23 research outputs found
Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight
In 1991, Weidong Fang and Huiling Li proved that there are only finitely many
non-trivial linear spaces that admit a line-transitive, point-imprimitive group
action, for a given value of gcd(k,r), where k is the line size and r is the
number of lines on a point. The aim of this paper is to make that result
effective. We obtain a classification of all linear spaces with this property
having gcd(k,r) at most 8. To achieve this we collect together existing theory,
and prove additional theoretical restrictions of both a combinatorial and group
theoretic nature. These are organised into a series of algorithms that, for
gcd(k,r) up to a given maximum value, return a list of candidate parameter
values and candidate groups. We examine in detail each of the possibilities
returned by these algorithms for gcd(k,r) at most 8, and complete the
classification in this case.Comment: 47 pages Version 1 had bbl file omitted. Apologie
Block-transitive 2-designs with a chain of imprimitive partitions
More than years ago, Delandtsheer and Doyen showed that the automorphism
group of a block-transitive -design, with blocks of size , could leave
invariant a nontrivial point-partition, but only if the number of points was
bounded in terms of . Since then examples have been found where there are
two nontrivial point partitions, either forming a chain of partitions, or
forming a grid structure on the point set. We show, by construction of infinite
families of designs, that there is no limit on the length of a chain of
invariant point partitions for a block-transitive -design. We introduce the
notion of an `array' of a set of points which describes how the set interacts
with parts of the various partitions, and we obtain necessary and sufficient
conditions in terms of the `array' of a point set, relative to a partition
chain, for it to be a block of such a design
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
Transitive and Co-Transitive Caps
A cap in PG(r,q) is a set of points, no three of which are collinear. A cap
is said to be transitive if its automorphism group in PGammaL(r+1,q) acts
transtively on the cap, and co-transitive if the automorphism group acts
transtively on the cap's complement in PG(r,q). Transitive, co-transitive caps
are characterized as being one of: an elliptic quadric in PG(3,q); a
Suzuki-Tits ovoid in PG(3,q); a hyperoval in PG(2,4); a cap of size 11 in
PG(4,3); the complement of a hyperplane in PG(r,2); or a union of Singer orbits
in PG(r,q) whose automorphism group comes from a subgroup of GammaL(1,q^{r+1}).Comment: To appear in The Bulletin of the Belgian Mathematical Society - Simon
Stevi
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
Inspired by some intriguing examples, we study uniform association schemes
and uniform coherent configurations, including cometric Q-antipodal association
schemes. After a review of imprimitivity, we show that an imprimitive
association scheme is uniform if and only if it is dismantlable, and we cast
these schemes in the broader context of certain --- uniform --- coherent
configurations. We also give a third characterization of uniform schemes in
terms of the Krein parameters, and derive information on the primitive
idempotents of such a scheme. In the second half of the paper, we apply these
results to cometric association schemes. We show that each such scheme is
uniform if and only if it is Q-antipodal, and derive results on the parameters
of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We
revisit the correspondence between uniform indecomposable three-class schemes
and linked systems of symmetric designs, and show that these are cometric
Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class
schemes in terms of only a few parameters, and show that any strongly regular
graph with a ("non-exceptional") strongly regular decomposition gives rise to
such a scheme. Hemisystems in generalized quadrangles provide interesting
examples of such decompositions. We finish with a short discussion of
five-class schemes as well as a list of all feasible parameter sets for
cometric Q-antipodal four-class schemes with at most six fibres and fibre size
at most 2000, and describe the known examples. Most of these examples are
related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions,
April 201
Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems
2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12