellular algebra by a permutation group which is analogous to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study for cellular algebras the notion of a base which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential.

Two asymptotic upper bounds on the information rate of a tree code as a function of its feedback decoding minimum distance are presented. These bounds are generalizations of the linear programming bounds for binary block codes proved by McEliece et al., and they are derived from linear programming problems based on Delsarte's theory of association schemes.

Given a finite nonabelian semisimple group G, we describe those groups that have the same holomorph as G, that is, those regular subgroups N ≃ G {N\simeq G} of S ⁢ ( G ) {S(G)} , the group of permutations on the set G, such that N S ⁢ ( G ) ⁢ ( N ) = N S ⁢ ( G ) ⁢ ( ρ ⁢ ( G ) ) {N_{S(G)}(N)=N_{S(G)}(\rho(G))} , where ρ is the right regular representation of G.

This book is a collection of articles, some introductory, some extended surveys, and some containing previously unpublished research, on a range of topics linking infinite permutation group theory and model theory. Topics covered include: oligomorphic permutation groups and omega-categorical structures; totally categorical structures and covers; automorphism groups of recursively saturated structures; Jordan groups; Hrushovski's constructions of pseudoplanes; permutation groups of finite Morley rank; applications of permutation group theory to models of set theory without the axiom of choice. There are introductory chapters by the editors on general model theory and permutation theory, recursively saturated structures, and on groups of finite Morley rank. The book is almost self-contained, and should be useful to both a beginning postgraduate student meeting the subject for the first time, and to an active researcher from either of the two main fields looking for an overview of the subject.

We consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.