Bipartite distance-regular graphs: The -polynomial property and pseudo primitive idempotents

Let Γ denote a bipartite distance-regular graph with diameter at least 4 and valency at least 3. Fix a vertex of Γ and let T denote the corresponding Terwilliger algebra. Suppose that Γ is Q -polynomial and there are two non-isomorphic irreducible T -modules with endpoint 2. We show that, unless the intersection numbers of Γ fit one exceptional case (which is not known to correspond to an actual graph), the entry-wise product of pseudo primitive idempotents associated with these modules is a linear combination of two pseudo primitive idempotents. This result relates to a conjecture of MacLean and Terwilliger.