2005/08/22 by Brian Hartwig, Hartwig, Brian · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #graph theory and CDMA systems #math.AC #math.CO #math.RT
paper · pdf · doi:10.48550/arxiv.math/0508415
19 pages. To be published in Linear Algebra and it Applications
arxiv created 2005/08/22 · openalex publication_date 2005/08/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (A,B) and (C,D) denote Leonard pairs on V. We say these pairs are adjacent whenever each basis for V which is standard for (A,B) (resp. (C,D)) is split for (C,D) (resp. (A,B)). Our main results are as follows: Theorem 1. There exists at most 3 mutually adjacent Leonard pairs on V provided the dimension of V is at least 2. Theorem 2. Let (A,B), (C,D), and (E,F) denote three mutually adjacent Leonard pairs on V. There for each of these pairs, the eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Theorem 3. Let (A,B) denote a Leonard pair on V whose eigenvalue sequence and dual eigenvalue sequence are in arithmetic progression. Then there exist Leonard pairs (C,D) and (E,F) on V such that (A,B), (C,D), and (E,F) are mutually adjacent.