89 research outputs found
On relative -designs in polynomial association schemes
Motivated by the similarities between the theory of spherical -designs and
that of -designs in -polynomial association schemes, we study two
versions of relative -designs, the counterparts of Euclidean -designs for
- and/or -polynomial association schemes. We develop the theory based on
the Terwilliger algebra, which is a noncommutative associative semisimple
-algebra associated with each vertex of an association scheme. We
compute explicitly the Fisher type lower bounds on the sizes of relative
-designs, assuming that certain irreducible modules behave nicely. The two
versions of relative -designs turn out to be equivalent in the case of the
Hamming schemes. From this point of view, we establish a new algebraic
characterization of the Hamming schemes.Comment: 17 page
New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
We use the Terwilliger algebra to provide a new approach to the
Assmus-Mattson theorem. This approach also includes another proof of the
minimum distance bound shown by Martin as well as its dual.Comment: 15 page
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
The eigenvalues of -Kneser graphs
In this note, we prove some combinatorial identities and obtain a simple form
of the eigenvalues of -Kneser graphs
Width and dual width of subsets in polynomial association schemes
AbstractThe width of a subset C of the vertices of a distance-regular graph is the maximum distance which occurs between elements of C. Dually, the dual width of a subset in a cometric association scheme is the index of the “last” eigenspace in the Q-polynomial ordering to which the characteristic vector of C is not orthogonal. Elementary bounds are derived on these two new parameters. We show that any subset of minimal width is a completely regular code and that any subset of minimal dual width induces a cometric association scheme in the original. A variety of examples and applications are considered
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