The shape of a tridiagonal pair
By Tatsuro Ito, Paul M. Terwilliger. 2003
87 citations recorded by the source corpus.
Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension and let A,A∗ denote a tridiagonal pair on V. We make an assumption about this pair. Let q denote a nonzero scalar in K that is not a root of unity. We assume A and A∗ satisfy the q-Serre relations A3A∗−[3]A2A∗A+[3]AA∗A2−A∗A3=0, A∗3A−[3]A∗2AA∗+[3]A∗AA∗2−AA∗3=0, where [3]=(q3−q−3)/(q−q−1). Let (ρ0,ρ1,…,ρd) denote the shape vector for A,A∗. We show the entries in this shape vector are bounded above by binomial coefficients as follows: ρi⩽di(0⩽i⩽d). We obtain this result by displaying a spanning set for V.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/8d75f2f4135a8ee2984bf0b5e8c08d788f09d836
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In Algebra and Number Theory
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Citations in the corpus
- An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials (2006, unknown): “See [46] for some partial results on Conjecture 9.”
- 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): “See [46] for some partial results on Conjecture 34.”
- LEONARD PAIRS AND THE ASKEY-WILSON RELATIONS (2003, methodology): “ determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey-Wilson relations. 1 Introduction We begin by recalling the notion of a Leonard pair [16, 17, 29, 30, 31, 32, 33, 34, 35, 36, 37]. We will use the following terms. Let X denote a square”
- The Tetrahedron algebra and its finite-dimensional irreducible modules (2006, background): “We now recall the concept of a tridiagonal pair [2], [9], [10], [11].”
- Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair (2006, unknown): “See [1], [2], [4], [5], [6], [7], [8], [10], [11], [20], [29] for related topics.”
- 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.”
- Finite-Dimensional Irreducible Modules for the Three-Point 𝔰𝔩2 Loop Algebra (2007, background): “display a nondegenerate symmetric bilinear form h,iσ on V such 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”
- TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS (2005, methodology): “10 A basis for Aq In this section we cite some results from [25] that we will use in our proof of Theorem 1.”
- Tridiagonal pairs of q-Racah type (2008, background): “cases [ 22 ,24,27,28]. The TD pair A, A∗ is called sharp whenever ρ0 = 1. By [45, Theorem 1.3], if F is algebraically closed then A, A∗ is sharp.”
- The augmented tridiagonal algebra (2009, methodology): “3 by using the triangular decomposition of L together with the basis of AIII given in [4].”