2014/04/27 by Kazumasa Nomura, Nomura, Kazumasa
Engineering · Mathematics · #05E30 #05E35 #33C45 #33D45 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #graph theory and CDMA systems #math.RA #msc:05E30 #msc:05E35 #msc:33C45 #msc:33D45
paper · pdf · doi:10.48550/arxiv.1404.6794
arxiv created 2014/04/27 · openalex publication_date 2014/04/27 · arxiv updated 2014/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix an algebraically closed field \mathbbF and an integer d ≥ 3. Let Matd+1(\mathbbF) denote the \mathbbF-algebra consisting of the (d+1) × (d+1) matrices that have all entries in \mathbbF. We consider a pair of diagonalizable matrices A,A^* in Matd+1(\mathbbF), each acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in Matd+1(\mathbbF). For a Leonard pair A,A^* there is a nonzero scalar q that is used to describe the eigenvalues of A and A^*. In the present paper we find all Leonard pairs A,A^* in Matd+1(\mathbbF) such that A is lower bidiagonal with subdiagonal entries all 1 and A^* is irreducible tridiagonal, under the assumption that q is not a root of unity. This gives a partial solution of a problem given by Paul Terwilliger.