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

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Abstract

We consider a bipartite distance-regular graph Γ with diameter D ≥ 4, valency k ≥ 3, intersection numbers bi, ci, distance matrices Ai, and eigenvalues θ0 > θ1 > ⋯ > θD. Let X denote the vertex set of Γ and fix x ∈ X. Let T = T (x) denote the subalgebra of MatX (C) generated by A, E0*, E1*, ..., ED*, where A = A1 and Ei* denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim Ei* W ≤ 1 for 0 ≤ 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* A2 E2*; let η denote the corresponding eigenvalue. It is known over(θ, ̃)1 ≤ η ≤ over(θ, ̃)d where over(θ, ̃)1 = - 1 - b2 b3 (θ12 - b2)- 1, over(θ, ̃)d = - 1 - b2 b3 (θd2 - b2)- 1, and d = ⌊ D / 2 ⌋. To describe the structure of W we distinguish four cases: (i) η = over(θ, ̃)1; (ii) D is odd and η = over(θ, ̃)d; (iii) D is even and η = over(θ, ̃)d; (iv) over(θ, ̃)1 < η < over(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 - e where e = 1 in case (iii) and e = 0 in case (iv). Let v denote a nonzero vector in E2* W. We show W has a basis Ei v (i ∈ S), where Ei denotes the primitive idempotent of A associated with θi and where the set S is { 1, 2, ..., d - 1 } ∪ { d + 1, d + 2, ..., D - 1 } in case (iii) and { 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* Ai v (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. © 2007 Elsevier B.V. All rights reserved.

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MacLean, M. S., & Terwilliger, P. (2008). The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two. Discrete Mathematics, 308(7), 1230–1259. https://doi.org/10.1016/j.disc.2007.03.071

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