The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model

Caughman, John S.; Wolff, Nadine

doi:10.1007/s10801-005-6913-1

Cited by 28 publications

(29 citation statements)

References 15 publications

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“…These vectors are nonzero by Lemma 11.7 and mutually orthogonal by (6), so they are linearly independent.The result follows. [28,Theorem 11.2] (i) Assume = −1.…”

Section: The Space MV For V ∈ U (˜ 1 <mentioning

confidence: 99%

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The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

MacLean

Terwilliger

2008

Discrete Mathematics

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We consider a bipartite distance-regular graph with diameter D 4, valency k 3, intersection numbers b i , c i , distance matrices A i , and eigenvalues 0 > 1 > · · · > 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 * 0 , E * 1 , . . . , E * D , where A = A 1 and E * i denotes the projection onto the ith subconstituent of with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of with respect to x.and d = D/2 . To describe the structure of W we distinguish four cases: (i) =˜ 1 ; (ii) D is odd and =˜ d ; (iii) D is even and =˜ d ; (iv)˜ 1 < <˜ d . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694-1721].Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D − 1 − e where e = 1 in case (iii) and e = 0 in case (iv). Let v denote a nonzero vector in E * 2 W . We show W has a basis E i v (i ∈ S), where E i denotes the primitive idempotent of A associated with i and where the set S is {1, 2, . . . , d − 1} ∪ {d + 1, d + 2, . . . , D − 1} in case (iii) and {1, 2, . . . , D − 1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis E * i+2 A i v (0 i D − 2 − e), and 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.

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“…These vectors are nonzero by Lemma 11.7 and mutually orthogonal by (6), so they are linearly independent.The result follows. [28,Theorem 11.2] (i) Assume = −1.…”

Section: The Space MV For V ∈ U (˜ 1 <mentioning

confidence: 99%

“…Since T is semi-simple, each T-module is a direct sum of irreducible T-modules. Describing the irreducible T-modules is an active area of research [4][5][6][7][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23][24]26,[28][29][30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

MacLean

Terwilliger

2008

Discrete Mathematics

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“…In [20] the subconstituent algebra was introduced, and used to investigate commutative association schemes. Since then the subconstituent algebra has received considerable attention; some notable papers are [3,5,7,8,10,11,12,13,15,16,17,18,19,23,25].…”

Section: Introductionmentioning

confidence: 99%

The quantum adjacency algebra and subconstituent algebra of a graph

Terwilliger

Žitnik

2019

Journal of Combinatorial Theory, Series A

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Let Γ denote a finite, undirected, connected graph, with vertex set X. Fix a vertex x ∈ X. Associated with x is a certain subalgebra T = T (x) of Mat X (C), called the subconstituent algebra. The algebra T is semisimple. Hora and Obata introduced a certain subalgebra Q ⊆ T , called the quantum adjacency algebra. The algebra Q is semisimple. In this paper we investigate how Q and T are related. In many cases Q = T , but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible T -modules. We show that the following are equivalent: (i) Q = T ; (ii) there exists a pair of quasi-isomorphic irreducible T -modules that have different endpoints. To illustrate this result we consider two examples. The first example concerns the Hamming graphs. The second example concerns the bipartite dual polar graphs. We show that for the first example Q = T , and for the second example Q = T .

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“…It has been studied for many examples such as P -and Q-polynomial association schemes [6], distance-regular graph that supports a spin model [7], group association schemes [4,5], strongly regular graphs [17].…”

Section: Introductionmentioning

confidence: 99%

Generalized quadrangles and subconstituent algebra ¹

Levstein¹,

Maldonado

2010

Cubo

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The point graph of a generalized quadrangle GQ(s, t) is a strongly regular graph Γ = srg(ν, κ, λ, µ) whose parameters depend on s and t. By a detailed analysis of the adjacency matrix we compute the Terwilliger algebra of this kind of graphs (and denoted it by T). We find that there are only two non-isomorphic Terwilliger algebras for all the generalized quadrangles. The two classes correspond to wether s 2 = t or not. We decompose the algebra into direct sum of simple ideals. Considering the action T × C X −→ C X we find the decomposition into irreducible T-submodules of C X (where X is the set of vertices of the Γ). RESUMEN El grafo de puntos de un cuadrángulo generalizado GQ(s, t) es un grafo fuertemente regular Γ = srg(ν, κ, λ, µ) cuyos parámetros dependen de s y t. Mediante un análisis detallado de

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The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model

Cited by 28 publications

References 15 publications

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

The quantum adjacency algebra and subconstituent algebra of a graph

Generalized quadrangles and subconstituent algebra ¹

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