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

Let Γ=(X,R)Γ=(X,R) denote a dd-bounded distance-regular graph with diameter d≥3d≥3. A regular strongly closed subgraph of ΓΓ is said to be a subspace of ΓΓ. For x∈Xx∈X, let P(x)P(x) be the set of all subspaces of ΓΓ containing xx. For each i=1,2,…,d−1i=1,2,…,d−1, let Δ0Δ0 be a fixed subspace with diameter d−id−i in P(x)P(x), and let ℒ(d,i)={Δ∈P(x)∣Δ+Δ0=Γ,d(Δ)=d(Δ∩Δ0)+i}∪{0̸}.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)ℒ(d,i) by ordinary inclusion (resp. reverse inclusion), then ℒ(d,i)ℒ(d,i) is a finite poset, denoted by ℒO(d,i)ℒO(d,i) (resp. ℒR(d,i)ℒR(d,i)). In the present paper we show that both ℒO(d,i)ℒO(d,i) and ℒR(d,i)ℒR(d,i) are atomic, and compute their characteristic polynomials.