2-Homogeneous bipartite distance-regular graphs and the quantum group U^ q(so₆)

We consider a 2-homogeneous bipartite distance-regular graph Γ Γ with diameter D ≥ 3 D ≥ 3 . We assume that Γ Γ is not a hypercube nor a cycle. We fix a Q -polynomial ordering of the primitive idempotents of Γ 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 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 Γ . We construct a subspace Λ Λ of V⊗ 3 V ⊗ 3 that has dimension ( arraycD+3\\ 3array) D + 3 3 , together with six linear maps from Λ Λ to Λ Λ . Using these maps we turn Λ Λ into an irreducible module for the nonstandard quantum group U^ q(so₆) U