Notes on the Leonard System Classification

Terwilliger, Paul

doi:10.1007/s00373-021-02357-y

Cited by 13 publications

(9 citation statements)

References 22 publications

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“…In (6.7), eliminate γ and ϱ using Lemma 6.21 and rearrange the terms. Compatibility and companions for Leonard pairs Lemma 6.26 (See [54,Lemma 19.13]). Referring to Definition 6.16,…”

Section: Tridiagonal Matrices and Diagonal Equivalencementioning

confidence: 99%

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Compatibility and companions for Leonard pairs

Nomura

Terwilliger²

2022

ELA

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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 $K = A-B$.

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“…In (6.7), eliminate γ and ϱ using Lemma 6.21 and rearrange the terms. Compatibility and companions for Leonard pairs Lemma 6.26 (See [54,Lemma 19.13]). Referring to Definition 6.16,…”

Section: Tridiagonal Matrices and Diagonal Equivalencementioning

confidence: 99%

“…We are using the description in [54,Theorem 18.1]. For the above parameter array, the sequence {θ i } d i=0 (resp.…”

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confidence: 99%

Compatibility and companions for Leonard pairs

Nomura

Terwilliger²

2022

ELA

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“…The classification shows that the orthogonal polynomial sequences that satisfy Askey-Wilson duality belong to the terminating branch of the Askey scheme; this branch consists of the q-Racah polynomials [1] along with their limiting cases [39]. The theory of Leonard pairs [29,71,73,74,76,78] provides a modern approach to Askey-Wilson duality.…”

Section: Our Next Goal Is To Show That a *mentioning

confidence: 99%

“…It is an ongoing project to describe the irreducible T -modules for a Q-polynomial distanceregular graph Γ. 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].…”

Section: Irreducible T -Modules and Tridiagonal Pairsmentioning

confidence: 99%

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

Terwilliger¹

2022

Preprint

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“…In (15), eliminate γ and ̺ using Lemma 6.21, and rearrange the terms. ✷ Lemma 6.26 (See [54,Lemma 19.13]. ) Referring to Definition 6.16, (i) The parameter array of…”

Section: Leonard Pairs and Leonard Systemsmentioning

confidence: 99%

Compatibility and companions for Leonard pairs

Nomura¹,

Terwilliger²

2021

Preprint

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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.

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Notes on the Leonard System Classification

Cited by 13 publications

References 22 publications

Compatibility and companions for Leonard pairs

Compatibility and companions for Leonard pairs

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

Compatibility and companions for Leonard pairs

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