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

Let 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 Δ 0 be a fixed subspace with diameter d − i in P ( x ) , and let ℒ ( d , i ) = { Δ ∈ P ( x ) ∣ Δ + Δ 0 = Γ , d ( Δ ) = d ( Δ ∩ Δ 0 ) + 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 ℒ O ( d , i ) (resp. ℒ R ( d , i ) ). In the present paper we show that both ℒ O ( d , i ) and ℒ R ( d , i ) are atomic, and compute their characteristic polynomials.