On bipartite QQ-polynomial distance-regular graphs

Let ΓΓ denote a bipartite QQ-polynomial distance-regular graph with vertex set XX, diameter d≥3d≥3 and valency k≥3k≥3. Let RXRX denote the vector space over RR consisting of column vectors with entries in RR and rows indexed by XX. For z∈Xz∈X, let zˆ denote the vector in RXRX with a 1 in the zz-coordinate, and 0 in all other coordinates. Fix x,y∈Xx,y∈X such that ∂(x,y)=2∂(x,y)=2, where ∂∂ denotes the path-length distance. For 0≤i,j≤d0≤i,j≤d define wij=∑zˆ, where the sum is over all z∈Xz∈X such that ∂(x,z)=i∂(x,z)=i and ∂(y,z)=j∂(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 MM is the Bose–Mesner algebra of ΓΓ. We observe that MWMW is the minimal AA-invariant subspace of RXRX which contains WW, where AA is the adjacency matrix of ΓΓ. We display a basis for MWMW that is orthogonal with respect to the dot product. We give the action of AA on this basis. We show that the dimension of MWMW is 3d−33d−3 if ΓΓ is 2-homogeneous, 3d−13d−1 if ΓΓ is the antipodal quotient of the 2d2d-cube, and 4d−44d−4 otherwise. We obtain our main result using Terwilliger’s “balanced set” characterization of the QQ-polynomial property.