An Action of the Tetrahedron Algebra on the Standard Module for the Hamming Graphs and Doob Graphs

Morales, John Vincent S.; Pascasio, Arlene A.

doi:10.1007/s00373-013-1366-0

Cited by 12 publications

(8 citation statements)

References 20 publications

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“…The paper [121] shows howĤ q is related to Leonard pairs. Additional results in the literature link tridiagonal pairs and Leonard pairs with the Lie algebra sl 2 (see [3, 7-9, 26, 81, 120]), the quantum algebras U q (sl 2 ) (see [2,29,30,88,147,162]), U q ( sl 2 ) (see [5,24,41,75,76,84,152]), the tetrahedron Lie algebra (see [25,56,79,96]), and its q-deformation q (see [42,73,77,78,80,164,165]).…”

Section: Casementioning

confidence: 99%

Totally bipartite tridiagonal pairs

Nomura

Terwilliger²

2021

ELA

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There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

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Section: Casementioning

confidence: 99%

Totally bipartite tridiagonal pairs

Nomura

Terwilliger²

2021

ELA

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“…Repeating the argument from the first paragraph again in this case gives Combining ( 29) - (41) we have that X ru , X su , X tu acts on V as a BD triad. It is immediate from (30), (31), (34), (35), (38), (39) that the first, second, and third eigenvalue sequences of X ru , X su , X tu are each {2i − d} d i=0 . Therefore, the BD triad X ru , X su , X tu is reduced.…”

Section: Proof Of Theorem 43: Letmentioning

confidence: 99%

“…For more information on how the tetrahedron algebra arises in representation theory see [2,6,11,25,32]. See [18,30,31] for how the tetrahedron algebra has been used in algebraic combinatorics and graph theory.…”

Section: Introductionmentioning

confidence: 99%

Bidiagonal Triads and the Tetrahedron Algebra

Funk-Neubauer¹

2021

Preprint

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We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other two. A triad differs from a triple in the way these bidiagonal actions are defined. We modify a number of theorems about bidiagonal triples to the case of bidiagonal triads. We also describe the close relationship between bidiagonal triads and the representation theory of the tetrahedron Lie algebra.

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“…Define C by (99). One verifies ( 97) and ( 98) using ( 94), ( 95), (96). ✷ Proposition 15.3 Assume that β = −2.…”

mentioning

confidence: 94%

Totally bipartite tridiagonal pairs

Nomura

Terwilliger²

2017

Preprint

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There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of Q-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or TB. Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey-Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the Z 3symmetric Askey-Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; (xi) an action of the modular group PSL 2 (Z) associated with a TB tridiagonal pair.

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An Action of the Tetrahedron Algebra on the Standard Module for the Hamming Graphs and Doob Graphs

Cited by 12 publications

References 20 publications

Totally bipartite tridiagonal pairs

Totally bipartite tridiagonal pairs

Bidiagonal Triads and the Tetrahedron Algebra

Totally bipartite tridiagonal pairs

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