Abstract We consider a 2-homogeneous bipartite distance-regular graph Γ with diameter D 3 D ≥ 3. We assume that Γ is not a hypercube nor a cycle. We fix a Q -polynomial ordering of the primitive idempotents of Γ. This Q -polynomial ordering is described using a nonzero parameter q C q ∈ C that is not a root of unity. We investigate Γ using an S₃ S 3 -symmetric approach. In this approach one considers V^ 3 = V V V V ⊗ 3 = V ⊗ V ⊗ V where V is the standard module of Γ. We construct a subspace Λ of V^ 3 V ⊗ 3 that has dimension (array{cD+3\\ 3array}) D + 3 3, together with six linear maps from Λ to Λ. Using these maps we turn Λ into an irreducible module for the nonstandard quantum group U^ q (so₆) U
Paul Terwilliger (Sat,) studied this question.