100,782 research outputs found
The Terwilliger Algebra of an Almost-Bipartite P- and Q-Polynomial Association Scheme
Get PDFLet Y denote a D-class symmetric association scheme with D≥3, and suppose Y is almost-bipartite P- and Q-polynomial. Let x denote a vertex of Y and let T=T(x) denote the corresponding Terwilliger algebra. We prove that any irreducible T-module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the dual endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the irreducible T-modules occur in the standard module. We compute this multiplicity for those irreducible T-modules which have diameter at least D−3
Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra
No full textAbstractLet Γ denote a distance-regular graph with diameter D≥3, intersection numbers ai, bi, ci and Bose–Mesner algebra M. For θ∈C∪∞ we define a one-dimensional subspace of M which we call M(θ). If θ∈C then M(θ) consists of those Y in M such that (A−θI)Y∈CAD, where A (resp. AD) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ=∞ then M(θ)=CAD. By a pseudo primitive idempotent for θ we mean a nonzero element of M(θ). We use these as follows. Let X denote the vertex set of Γ and fix x∈X. Let T denote the subalgebra of MatX(C) generated by A, E0∗,E1∗,…,ED∗, where Ei∗ denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the Terwilliger algebra. Let W denote an irreducible T-module. By the endpoint of W we mean min{i∣Ei∗W≠0}. W is called thin whenever dim(Ei∗W)≤1 for 0≤i≤D. Let V=CX denote the standard T-module. Fix 0≠v∈E1∗V with v orthogonal to the all ones vector. We define (M;v):={P∈M∣Pv∈ED∗V}. We show the following are equivalent: (i) dim(M;v)≥2; (ii) v is contained in a thin irreducible T-module with endpoint 1. Suppose (i), (ii) hold. We show (M;v) has a basis J, E where J has all entries 1 and E is defined as follows. Let W denote the T-module which satisfies (ii). Observe E1∗W is an eigenspace for E1∗AE1∗; let η denote the corresponding eigenvalue. Define η=−1−b1(1+η)−1 if η≠−1 and η=∞ if η=−1. Then E is a pseudo primitive idempotent for η
The terwilliger algebra of a distance-regular graph
No full textThis dissertation deals with the Terwilliger algebra of a distance-regular graph.The study has two main parts. The first part studies the Terwilliger algebra of the D-cube QD, also known as hypercube. Let X denote the vertex set of QD. Fix x e X, and let T=T(x) denote its associated Terwilliger algebra. T is shown as the subalgebra of Matx (C) generated by the adjacency matrix A and a diagonal matrix A*=A*(x), where A* has yy entry D-2a(x,y) for all y e X. A, A* satisfyA2A*-2AA*+A*A2 = 4A*,A*2-2A*AA*+AA*2 = 4AUsing the above equations, the irreducible T-modules is found. For each irreducible T-module W, two orthogonal bases are displayed, the standard basis and the dual standard basis. Action of A and A* are described on these basis. The transition matrix is given from the standard basis to the dual standard basis. The multiplicity with which each irreducible T-module W appears in is computed. An elementary proof that QD has the Q-polynomial property is given. T, a homomorphic image of the universal enveloping algebra of the Lie algebra sl2 (C) is shown. The center of T is described.
The second part of this dissertation studies the Terwilliger algebra of a tight distance-regular graph. Let r = (X,R) denote a distance-regular graph with diameter D greater than or equal to 3. Fix x e C, and let T - T(x) denote its associated Terwilliger algebra. We associate two integer parameters: the endpoint and the diameter, to each irreducible T-module. It turns out that the dimension of such a module is at least one more than its diameter. Whenever equality is attained, the module is said to be thin. To each irreducible T-module of endpoint 1 and diameter D-2, another real parameter, the type is associated. The assumption now is that r is tight. The r is shown to have at least one irreducible T-module of type 1, at least one irreducible T-module of type D, and up to isomorphism, no other irreducible T-modules of endpoint 1. Each type is shown to be thin and has diameter D-2. The multiplicity with which each module appears inCx is computed
The Terwilliger Algebra of the Hypercube
No full textAbstractWe give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube QDof dimension D. Let X denote the vertex set ofQD . Fix a vertex x∈X, and letT=T(x) denote the associated Terwilliger algebra. We show thatT is the subalgebra of MatX(C) generated by the adjacency matrixA and a diagonal matrix A*=A* (x), where A*has yy entryD− 2 ∂(x, y) for all y∈X , and where ∂ denotes the path-length distance function. We show that A andA* satisfy A2A*− 2AA*A+A*A2&=& 4A* , A*2A− 2A*AA*+AA*2&=& 4 A. Using the above equations, we find the irreducible T -modules. For each irreducible T -module W, we display two orthogonal bases, which we call the standard basis and the dual standard basis. We describe the action of A andA* on each of these bases. We give the transition matrix from the standard basis to the dual standard basis for W. We compute the multiplicity with which each irreducible T -module W appears inCX . We give an elementary proof that QDhas the Q -polynomial property. We show that T is a homomorphic image of the universal enveloping algebra of the Lie algebrasl2 (C). We obtain an element φ of T that generates the center ofT . We obtain the central primitive idempotents of T as polynomials in φ
The Generalized Terwilliger Algebra of the Hypercube
Get PDFIn the year 2000, Eric Egge introduced the generalized Terwilliger algebra
of a distance-regular graph . For any vertex of
there is a surjective algebra homomorphism from to the Terwilliger algebra . If is complete, then
is an isomorphism. If is not complete, then may or may not
be an isomorphism, and in general the details are unknown. We show that if
is a hypercube, then the algebra homomorphism is an isomorphism for all vertices of .Comment: 24 page
ISR "Terwilliger" Quadrupole
No full textThere were 48 of these Quadrupoles in the ISR. They were distributed around the rings according to the so-called Terwilliger scheme. Their aperture was 184 mm, their core length 300 mm, their gradient 5 T/m. Due to their small length as compared to the aperture, the end fringe field errors had to be compensated by suitably shaping the poles
A Generalization of the Terwilliger Algebra
No full textAbstractP. M. Terwilliger (1992, J. Algebraic Combin.1, 363–388) considered the C-algebra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M*. This algebra is now known as the Terwilliger algebra and is usually denoted by T. Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T. These relations determine much of the structure of T, thought not all of it in general. To illuminate the role these relations play, we consider a certain generalization T of T. To go from T to T, we replace M and M* with a pair of dual character algebras C and C*. We define T by generators and relations; intuitively T is the associative C-algebra with identity generated by C and C* subject to the analogues of Terwilliger's relations. T is infinite dimensional and noncommutative in general. We construct an irreducible T-module which we call the primary module; the dimension of this module is the same as that of C and C*. We find two bases of the primary module; one diagonalizes C and the other diagonalizes C*. We compute the action of the generators of T on these bases. We show T is a direct sum of two sided ideals T0 and T1 with T0 isomorphic to a full matrix algebra. We show that the irreducible module associated with T0 is isomorphic to the primary module. We compute the central primitive idempotent of T associated with T0 in terms of the generators of T
The Terwilliger algebras of bipartite P- and Q-polynomial schemes
No full textAbstractLet Y denote a D-class symmetric association scheme with D ⩾ 3, and suppose Y is bipartite P- and Q-polynomial. Let T denote the Terwilliger algebra with respect to any vertex x.We prove that any irreducible T-module W is both thin and dual-thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T-module is determined by two parameters, the endpoint and diameter of W. We find a recurrence which gives the multiplicities with which the irreducible T-modules occur in the standard module. Using this recurrence, we produce formulas for the multiplicities of the irreducible T-modules with endpoint at most four
Almost Commutative Terwilliger Algebras of Group Association Schemes II: Primitive Idempotents
Get PDFThis paper is a continuation of Almost Commutative Terwilliger Algebras of Group Association Schemes I: Classification [1]. In that paper, we found all groups G for which the Terwilliger algebra of the group association scheme, denoted T (G), is almost commutative. We also found the primitive idempotents for T (G) for three of the four types of such groups. In this paper, we determine the primitive idempotents for the fourth type.Submitted to Discrete Mathematics Journa
U(sl₂) AND THE TERWILLIGER ALGEBRAS (Research on finite groups, algebraic combinatorics, and vertex algebras)
No full textThe universal enveloping algebra U(sl₂) of sl₂ is a unital associative algebra over ℂ generated by E, F, H subject to the relations [H, E] =2E, [H, F] = -2F, [E, F]=H. In 2002, Junie T. Go showed that the Terwilliger algebra of H(D, 2) is a homomorphic image of U(sl₂). Firstly, I will present a connection of the even subalgebra of U(sl₂) with the Terwilliger algebra of ½H(D, 2). Secondly, I will show how the Clebsch-Gordan rule of U(sl₂) is related to the Terwilliger algebra of H(D, q). Thirdly, I will give an algebraic connection between the Clebsch-Gordan coefficients of U(sl₂) and the Terwilliger algebra of J(D, k). The first part is a joint work with Chia-Yi Wen
- …
