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

Abstract

We consider the Norton algebra associated with a Q-polynomial primitive idempotent of the adjacency matrix for a distance-regular graph. We obtain a formula for the Norton algebra product that we find attractive.

Leonard pairs, spin models, and distance-regular graphs

“…Conjecture 17.7. The article [50] discusses the Norton algebra of a distance-regular graph Γ = (X, R) that is Q-polynomial with respect to a primitive idempotent E. Let x, y ∈ X be distinct. In [50,Theorem 4.4] the Norton algebra product of E x ⋆ E ŷ is expressed as a linear combination of three vectors C(x, y), B(x, y),…”

Norton algebras of the Hamming Graphs via linear characters

“…When Γ belongs to certain important families of distance regular graphs (i.e., the Johnson graphs, Grassmann graphs, dual polar graphs, and hypercube graphs), Levstein, Maldonado and Penazzi [19,21] constructed the eigenspaces from a filtration of vector spaces corresponding to a graded lattice associated with Γ, and derived an explicit formula for the Norton product on the eigenspace of V 1 . Recently Terwilliger [27] obtained a more general formula for Q-polynomial distance-regular graphs. But for i ≥ 2 the Norton algebra structure on V i has not been determined.…”