Notes on the Leonard system classification

Abstract

Around 2001 we classified the Leonard systems up to isomorphism. The proof was lengthy and involved considerable computation. In this paper we give a proof that is shorter and involves minimal computation. We also give a comprehensive description of the intersection numbers of a Leonard system.

The Norton algebra of a $Q$-polynomial distance-regular graph

“…The eigenvalue θ 2 appears in the above results. By [12,Lemma 19.21] we find that 1 + θ 1 , 1 + θ * 1 are nonzero and…”

“…To verify (11), note that each side is equal to z∈Γ(x) ẑ. To get (12), apply E to each side of (11), and use EA = θ 1 E.…”

“…In ( 16), eliminate the term Ex 0 y using (12). In the resulting equation, solve for E x ⋆ E ŷ and we are done.…”

“…The algebraic structure of a Q-polynomial distance-regular graph can be described using the concept of a Leonard system [11,Definition 1.4]; this concept was motivated by a theorem of D. A. Leonard [1, p. 260], [6]. We refer the reader to [11,12] for the standard notation and basic results about Leonard systems. The equations below are routinely obtained using the formulas in [12,Sections 19,20].…”

“…We refer the reader to [11,12] for the standard notation and basic results about Leonard systems. The equations below are routinely obtained using the formulas in [12,Sections 19,20]. Let Φ denote any Leonard system with diameter d ≥ 2.…”