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

We consider a $Q$-polynomial distance-regular graph $\mathrm{\Gamma }$ with vertex set $X$ and diameter $D\ge 3$. For $\mu, \nu \in \lbrace \downarrow, \uparrow \rbrace$wedefineadirectsumdecompositionofthestandardmodule$V=\C X$, called the $(\mu ,\nu )$–split decomposition. For this decomposition we compute the complex conjugate and transpose of the associated primitive idempotents. Now fix $b,\beta \in \mathbb{C}$ such that $b\mathrm{\ne }1$ and assume $\mathrm{\Gamma }$ has classical parameters $(D,b,\alpha ,\beta )$ with $\alpha =b-1$. Under this assumption Ito and Terwilliger displayed an action of the $q$-tetrahedron algebra ${⊠}_q$ on the standard module of $\mathrm{\Gamma }$. To describe this action they defined eight matrices in ${\text{Mat}}_X(\mathbb{C})$, called \begin{eqnarray*} \label{eq:list} A,\quad A^,\quad B,\quad B^, \quad K,\quad K^,\quad \Phi,\quad \Psi. \end{eqnarray} 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 $V$.