2014/08/10 by Nomura, Kazumasa
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1408.2180
Fix an algebraically closed field \F and an integer d ≥ 3. Let V be a vector space over \F with dimension d+1. A Leonard pair on V is a pair of diagonalizable linear transformations A: V → V and A^* : V → V, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars (\þi\i=0d, \þ^*i\i=0d, \\vphii\i=1d, \ϕi\i=1d), called its parameter array. The scalars \þi\i=0d (resp. \þ^*i\i=0d) are mutually distinct, and the expressions (þi-2 - þi+1)/(þi-1-þi), (þ^*i-2 - þ^*i+1)/(þ^*i-1-þ^*i) are equal and independent of i for 2 ≤ i ≤ d-1. Write this common value as β+1. In the present paper, we consider the "end-parameters" þ0, þd, þ^*0, þ^*d, \vphi1, \vphid, ϕ1, ϕd of the parameter array. We show that a Leonard system is determined up to isomorphism by the end-parameters and β. We display a relation between the end-parameters and β. Using this relation, we show that there are up to inverse at most \lfloor (d-1)/2 \rfloor Leonard systems that have specified end-parameters. The upper bound \lfloor (d-1)/2 \rfloor is best possible.