2021/12/26 by Nomura, Kazumasa, Terwilliger, Paul
#05E30 #15A21 #15B10 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2112.13326
In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let \mathbbF denote a field, and let V denote a vector space over \mathbbF with finite positive dimension. A Leonard pair on V is an ordered pair of diagonalizable \mathbbF-linear maps A : V → V and A^* : V → V that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs A,A^* and B,B^* on V are said to be compatible whenever A^* = B^* and [A,A^*] = [B,B^*], where [r,s] = r s - s r. For a Leonard pair A,A^* on V, by a companion of A,A^* we mean an \mathbbF-linear map K: V → V such that K is a polynomial in A^* and A-K, A^* is a Leonard pair on V. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs A,A^* and B,B^* on V, define K = A-B. Then K is a companion of A,A^*. For a Leonard pair A,A^* on V and a companion K of A,A^*, define B = A-K and B^* = A^*. Then B,B^* is a Leonard pair on V that is compatible with A,A^*. Let A,A^* denote a Leonard pair on V. We find all the Leonard pairs B, B^* on V that are compatible with A,A^*. For each solution B, B^* we describe the corresponding companion K = A-B.