The Norton-Balanced Condition for $Q$-Polynomial Distance-Regular Graphs

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. The standard module $V$ has a basis $\lbrace {\hat x} \vert x \in X\rbrace$, where ${\hat x}$ denotes column $x$ of the identity matrix $I \in {\rm Mat}_X(\mathbb C)$. Let $E$ denote a $Q$-polynomial primitive idempotent of $\Gamma$. The eigenspace $EV$ is spanned by the vectors $\lbrace E {\hat x} \vert x \in X\rbrace$. 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 $\Gamma$ to be Norton-balanced whenever $\Gamma$ has a $Q$-polynomial primitive idempotent $E$ such that the set $\lbrace E {\hat x} \vert x \in X\rbrace$ is Norton-balanced. We show that $\Gamma$ is Norton-balanced in the following cases: (i) $\Gamma$ is bipartite; (ii) $\Gamma$ is almost bipartite; (iii) $\Gamma$ is dual-bipartite; (iv) $\Gamma$ is almost dual-bipartite; (v) $\Gamma$ is tight; (vi) $\Gamma$ is a Hamming graph; (vii) $\Gamma$ is a Johnson graph; (viii) $\Gamma$ is the Grassmann graph $J_q(2D,D)$; (ix) $\Gamma$ is a halved bipartite dual-polar graph; (x) $\Gamma$ is a halved Hemmeter graph; (xi) $\Gamma$ is a halved hypercube; (xii) $\Gamma$ is a folded-half hypercube; (xiii) $\Gamma$ 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.