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

Abstract We consider a 2-homogeneous bipartite distance-regular graph $$\Gamma $$ Γ with diameter $$D \ge 3$$ D ≥ 3 . We assume that $$\Gamma $$ Γ is not a hypercube nor a cycle. We fix a Q -polynomial ordering of the primitive idempotents of $$\Gamma $$ Γ . This Q -polynomial ordering is described using a nonzero parameter $$q \in \mathbb {C}$$ q ∈ C that is not a root of unity. We investigate $$\Gamma $$ Γ using an $$S_3$$ S 3 -symmetric approach. In this approach one considers $$V^{\otimes 3} = V \otimes V \otimes V$$ V ⊗ 3 = V ⊗ V ⊗ V where V is the standard module of $$\Gamma $$ Γ . We construct a subspace $$\Lambda $$ Λ of $$V^{\otimes 3}$$ V ⊗ 3 that has dimension $$\left( {\begin{array}{c}D+3\\ 3\end{array}}\right) $$ D + 3 3 , together with six linear maps from $$\Lambda $$ Λ to $$\Lambda $$ Λ . Using these maps we turn $$\Lambda $$ Λ into an irreducible module for the nonstandard quantum group $$U^\prime _q(\mathfrak {so}_6)$$ U