157 research outputs found
Nonexistence of Certain Skew-symmetric Amorphous Association Schemes
An association scheme is amorphous if it has as many fusion schemes as
possible. Symmetric amorphous schemes were classified by A. V. Ivanov [A. V.
Ivanov, Amorphous cellular rings II, in Investigations in algebraic theory of
combinatorial objects, pages 39--49. VNIISI, Moscow, Institute for System
Studies, 1985] and commutative amorphous schemes were classified by T. Ito, A.
Munemasa and M. Yamada [T. Ito, A. Munemasa and M. Yamada, Amorphous
association schemes over the Galois rings of characteristic 4, European J.
Combin., 12(1991), 513--526]. A scheme is called skew-symmetric if the diagonal
relation is the only symmetric relation. We prove the nonexistence of
skew-symmetric amorphous schemes with at least 4 classes. We also prove that
non-symmetric amorphous schemes are commutative.Comment: 10 page
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
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
Four-class Skew-symmetric Association Schemes
An association scheme is called skew-symmetric if it has no symmetric
adjacency relations other than the diagonal one. In this paper, we study
4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of
skew-symmetric amorphous association schemes, submitted for publication], we
discovered that their character tables fall into three types. We now determine
their intersection matrices. We then determine the character tables and
intersection numbers for 4-class skew-symmetric pseudocyclic association
schemes, the only known examples of which are cyclotomic schemes. As a result,
we answer a question raised by S. Y. Song [Commutative association schemes
whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996].
We characterize and classify 4-class imprimitive skew-symmetric association
schemes. We also prove that no 2-class Johnson scheme can admit a 4-class
skew-symmetric fission scheme. Based on three types of character tables above,
a short list of feasible parameters is generated.Comment: 12 page
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