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

Let 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. 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 to be Norton-balanced whenever has a Q-polynomial primitive idempotent E such that the set E x x X is Norton-balanced. We show that is Norton-balanced in the following cases: (i) is bipartite; (ii) is almost bipartite; (iii) is dual-bipartite; (iv) is almost dual-bipartite; (v) is tight; (vi) is a Hamming graph; (vii) is a Johnson graph; (viii) is the Grassmann graph Jq (2D, D) ; (ix) is a halved bipartite dual-polar graph; (x) is a halved Hemmeter graph; (xi) is a halved hypercube; (xii) is a folded-half hypercube; (xiii) 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.