Leonard pairs, spin models, and distance-regular graphs

A Leonard pair is an ordered of diagonalizable linear maps on a finite-dimensional vector space, that each act eigenbasis for the other one in irreducible tridiagonal fashion. In present paper we consider type pair, said to have spin. The notion spin model was introduced by V.F.R. Jones construct link invariants. symmetric matrix over C satisfies two conditions, called II and III conditions. It known W contained certain algebra N(W), Nomura algebra. often happens W∈M⊆N(W), where M Bose-Mesner distance-regular graph Γ; this case say Γ affords W. If model, then module every Terwilliger takes form, recently described Caughman, Curtin, Nomura, Wolff. show converse true; if model. We explicitly when has q-Racah type. proof our main result relies heavily theory pairs.