10 research outputs found
Terwilliger Algebras of Cyclotomic Schemes and Jacobi Sums
AbstractWe show that theT-module structure of a cyclotomic scheme is described in term of Jacobi sums. It holds that an irreducibleT-module of a cyclotomic scheme fails to have maximal dimension if and only if Jacobi sums satisfy certain kind of equations, which are of some number theoretical interest in themselves
Terwilliger algebraの表現とその応用
金沢大学理学部本研究の最大の研究成果はTerwilliger algebraのnonthin表現の研究においてbreakthroughがあったことである。このbreakthoughに的をしぼり、いかにしてnonthin表現の表現論を完成しようとしているかを説明する。その他の研究成果については、代表的なものとして、cyclotomic schemeのTerwilliger algebraの研究が整数論との関連のもとに深まったこと、spin modelとquantum groupとTerwilliger algebraとの係わりが見出されたこと、type II matrixの構造に新知見が得られたことなどの先駆的仕事をあげるにとどめる。本研究ではnonthincaseにおいて、classical parameterを持つP-and Q-polynomial schemaのT-algerbaの表現について、以下のような成果が得られた(ただし、ここではclassical parameterを通常より変数が1個少ない意味に解釈している)。endpoint 1のirreducible T-moduleはladder basisという非常に良い性質をもつ(Hobart-伊藤)。最も簡単なparameterの場合、irreducible T-moduleは、Onsager algebraの有限次元既約表現から求まる(伊藤-田辺-Terwilliger)。以上の結果を一般の場合に拡張するための基本となる構造定理がT-moduleに対して得られ、Onsager algebraのq-analogue(q-Onsager algebra)が定義された(伊藤-田辺-Terwilliger)。以上により、classical parameterの場合、問題はq-Onsager algebraの有限次元既約表現に帰着する。diameterが3のとき、q-Onsager algebraの有限次元既約表現は、affine quantom algebra U_g(sl_2)のtype(1,1)表現から求まる(伊藤-田辺-Benkart-Terwilliger、論文準備中)。これが初めに述べたbreakthroughであり、この結果を一般のdiameterに拡張するのが、今後の研究の目標となる。The major outcome of this research project, which will be discussed in detail later, is that there was a breakthrough in the area of the non thin representations of Terwilliger algebras of P- and Q- polynomial type. Among others are some pioneering works on the Terwilliger algebras of cyclotomic schemes and the Jacobi sums, on relations between spin models, quantum groups and Terwilliger algebras, on the structure of type II matrices. Let T be a Terwilliger algebra of P- and Q-polynomial type with classical parameters, where the classical parameters mean the ones with one less variables than usual. We obtained the following results.The irreducible T-modules of endpoint 1 have a ladder basis (Hobart-Ito). In the simplest case of parameters, irreducible T-modules are determined via finite dimensional irreducible representations of On sager algebras (Ito-Tanabe-Terwilliger). A basic theorem is obtained for the structure of T-modules, enabling us to deal with the general case by defining the q-analogue of an On sager algebra (q-On sager algebra) (Ito-Tanabe-Terwilliger).Thus in the case of classical parameters, the problem of irreducible T-modules is reduced to the determination of finite dimensional irreducible representations of q-Onsager algebras. If the diameter is 3, finite dimensional irreducible representations of q-0n sager algebras are determined via the type (1,1) representations of the affine quantum algebra U_q (sl_2). This is the breakthrough mentioned at the beginning and we are aiming at generalizing it to arbitrary diameters.研究課題/領域番号:10440003, 研究期間(年度):1998 – 2001出典:「Terwilliger algebraの表現とその応用」研究成果報告書 課題番号10440003(KAKEN:科学研究費助成事業データベース(国立情報学研究所))(https://kaken.nii.ac.jp/ja/grant/KAKENHI-PROJECT-10440003/)を加工して作
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
Permutation group approach to association schemes
AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed
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 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
Invariant Differential Operators
With applications in quantum field theory, general relativity and elementary particle physics, this three-volume work studies the invariance of differential operators under Lie algebras, quantum groups and superalgebras. This second volume covers quantum groups in their two main manifestations: quantum algebras and matrix quantum groups. The exposition covers both the general aspects of these and a great variety of concrete explicitly presented examples. The invariant q-difference operators are introduced mainly using representations of quantum algebras on their dual matrix quantum groups as carrier spaces. This is the first book that covers the title matter applied to quantum groups
On the pure Jacobi Sums
this paper, we assume that ord( ) = 2 and ord() = n 3. This special type of Jacobi sums play an important role in evaluating the argument of Gauss sum: G() = X x2Fq (x) Tr Fq =Fp (x) p ; where p is a primitive p-th root of unity (see [3], [4] ). Moreover, recently the rationality of this Jacobi sum is used to characterize the irreducible module of the Terwilliger algebras of cyclotomic association schemes (see [10]