The Classification of Leonard Triples of Bannai/Ito Type with Even Diameters

Abstract

Let 𝕂 denote an algebraically closed field of characteristic zero. Let V denote a vector space over 𝕂 with finite positive dimension. By a Leonard triple on V, we mean an ordered triple of linear transformations A, A*, A ^ϵ in End(V) such that for each B ∈ {A, A*, A ^ϵ} there exists a basis for V with respect to which the matrix representing B is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. The diameter of the Leonard triple (A, A*, A ^ϵ) is defined to be one less than the dimension of V. In this paper we define a family of Leonard triples said to be Bannai/Ito type and classify these Leonard triples with even diameters up to isomorphism. Moreover, we show that each of them satisfies the ℤ₃-symmetric Askey–Wilson relations.

Key Words:

• Leonard pair
• Leonard system
• Leonard triple
• Bannai/Ito type

2010 Mathematics Subject Classification:

• 15A04
• 33D45

ACKNOWLEDGMENT

The authors would like to thank the referee for many valuable comments and useful suggestions. The authors are also grateful to professor P. Terwilliger and professor T. Ito for the advice they offered during their study of the q-tetrahedron algebra.

Notes

Communicated by M. Cohen.