Posets associated with subspaces in a d-bounded distance-regular graph

AbstractLet Γ=(X,R) denote a d-bounded distance-regular graph with diameter d≥3. A regular strongly closed subgraph of Γ is said to be a subspace of Γ. For x∈X, let P(x) be the set of all subspaces of Γ containing x. For each i=1,2,…,d−1, let Δ0 be a fixed subspace with diameter d−i in P(x), and let ℒ(d,i)={Δ∈P(x)∣Δ+Δ0=Γ,d(Δ)=d(Δ∩Δ0)+i}∪{0̸}. If we define the partial order on ℒ(d,i) by ordinary inclusion (resp. reverse inclusion), then ℒ(d,i) is a finite poset, denoted by ℒO(d,i) (resp. ℒR(d,i)). In the present paper we show that both ℒO(d,i) and ℒR(d,i) are atomic, and compute their characteristic polynomials