43,806 research outputs found
Characterization of Balanced Coherent Configurations
Let be a group acting on a finite set . Then acts on
by its entry-wise action and its orbits form the basis
relations of a coherent configuration (or shortly scheme). Our concern is to
consider what follows from the assumption that the number of orbits of on
is constant whenever and are
orbits of on . One can conclude from the assumption that the
actions of on 's have the same permutation character and are
not necessarily equivalent. From this viewpoint one may ask how many
inequivalent actions of a given group with the same permutation character there
exist. In this article we will approach to this question by a purely
combinatorial method in terms of schemes and investigate the following topics:
(i) balanced schemes and their central primitive idempotents, (ii)
characterization of reduced balanced schemes
A characterization of Q-polynomial association schemes
We prove a necessary and sufficient condition for a symmetric association
scheme to be a Q-polynomial scheme.Comment: 8 pages, no figur
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
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