The Subconstituent Algebra of a Strongly Regular Graph

Tomiyama, Masato; Yamazaki, Norio

doi:10.2206/kyushujm.48.323

Cited by 42 publications

(35 citation statements)

References 5 publications

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“…By (27), the left-hand side of (88) equals k i . Concerning the right-hand side of (88), observe for each h (0 ≤ h ≤ n) that dim E * i W h is unity if the endpoint of W h is at most i, and zero if the endpoint of W h is greater than i.…”

Section: Multiplicities Of the Irreducible T -Modules In C Xmentioning

confidence: 99%

“…The algebra is a finite dimensional, semi-simple C-algebra, and is noncommutative in general. The Terwilliger algebra has been used to study P-and Q-polynomial association schemes [5,8,9] group schemes [1,3], strongly regular graphs [27], Doob schemes [23] and schemes over the Galois rings of characteristic four [20]. Other works involving this algebra can be found in [6, 7, 10, 15-17, 19, 25] and [26].…”

Section: Introductionmentioning

confidence: 99%

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The Terwilliger Algebra of the Hypercube

Go¹

2002

European Journal of Combinatorics

100

114

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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.

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Section: Multiplicities Of the Irreducible T -Modules In C Xmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Terwilliger Algebra of the Hypercube

Go¹

2002

European Journal of Combinatorics

100

114

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“…We recall the subconstituent algebra of a strongly regular graph [17]. We use the following notation.…”

Section: Strongly Regular Graphsmentioning

confidence: 99%

The subconstituent algebra of strongly regular graphs associated with a Latin square

Curtin

Daqqa

2009

Des. Codes Cryptogr.

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“…The subalgebra T of M n generated by M and M * is known as the subconstituent algebra or the Terwilliger algebra [39]. It has been used to study P-and Q-polynomial association schemes [18,39], group association schemes [8,10], strongly regular graphs [42], Doob schemes [38], and association schemes over the Galois rings of characteristic 4 [32]. Other work involving the Terwilliger algebra can be found in [19-25, 27, 28, 30, 40, 41].…”

Section: Introductionmentioning

confidence: 99%

“…We remark that Tomiyama and Yamazaki [42] have studied T and its finite-dimensional modules when d = 2. In contrast, our results involve and its finite-dimensional modules when d = 2.…”

Section: Introductionmentioning

confidence: 99%

The Generalized Terwilliger Algebra and its Finite-dimensional Modules when d=2

Egge

2002

Journal of Algebra

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Terwilliger [J. Algebraic Combin. 1 (1992), considered the -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 are often called the triple product relations. They determine much of the structure of T , though not all of it in general. To illuminate the role these relations play, the current author introduced [J. Algebra 233 (2000), 213-252] a generalization of T . To go from T to , we replace M and M * with a pair of dual character algebras C and C * . The dimensions of C and C * are equal; let d + 1 denote this common dimension. Intuitively, is the associative -algebra with identity generated by C and C * subject to the analogues of Terwilliger's triple product relations. is infinite-dimensional and noncommutative in general. In this paper we study and its finite-dimensional modules when d = 2 and has no "extra" vanishing intersection numbers or dual intersection numbers. In this case we show is -algebra isomorphic to M 3 ⊕ , where M 3 denotes the -algebra consisting of all 3-by-3 matrices with entries in and denotes the associative -algebra with identity generated by the symbols e and f subject to the relations e 2 = e and f 2 = f . We find a basis for and we determine the center of . We classify the finite-dimensional indecomposable -modules up to isomorphism. There are four such -modules in every odd dimension, and in every even dimension these modules are parameterized by a single complex number. We also classify the finite-dimensional irreducible -modules up to isomorphism. Using our results concerning , we find a basis for , we describe the center of , and we classify both the finite-dimensional indecomposable and the finite-dimensional irreducible -modules up to isomorphism.  2002 Elsevier Science (USA)

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The Subconstituent Algebra of a Strongly Regular Graph

Cited by 42 publications

References 5 publications

The Terwilliger Algebra of the Hypercube

The Terwilliger Algebra of the Hypercube

The subconstituent algebra of strongly regular graphs associated with a Latin square

The Generalized Terwilliger Algebra and its Finite-dimensional Modules when d=2

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