Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition

By Paul M. Terwilliger. 2003

55 citations recorded by the source corpus.

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and A*: V → V that satisfy both conditions 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. Referring to the above Leonard pair, it is known there exists a decomposition of V into a direct sum of one-dimensional subspaces, on which A acts in a lower bidiagonal fashion and A* acts in an upper bidiagonal fashion. This is called the split decomposition. In this paper, we give two characterizations of a Leonard pair that involve the split decomposition.

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

View Paper: https://www.semanticscholar.org/paper/f474e449520af5a5e63114e885e0e817ba92f5fd

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Citations in the corpus

LEONARD PAIRS AND THE ASKEY-WILSON RELATIONS (2003, methodology): “We begin by recalling the notion of a Leonard pair [16, 17, 29, 30, 31, 32, 33, 34, 35, 36, 37].”
An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials (2006, unknown): “These notes are based on the papers [45], [46], [47], [79], [80], [81], [82], [83], [84], [ 85 ], [86], [87], [88].”
TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS (2005, unknown): “A recent topic of research concerns the Leonard pairs [23], [37], [39], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [53], [54] and the closely related tridiagonal pairs [1], [2], [24], [25], [26], [34], [35], [36].”
Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair (2006, unknown): “We refer the reader to [3], [9], [12], [13], [14], [15], [16], [17], [18], [19], [21], [22], [23], [24], [25], [26], [27], [28], [30], [31] for background on Leonard pairs.”
Tridiagonal pairs and the quantum affine algebra Uq(sl2) (代数的組合せ論) (2005, unknown): “…for 0 ≤ i ≤ d. [0∗0]: Apply the present Lemma, row [0∗D], with Vi replaced by Vd−i for 0 ≤ i ≤ d. [D∗0]: Apply the present Lemma, row [0∗D], with Vi replaced by Vd−i and V ∗i replaced by V ∗d−i for 0 ≤ i ≤ d. [D∗D]: Apply the present Lemma, row [0∗D], with V ∗i replaced by V ∗ d−i for 0 ≤ i ≤ d.”
Two linear transformations each tridiagonal with respect to an eigenbasis of the other: An Algebraic approach to the Askey scheme of orthogonal polynomials (2004, unknown): “These notes are based on the papers [45], [46], [47], [79], [80], [81], [82], [83], [84], [ 85 ], [86], [87], [88].”