A Generalization of the Terwilliger Algebra

Egge, Eric S.

doi:10.1006/jabr.2000.8420

Cited by 51 publications

(53 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We call this module V 0 . The module V 0 is described in [8,17]. We summarize some details below in order to motivate the results that follow.…”

Section: The Subconstituent Algebra and Its Modulesmentioning

confidence: 99%

“…With reference to Definition 5.1, there exists a unique irreducible T -module with endpoint 0 [17,Proposition 8.4]. We call this module V 0 .…”

Section: The Subconstituent Algebra and Its Modulesmentioning

confidence: 99%

“…We mentioned earlier that p D (A) = J . Multiplying both sides of this equation by E h , and recalling J is a scalar multiple of E 0 , we find p D (A)E h = 0 in view of (17). (27), let the integers i, j be given.…”

Section: (We Recall M H Denotes the Multiplicity Of θ H Formentioning

confidence: 99%

“…The module V 0 is thin; in fact E * i V 0 and E i V 0 have dimension 1 for 0 ≤ i ≤ D [38,Lemma 3.6]. For a detailed description of V 0 see [8,17]. In this paper we are concerned with the thin irreducible T -modules with endpoint 1.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Tight Distance-regular Graphs and the Subconstituent Algebra

Terwilliger

2002

European Journal of Combinatorics

View full text Add to dashboard Cite

We consider a distance-regular graph with diameter D ≥ 3, intersection numbers a i , b i , c i and eigenvalues k = θ 0 > θ 1 > · · · > θ D . Let X denote the vertex set of and fix x ∈ X. Let T = T (x) denote the subalgebra of Mat X (C) generated by A, E , where E i denotes the primitive idempotent of A associated with θ i . We show this basis is orthogonal (with respect to the Hermitean dot product) and we compute the square norm of each basis vector. We show, where A i denotes the ith distance matrix for . We find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square norm of each basis vector. We find the transition matrix relating our two bases for W . For notational convenience, we say is 1-thin with respect to x whenever every irreducible T -module with endpoint 1 is thin. Similarly, we say is tight with respect to x whenever every irreducible T -module with endpoint 1 is thin with local eigenvalueθ 1 orθ D . In [J. Algebr. Comb., 12, (2000), Jurišić, Koolen and Terwilliger showedThey defined to be tight whenever is nonbipartite and equality holds above. We show the following are equivalent: (i) is tight; (ii) is tight with respect to each vertex; (iii) is tight with respect to at least one vertex. We show the following are equivalent: (i) is tight; (ii) is nonbipartite, a D = 0, and is 1-thin with respect to each vertex; (iii) is nonbipartite, a D = 0, and is 1-thin with respect to at least one vertex.

show abstract

“…We call this module V 0 . The module V 0 is described in [8,17]. We summarize some details below in order to motivate the results that follow.…”

Section: The Subconstituent Algebra and Its Modulesmentioning

confidence: 99%

“…With reference to Definition 5.1, there exists a unique irreducible T -module with endpoint 0 [17,Proposition 8.4]. We call this module V 0 .…”

Section: The Subconstituent Algebra and Its Modulesmentioning

confidence: 99%

Section: (We Recall M H Denotes the Multiplicity Of θ H Formentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Tight Distance-regular Graphs and the Subconstituent Algebra

Terwilliger

2002

European Journal of Combinatorics

View full text Add to dashboard Cite

show abstract

“…Moreover T is semi-simple since it is closed under the conjugate transponse map [12, p. 157]. See [7,8,11,16,17,18,20,26,29,30,31] for more information on the subconstituent algebra.…”

Section: Preliminaries Concerning Distance-regular Graphsmentioning

confidence: 99%

The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

Terwilliger

2005

Graphs and Combinatorics

View full text Add to dashboard Cite

show abstract

The Terwilliger Algebra of the Hypercube

Go¹

2002

European Journal of Combinatorics

102

114

View full text Add to dashboard Cite

We give an introduction to the Terwilliger algebra of a distance-regular graph, focusing on the hypercube Q D of dimension D. Let X denote the vertex set of Q D . Fix a vertex x ∈ X , and let T = T (x) denote the associated Terwilliger algebra. We show that T is the subalgebra of Mat X (C) generated by the adjacency matrix A and a diagonal matrix A * = A * (x), where A * has yy entry D − 2∂(x, y) for all y ∈ X , and where ∂ denotes the path-length distance function. We show that A and A * satisfyUsing 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 and A * 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 in C X . We give an elementary proof that Q D has the Q-polynomial property. We show that T is a homomorphic image of the universal enveloping algebra of the Lie algebra sl 2 (C). We obtain an element φ of T that generates the center of T . We obtain the central primitive idempotents of T as polynomials in φ.

show abstract

A Generalization of the Terwilliger Algebra

Cited by 51 publications

References 31 publications

Tight Distance-regular Graphs and the Subconstituent Algebra

Tight Distance-regular Graphs and the Subconstituent Algebra

The Displacement and Split Decompositions for a Q-Polynomial Distance-regular Graph

The Terwilliger Algebra of the Hypercube

Contact Info

Product

Resources

About