On bipartite Q-polynomial distance-regular graphs

Let Γ denote a bipartite Q-polynomial distance-regular graph with vertex set X, diameter d ≥ 3 and valency k ≥ 3. Let RX denote the vector space over R consisting of column vectors with entries in R and rows indexed by X. For z ∈ X, let ẑ denote the vector in RX with a 1 in the z-coordinate, and 0 in all other coordinates. Fix x, y ∈ X such that ∂(x, y) = 2, where ∂ denotes path-length distance. For 0 ≤ i, j ≤ d we define wij = ∑ ẑ, where the sum is over all z ∈ X such that ∂(x, z) = i and ∂(y, z) = j. We define W = span{wij | 0 ≤ i, j ≤ d}. In this paper we consider the space MW = span{mw | m ∈ M,w ∈ W}, where M is the Bose-Mesner algebra of Γ. We observe MW is the minimal A-invariant subspace of RX which contains W , where A is the adjacency matrix of Γ. We display a basis for MW that is orthogonal with respect to the dot product. We give the action of A on this basis. We show that the dimension of MW is 3d− 3 if Γ is 2-homogeneous, 3d − 1 if Γ is the antipodal quotient of the 2d-cube, and 4d − 4 otherwise. We obtain our main result using Terwilliger’s “balanced set” characterization of the Q-polynomial property.