2-Homogeneous bipartite distance-regular graphs and the quantum group $U^\prime_q(\mathfrak{so}_6)$

We consider a 2-homogeneous bipartite distance-regular graph $\Gamma$ with diameter $D \geq 3$. We assume that $\Gamma$ is not a hypercube nor a cycle. We fix a $Q$-polynomial ordering of the primitive idempotents of $\Gamma$. This $Q$-polynomial ordering is described using a nonzero parameter $q \in \mathbb C$ that is not a root of unity. We investigate $\Gamma$ 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 $\Gamma$. We construct a subspace $\Lambda$ of $V^{\otimes 3}$ that has dimension $\binom{D+3}{3}$, together with six linear maps from $\Lambda$ to $\Lambda$. Using these maps we turn $\Lambda$ into an irreducible module for the nonstandard quantum group $U^\prime_q(\mathfrak{so}_6)$ introduced by Gavrilik and Klimyk in 1991.