2-Homogeneous bipartite distance-regular graphs and the quantum group $U_q^{\mathrm{\prime }}({\mathfrak{s}\mathfrak{o}}_6)$

We consider a 2-homogeneous bipartite distance-regular graph $\mathrm{\Gamma }$ with diameter $D\ge 3$. We assume that $\mathrm{\Gamma }$ is not a hypercube nor a cycle. We fix a $Q$-polynomial ordering of the primitive idempotents of $\mathrm{\Gamma }$. This $Q$-polynomial ordering is described using a nonzero parameter $q\in \mathbb{C}$ that is not a root of unity. We investigate $\mathrm{\Gamma }$ using an $S_3$-symmetric approach. In this approach one considers $V^{\otimes 3}=V\otimes V\otimes V$ where $V$ is the standard module of $\mathrm{\Gamma }$. We construct a subspace $\mathrm{\Lambda }$ of $V^{\otimes 3}$ that has dimension $\left(\genfrac{}{}{0px}{}{D+3}{3}\right)$, together with six linear maps from $\mathrm{\Lambda }$ to $\mathrm{\Lambda }$. Using these maps we turn $\mathrm{\Lambda }$ into an irreducible module for the nonstandard quantum group $U_q^{\mathrm{\prime }}({\mathfrak{s}\mathfrak{o}}_6)$ introduced by Gavrilik and Klimyk in 1991.