The Norton-balanced condition for Q-polynomial distance-regular graphs

Let Γ denote a Q-polynomial distance-regular graph, with vertex set X and diameter D≥ 3. The standard module V has a basis x x ∈ X, where x denotes column x of the identity matrix I ∈ MatX( C). Let E denote a Q-polynomial primitive idempotent of Γ. The eigenspace $EV$ is spanned by the vectors E x x ∈ X. It was previously known that these vectors satisfy a condition called the balanced set condition. In this paper, we introduce a variation on the balanced set condition called the Norton-balanced condition. The Norton-balanced condition involves the Norton algebra product on $EV$. We define Γ to be Norton-balanced whenever Γ has a Q-polynomial primitive idempotent E such that the set E x x ∈ X is Norton-balanced. We show that Γ is Norton-balanced in the following cases: (i) Γ is bipartite; (ii) Γ is almost bipartite; (iii) Γ is dual-bipartite; (iv) Γ is almost dual-bipartite; (v) Γ is tight; (vi) Γ is a Hamming graph; (vii) Γ is a Johnson graph; (viii) Γ is the Grassmann graph Jq(2D,D); (ix) Γ is a halved bipartite dual-polar graph; (x) Γ is a halved Hemmeter graph; (xi) Γ is a halved hypercube; (xii) Γ is a folded-half hypercube; (xiii) Γ has q-Racah type and affords a spin model. Some theoretical results about the Norton-balanced condition are obtained, and some open problems are given.