71 research outputs found
Bose-Mesner Algebras attached to Invertible Jones Pairs
In 1989, Vaughan Jones introduced spin models and showed that they could be
used to form link invariants in two different ways--by constructing
representations of the braid group, or by constructing partition functions.
These spin models were subsequently generalized to so-called 4-weight spin
models by Bannai and Bannai; these could be used to construct partition
functions, but did not lead to braid group representations in any obvious way.
Jaeger showed that spin models were intimately related to certain association
schemes. Yamada gave a construction of a symmetric spin model on vertices
from each 4-weight spin model on vertices.
In this paper we build on recent work with Munemasa to give a different proof
to Yamada's result, and we analyse the structure of the association scheme
attached to this spin model.Comment: 23 page
Hamming graphs in Nomura Algebras
Let A be an association scheme on q\geq 3 vertices. We show that the
Bose-Mesner algebra of the generalized Hamming scheme H(n,A), for n\geq 2, is
not the Nomura algebra of a type II matrix. This result gives examples of
formally self-dual Bose-Mesner algebras that are not the Nomura algebras of
type II matrices.Comment: 15 pages, minor revisio
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-Weight Spin Models and Jones Pairs
We introduce and discuss Jones pairs. These provide a generalization and a
new approach to the four-weight spin models of Bannai and Bannai. We show that
each four-weight spin model determines a ``dual'' pair of association schemes
Subfactors from regular graphs induced by association schemes
We clarify the relations between the mathematical structures that enable
fashioning quantum walks on regular graphs and their realizations in anyonic
systems. Our protagonist is association schemes that may be synthesized from
type-II matrices which have a canonical construction of subfactors. This way we
set up quantum walks on growing distance-regular graphs induced by association
schemes via interacting Fock spaces and relate them to anyon systems described
by subfactors. We discuss in detail a large family of graphs that may be
treated within this approach. Classification of association schemes and
realizable anyon systems are complex combinatorial problems and we tackle a
part of it with a quantum walk application based approach.Comment: arXiv admin note: text overlap with arXiv:2109.10934,
arXiv:1902.0866
Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra
Let denote a distance-regular graph with diameter and
Bose-Mesner algebra . For we define a 1 dimensional
subspace of which we call . If then
consists of those in such that , where
(resp. ) is the adjacency matrix (resp. th distance matrix) of
If then . By a {\it pseudo
primitive idempotent} for we mean a nonzero element of . We
use pseudo primitive idempotents to describe the irreducible modules for the
Terwilliger algebra, that are thin with endpoint one.Comment: 17 page
Spin Models, Association Schemes and the Nakanishi–Montesinos Conjecture
AbstractA 3-transformation of a link is a local change which replaces two strings that are three times half twisted around each other by two untwisted strings (and vice versa). The Nakanishi–Montesinos (NM) conjecture asserts that this 3-transformation can unknot any link. We introduce the notion of the NM-spin model, which gives a link invariant preserved by 3-transformation. We try to classify such spin models and determine the corresponding link invariant. It is proved that the dimension of the Bose–Mesner algebra generated by the spin model is ≤4. For dimension 1 and 2, there is no such spin model, but for dimension 3, there exists a unique one. Its link invariant is a non-trivial specialization of the Kauffman polynomial, but does not distinguish trivial links from the others, and hence cannot disprove the NM conjecture. For dimension 4, we give a family of NM-spin models. The corresponding link invariant is identified and does not distinguish trivial links from the others. Strong regularity and triple regularity of the Bose–Mesner algebra and its fusions are studied
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