Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

By Kazumasa Nomura, Paul M. Terwilliger. 2006

58 citations recorded by the source corpus.

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A∗ : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A∗ is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A∗ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by I,A,A∗,AA∗,A∗A and these elements form a basis for X provided the dimension of V is at least 3.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/0ba6361de550e253100865372cf6d385e9a2bbb7

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In Computational Theory and Mathematics

1. Tatsuro Ito
2. Hau-wen Huang
3. Paul M. Terwilliger
4. Plamen Iliev
5. Alexei S. Zhedanov
6. Luc Vinet
7. Raimundas Vidunas
8. 伊藤 達郎

In Discrete Mathematics and Combinatorics

1. Paul M. Terwilliger
2. Tatsuro Ito
3. Hau-wen Huang
4. Raimundas Vidunas
5. B. Hartwig

In Electrical and Electronic Engineering

1. Tatsuro Ito
2. Hau-wen Huang
3. Paul M. Terwilliger
4. Raimundas Vidunas
5. Alexei S. Zhedanov
6. Luc Vinet

Citations in the corpus

• Finite-Dimensional Irreducible Modules for the Three-Point 𝔰𝔩2 Loop Algebra (2007, background): “h that hξ.u,viσ = −hu,σ(ξ).viσ ξ∈ ⊠, u,v∈ V. We remark on our motivation to investigate ⊠. This has to do with the tridiagonal pairs [19], [20], [21], [22] and the closely related Leonard pairs [27], [28], [29], [35], [36], [37], [39], [40]. A Leonard pair is a pair of semisimple linear transformati”
• The classification of Leonard triples of QRacah type (2011, background): “The group generated by the symbols ∗,↓,⇓ subject to the relations (1), (2) is the dihedral group D4.”