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

Abstract

We consider a $Q$-polynomial distance-regular graph $\Gamma$ with vertex set $X$ and diameter $D \geq 3$. For $\mu, \nu \in \lbrace \downarrow, \uparrow \rbrace$ we define a direct sum decomposition of the standard module $V=\C X$, 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 $b \neq 1$ and assume $\Gamma$ 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 $\boxtimes_q$ on the standard module of $\Gamma$. To describe this action they defined eight matrices in $\hbox{Mat}_X(\mathbb C)$, 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 $\boxtimes_q$ on $V$.