A characterization of Leonard pairs using the parameters \a_i\_i=0^d

2012/05/19 by Edward Hanson, Hanson, Edward
Computer Science · Engineering · Mathematics · #05E30 (Secondary) #15A21 (Primary) #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Matrix Theory and Algorithms #Polynomial and algebraic computation #Rings and Algebras (math.RA) #graph theory and CDMA systems #math.RA #msc:05E30 #msc:15A21

paper · pdf · doi:10.48550/arxiv.1205.4368

21 pages. arXiv admin note: substantial text overlap with arXiv:0911.0098

openalex publication_date 2012/05/19 · arxiv created 2012/05/20 · arxiv updated 2012/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V denote a vector space with finite positive dimension. We consider an ordered 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. Arlene Pascasio recently obtained a characterization of the Q-polynomial distance-regular graphs using the intersection numbers a_i. In this paper, we extend her results to a linear algebraic level and obtain a characterization of Leonard pairs. Pascasio's argument appears to rely on the underlying combinatorial assumptions, so we take a different approach that is algebraic in nature.

Citations

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
• A characterization of Leonard pairs using the notion of a tail

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