Dual polar graphs, the quantum algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">sl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>, and Leonard systems of dual q-Krawtchouk type

Worawannotai, Chalermpong

doi:10.1016/j.laa.2012.08.016

Cited by 24 publications

(4 citation statements)

References 23 publications

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“…In the right-hand side of ( 52), eliminate ϕ 1 , φ 1 using ( 47), (48), and simplify the result using (50) to get (52). The line ( 53) is similarly obtained.…”

Section: Proposition 101 the Pair A A * Is Leonard Pair Over F If And...mentioning

confidence: 99%

“…We just mentioned how Leonard pairs are related to orthogonal polynomials. Leonard pairs have applications to many other areas of mathematics and physics, such as Lie theory [3,21,22,25,29,37], quantum groups [1,2,[13][14][15][26][27][28], spin models [17][18][19]41], double affine Hecke algebras [23,24,30,31,38], partially ordered sets [33,34,44,52], and exactly solvable models in statistical mechanics [6][7][8][9][10][11][12]. For more information about Leonard pairs and related topics, see [4,37,39,40,42,43,45,47,51].…”

Section: Introductionmentioning

confidence: 99%

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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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“…In the right-hand side of ( 52), eliminate ϕ 1 , φ 1 using ( 47), (48), and simplify the result using (50) to get (52). The line ( 53) is similarly obtained.…”

Section: Proposition 101 the Pair A A * Is Leonard Pair Over F If And...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Totally bipartite tridiagonal pairs

Nomura

Terwilliger²

2021

ELA

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“…The equitable presentation of this algebra was introduced in [1], where its relationship to the usual presentation in terms of the Chevalley generators [2] is discussed. The equitable presentation has been studied in connection with tridiagonal pairs [3,4], Leonard pairs [5], the q-tetrahedron algebra [6][7][8][9], bidiagonal pairs [10], Q-polynomial distance-regular graphs [11][12][13], in Poisson algebras [14], and the universal Askey-Wilson algebra [15].…”

Section: Lemma 1 [Theorem 21] [1]mentioning

confidence: 99%

Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)

Alnajjar

Curtin

2020

Mathematics

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Let F denote an algebraically closed field; let q be a nonzero scalar in F such that q is not a root of unity; let d be a nonnegative integer; and let X, Y, Z be the equitable generators of Uq(sl2) over F. Let V denote a finite-dimensional irreducible Uq(sl2)-module with dimension d+1, and let R denote the set of all linear maps from V to itself that act tridiagonally on the standard ordering of the eigenbases for each of X, Y, and Z. We show that R has dimension at most seven. Indeed, we show that the actions of 1, X, Y, Z, XY, YZ, and ZX on V give a basis for R when d≥3.

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“…In the present paper, we introduce a family of Leonard pairs called near-We just mentioned how Leonard pairs are related to orthogonal polynomials. Leonard pairs have applications to many other areas of mathematics and physics, such as Lie theory [3,21,22,25,29,37], quantum groups [1,2,[13][14][15][26][27][28], spin models [17][18][19]41], double affine Hecke algebras [23,24,30,31,38], partially ordered sets [33,34,44,52], and exactly solvable models in statistical mechanics [6][7][8][9][10][11][12]. For more information about Leonard pairs and related topics, see [4,37,39,40,42,43,45,47,51].…”

mentioning

confidence: 99%

Near-bipartite Leonard pairs

Nomura,

Terwilliger

2024

ELA

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Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A: V \to V$ and $A^* : V \to V$ that each act on an eigenbasis for the other in an irreducible tridiagonal fashion. Let $A,A^*$ denote a Leonard pair on $V$. Let $\{v_i\}_{i=0}^d$ denote an eigenbasis for $A^*$ on which $A$ acts in an irreducible tridiagonal fashion. For $0 \leq i \leq d$, define an $\mathbb{F}$-linear map $E^*_i : V \to V$ such that $E^*_i v_i = v_i$ and $E^*_i v_j = 0$ if $j \neq i$ $(0 \leq j \leq d)$. The map $F = \sum_{i=0}^d E^*_i A E^*_i$ is called the flat part of $A$. The Leonard pair $A,A^*$ is bipartite whenever $F=0$. The Leonard pair $A,A^*$ is said to be near-bipartite whenever the pair $A-F, A^*$ is a Leonard pair on $V$. In this case, the Leonard pair $A-F, A^*$ is bipartite and called the bipartite contraction of $A,A^*$. Let $B,B^*$ denote a bipartite Leonard pair on $V$. By a near-bipartite expansion of $B,B^*$, we mean a near-bipartite Leonard pair on $V$ with bipartite contraction $B,B^*$. In the present paper, we have three goals. Assuming $\mathbb{F}$ is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over $\mathbb{F}$; (ii) for each near-bipartite Leonard pair over $\mathbb{F}$ we describe its bipartite contraction; (iii) for each bipartite Leonard pair over $\mathbb{F}$ we describe its near-bipartite expansions. Our classification (i) is summarized as follows. We identify two families of Leonard pairs, said to have Krawtchouk type and dual $q$-Krawtchouk type. A Leonard pair of dual $q$-Krawtchouk type is said to be reinforced whenever $q^{2i} \neq -1$ for $1 \leq i \leq d-1$. A Leonard pair $A,A^*$ is said to be essentially bipartite whenever the flat part of $A$ is a scalar multiple of the identity. Assuming $\mathbb{F}$ is algebraically closed, we show that a Leonard pair $A,A^*$ over $\mathbb{F}$ with $d \geq 3$ is near-bipartite if and only if at least one of the following holds: (i) $A,A^*$ is essentially bipartite; (ii) $A,A^*$ has reinforced dual $q$-Krawtchouk type; and (iii) $A,A^*$ has Krawtchouk type.

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Dual polar graphs, the quantum algebra Uq(sl2), and Leonard systems of dual q-Krawtchouk type

Cited by 24 publications

References 23 publications

Totally bipartite tridiagonal pairs

Totally bipartite tridiagonal pairs

Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)

Near-bipartite Leonard pairs

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