Association schemes on 28 points as mergings of a half-homogeneous coherent configuration. (English) Zbl 1145.05056
The key contribution of this article, as intended by the authors, is providing a unified explanation of various association schemes on \(28\) points. They are all thought of as mergings of one specific coherent configuration, a rank 112 coherent configuration on \(28\) points with \(7\) fibers of size \(4\) arising from the regular action of \(Z_2^3\) on the set of \(2\)-element subsets of an \(8\)-element set. This configuration is constructed and inspected from various sides, with a whole section devoted to its automorphism groups.
Various association schemes, including those of pseudocyclic, quasithin, and pseudotriangular types, are then shown to be the results of mergings which arise from suitable subgroups of the group of algebraic automorphisms of the main configuration. Particular attention is paid to the curious concept of algebraic twins, special pairs of equivalence classes, some of which contain both Schurian and non-Schurian fusions.
Although the article contains a wealth of data and includes several pages of computer output, the reader may occasionally feel that some of the concepts have not been given enough space, as if the article was created by leaving out parts from a larger, more detailed text.
Various association schemes, including those of pseudocyclic, quasithin, and pseudotriangular types, are then shown to be the results of mergings which arise from suitable subgroups of the group of algebraic automorphisms of the main configuration. Particular attention is paid to the curious concept of algebraic twins, special pairs of equivalence classes, some of which contain both Schurian and non-Schurian fusions.
Although the article contains a wealth of data and includes several pages of computer output, the reader may occasionally feel that some of the concepts have not been given enough space, as if the article was created by leaving out parts from a larger, more detailed text.
Reviewer: Robert Jajcay (Terre Haute)
MSC:
| 05E30 | Association schemes, strongly regular graphs |
Keywords:
coherent configuration; association scheme; algebraic automorphism; color automorphism; mergingReferences:
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