1,340 research outputs found
The recognition problem for table algebras and reality-based algebras
Given a finite-dimensional noncommutative semisimple algebra with
involution, we show that always has an RBA-basis. We look for an RBA-basis
that has integral or rational structure constants, and ask if the RBA admits a
positive degree map. For RBAs that have a positive degree map, we try to find
an RBA-basis with nonnegative structure constants to determine if there is a
generalized table algebra structure. We settle these questions for the algebras
, .Comment: 16 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
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