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

We consider a QQ-polynomial distance-regular graph ΓΓ with vertex set XX and diameter D≥3D≥3. For μ,ν∈{↓,↑}μ,ν∈{↓,↑} we define a direct sum decomposition of the standard module V=CXV=CX, called the (μ,ν)(μ,ν)-split decomposition. For this decomposition we compute the complex conjugate and transpose of the associated primitive idempotents. Now fix b,β∈Cb,β∈C such that b≠1b≠1 and assume ΓΓ has classical parameters (D,b,α,β)(D,b,α,β) with α=b−1α=b−1. Under this assumption Ito and Terwilliger displayed an action of the qq-tetrahedron algebra ⊠q⊠q on the standard module of ΓΓ. 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⊠q on VV.