2024/04/14 by Nomura, Kazumasa, Terwilliger, Paul
#05E30 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2404.09346
Let Γ denote a Q-polynomial distance-regular graph, with vertex set X and diameter D≥ 3. The standard module V has a basis \lbrace x \vert x ∈ X\rbrace, where x denotes column x of the identity matrix I ∈ \rm MatX(\mathbb C). Let E denote a Q-polynomial primitive idempotent of Γ. The eigenspace EV is spanned by the vectors \lbrace E x \vert x ∈ 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 Γ to be Norton-balanced whenever Γ has a Q-polynomial primitive idempotent E such that the set \lbrace E x \vert x ∈ X\rbrace 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.