The shape of a tridiagonal pair

2003/04/17 by Tatsuro Ito, Paul Terwilliger, Ito, Tatsuro +1 · 2 citations
Engineering · Mathematics · #05E35 #17B37 #33C45 #33D45 #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #Manufacturing Process and Optimization #Quantum Algebra (math.QA) #math.QA #msc:05E35 #msc:17B37 #msc:33C45 #msc:33D45

paper · pdf · doi:10.48550/arxiv.math/0304244

17 pages

arxiv created 2003/04/17 · openalex publication_date 2003/04/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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,B denote a tridiagonal pair on V. We make an assumption about this pair. Let q denote a nonzero scalar in K which is not a root of unity. We assume A and B satisfy the q-Serre relations (i) A³B - [3]A²BA + [3]ABA² - BA³=0; (ii) B³A - [3]B²AB + [3]BAB² - AB³=0, where [3]=(q³-q^-3)/(q-q^-1). Let (ρ₀, ρ₁,...,ρ_d) denote the shape vector for A,B. We show the entries in this shape vector are bounded above by binomial coefficients. Indeed we show ρ_i is at most (d \atop i) for 0 ≤ i ≤ d. We obtain this result by displaying a spanning set for V.

Cited by

• Every Q-polynomial distance-regular graph is sharp over ℝ
• Raising and lowering maps for tridiagonal pairs

Related

• On the shape of a tridiagonal pair
• Tridiagonal pairs of shape (1,2,1)
• Tridiagonal pairs of q-Serre type and their linear perturbations
• Tridiagonal pairs of Krawtchouk type
• The structure of a tridiagonal pair