The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

We consider a bipartite distance-regular graph ΓΓ with diameter D⩾4D⩾4, valency k⩾3k⩾3, intersection numbers bi,cibi,ci, distance matrices AiAi, and eigenvalues θ0>θ1>⋯>θDθ0>θ1>⋯>θD. Let X   denote the vertex set of ΓΓ and fix x∈Xx∈X. Let T=T(x)T=T(x) denote the subalgebra of MatX(C)MatX(C) generated by A,E0*,E1*,…,ED*, where A=A1A=A1 and Ei* denotes the projection onto the iith subconstituent of ΓΓ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra  ) of ΓΓ with respect to x. An irreducible T-module W is said to be thin   whenever dimEi*W⩽1 for 0⩽i⩽D0⩽i⩽D. By the endpoint of W   we mean min{i|Ei*W≠0}. Assume W   is thin with endpoint 2. Observe E2*W is a one-dimensional eigenspace for E2*A2E2*; let ηη denote the corresponding eigenvalue. It is known θ˜1⩽η⩽θ˜d where θ˜1=-1-b2b3(θ12-b2)-1,θ˜d=-1-b2b3(θd2-b2)-1, and d=⌊D/2⌋d=⌊D/2⌋. To describe the structure of W   we distinguish four cases: (i) η=θ˜1; (ii) D   is odd and η=θ˜d; (iii) D   is even and η=θ˜d; (iv) θ˜1<η<θ˜d. We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694–1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W   is D-1-eD-1-e where e=1e=1 in case (iii) and e=0e=0 in case (iv). Let vv denote a nonzero vector in E2*W. We show W   has a basis Eiv(i∈S), where EiEi denotes the primitive idempotent of A   associated with θiθi and where the set S   is {1,2,…,d-1}∪{d+1,d+2,…,D-1}{1,2,…,d-1}∪{d+1,d+2,…,D-1} in case (iii) and {1,2,…,D-1}{1,2,…,D-1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W   has a basis Ei+2*Aiv(0⩽i⩽D-2-e), and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W.