246 research outputs found
Terwilliger Algebras of Some Group Association Schemes
The Terwilliger algebra plays an important role in the theory of association schemes. The present paper gives the explicit structures of the Terwilliger algebras of the group association schemes of the finite groups PSL(2, 7), A6, and S6
Investigation of continuous-time quantum walk on root lattice and honeycomb lattice
The continuous-time quantum walk (CTQW) on root lattice (known as
hexagonal lattice for ) and honeycomb one is investigated by using
spectral distribution method. To this aim, some association schemes are
constructed from abelian group and two copies of finite
hexagonal lattices, such that their underlying graphs tend to root lattice
and honeycomb one, as the size of the underlying graphs grows to
infinity. The CTQW on these underlying graphs is investigated by using the
spectral distribution method and stratification of the graphs based on
Terwilliger algebra, where we get the required results for root lattice
and honeycomb one, from large enough underlying graphs. Moreover, by using the
stationary phase method, the long time behavior of CTQW on infinite graphs is
approximated with finite ones. Also it is shown that the Bose-Mesner algebras
of our constructed association schemes (called -variable -polynomial) can
be generated by commuting generators, where raising, flat and lowering
operators (as elements of Terwilliger algebra) are associated with each
generator. A system of -variable orthogonal polynomials which are special
cases of \textit{generalized} Gegenbauer polynomials is constructed, where the
probability amplitudes are given by integrals over these polynomials or their
linear combinations. Finally the suppersymmetric structure of finite honeycomb
lattices is revealed. Keywords: underlying graphs of association schemes,
continuous-time quantum walk, orthogonal polynomials, spectral distribution.
PACs Index: 03.65.UdComment: 41 pages, 4 figure
Terwilliger algebras of wreath products by quasi-thin schemes
The structure of Terwilliger algebras of wreath products by thin schemes or
one-class schemes was studied in [A. Hanaki, K. Kim, Y. Maekawa, Terwilliger
algebras of direct and wreath products of association schemes, J. Algebra 343
(2011) 195--200]. In this paper, we will consider the structure of Terwilliger
algebras of wreath products by quasi-thin schemes. This gives a generalization
of their result
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
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
The Terwilliger algebra of an almost-bipartite P- and Q-polynomial association scheme
Let denote a -class symmetric association scheme with , and
suppose is almost-bipartite P- and Q-polynomial. Let denote a vertex of
and let denote the corresponding Terwilliger algebra. We prove
that any irreducible -module is both thin and dual thin in the sense of
Terwilliger. We produce two bases for and describe the action of on
these bases. We prove that the isomorphism class of as a -module is
determined by two parameters, the dual endpoint and diameter of . We find a
recurrence which gives the multiplicities with which the irreducible
-modules occur in the standard module. We compute this multiplicity for
those irreducible -modules which have diameter at least .Comment: 22 page
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