Leonard pairs, spin models, and distance-regular graphs

Nomura, Kazumasa; Terwilliger, Paul

doi:10.1016/j.jcta.2020.105312

Cited by 11 publications

(7 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Comprehensive treatments can be found in [3,32,36,37,[67][68][69]78]. In addition, there are papers about the thin condition [10,21,24,64,69,82], irreducible T -modules with endpoint one [30], Γ being bipartite [6,13,14,41], Γ being almost-bipartite [9,42], Γ being dual bipartite [22], Γ being almost dual bipartite [23], Γ being 2-homogeneous [15,17,18,53], Γ being tight [56], Γ being a hypercube [27], Γ being a Doob graph [63], Γ being a Johnson graph [49,62], Γ being a Grassmann graph [48], Γ being a dual polar graph [84], Γ having a spin model in the Bose-Mesner algebra [16,52]. Some miscellaneous topics about irreducible T -modules can be found in [26,34,35,40,43,44,55,59,60,…”

Section: Irreducible T -Modules and Tridiagonal Pairsmentioning

confidence: 99%

“…Let h be given, and pick z ∈ X such that ∂(x, z) = h. We compute the (x, z)-entry of each term in (53). We do this using Lemma 19.1 (with ℓ = 1 and y = x) along with (52). A brief calculation yields p h i,j (θ * i − θ * j ) = r h i,j (θ * 0 − θ * h ), and (54) follows.…”

Section: The Tridiagonal Relationsmentioning

confidence: 99%

“…We assume that E is nondegenerate, so θ * n = θ * 0 for 1 ≤ n ≤ d. Claim 1. Pick an integer h (1 ≤ h ≤ D) and y, z ∈ X such that ∂(y, z) = h. Then For each term in the above equation, take the inner product with E ŷ using (52). A brief calculation yields…”

Section: The Tridiagonal Relationsmentioning

confidence: 99%

See 2 more Smart Citations

Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property

Terwilliger¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Irreducible T -Modules and Tridiagonal Pairsmentioning

confidence: 99%

Section: The Tridiagonal Relationsmentioning

confidence: 99%

See 1 more Smart Citation

Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property

Terwilliger¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…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 [4,21,22,25,29,39], quantum groups [1, 2, 12-14, 26, 28, 30], spin models [16][17][18]41], double affine Hecke algebras [23,24,32,33,40], partially ordered sets [35,36,45,55], and exactly solvable models in statistical mechanics [5][6][7][8][9][10][11]. For more information about Leonard pairs and related topics, see [39,42,44,46,48].…”

Section: Introductionmentioning

confidence: 99%

Compatibility and companions for Leonard pairs

Nomura¹,

Terwilliger²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let F denote a field, and let V denote a vector space over F with finite positive dimension. A Leonard pair on V is an ordered pair of diagonalizable F-linear maps A : V → V and A * : V → V that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs A, A * and B, B * on V are said to be compatible wheneverFor a Leonard pair A, A * on V , by a companion of A, A * we mean an F-linear map K : V → V such that K is a polynomial in A * and A − K, A * is a Leonard pair on V . The concepts of compatibility and companion are related as follows. For compatible Leonard pairs A, A * and B, B * on V , define K = A − B. Then K is a companion of A, A * . For a Leonard pair A, A * on V and a companion K of A, A * , define B = A − K and B * = A * . Then B, B * is a Leonard pair on V that is compatible with A, A * . Let A, A * denote a Leonard pair on V . We find all the Leonard pairs B, B * on V that are compatible with A, A * . For each solution B, B * we describe the corresponding companion K = A − B.

show abstract

“…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 [25,39,4,21,22,29], quantum groups [1,12,28,30,13,2,14,26], spin models [17,41,18,16], double affine Hecke algebras [40,23,24,32,33], partially ordered sets [35,45,55,36], and exactly solvable models in statistical mechanics [5,6,7,8,9,10,11]. For more information about Leonard pairs and related topics, see [48,46,39,42,44].…”

mentioning

confidence: 99%

Compatibility and companions for Leonard pairs

Nomura

Terwilliger²

2022

ELA

Self Cite

View full text Add to dashboard Cite

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. 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 in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r$. For a Leonard pair $A,A^*$ on $V$, by a companion of $A,A^*$ we mean an $\mathbb{F}$-linear map $K: V \to V$ such that $K$ is a polynomial in $A^*$ and $A-K, A^*$ is a Leonard pair on $V$. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs $A,A^*$ and $B,B^*$ on $V$, define $K = A-B$. Then $K$ is a companion of $A,A^*$. For a Leonard pair $A,A^*$ on $V$ and a companion $K$ of $A,A^*$,define $B = A-K$ and $B^* = A^*$. Then $B,B^*$ is a Leonard pair on $V$ that is compatible with $A,A^*$. Let $A,A^*$ denote a Leonard pair on $V$. We find all the Leonard pairs $B, B^*$ on $V$ that are compatible with $A,A^*$.For each solution $B, B^*$, we describe the corresponding companion $K = A-B$.

show abstract

Leonard pairs, spin models, and distance-regular graphs

Cited by 11 publications

References 48 publications

Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property

Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property

Compatibility and companions for Leonard pairs

Compatibility and companions for Leonard pairs

Contact Info

Product

Resources

About