Some matrices associated with the split decomposition for a Q-polynomial distance-regular graph

We consider a Q-polynomial distance-regular graph Γ with vertex set X and diameter D ≥ 3. For μ, ν ∈ {↓, ↑} we define a direct sum decomposition of the standard module V = CX, called the (μ, ν)–split decomposition. For this decomposition we compute the complex conjugate and transpose of the associated primitive idempotents. Now fix b, β ∈ C such that b 6= 1 and assume Γ has classical parameters (D, b, α, β) with α = b− 1. Under this assumption Ito and Terwilliger displayed an action of the q-tetrahedron algebra ⊠q on the standard module of Γ. To describe this action they defined eight matrices in MatX(C), called A, A, B, B, K, K, Φ, Ψ. For each matrix in the above list we compute the transpose and complex conjugate. Using this information we compute the transpose and complex conjugate for each generator of ⊠q on V .