On thin irreducible t-modules with endpoint 1

• Main Authors: Bautista, Paolo Lorenzo Y., Pascasio, Arlene A.
• Format: text
• Published: Animo Repository 2012
• Subjects: Graph theory; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/faculty_research/11220
• Institution: De La Salle University

Consider a distance-regular graph Γ = (X, R) with D ≥ 3 and adjacency matrix A. The subalgebra of MatX (C) generated by A is called the Bose-Mesner algebra M of Γ. Fix a vertex x ∈ X. Let E0∗, . . . , E∗ denote the dual primitive idempotents of Γ with respect to x. The subalgebra of MatX (C) generated by A, E0∗, . . . , E∗ is called the subconstituent algebra or Terwilliger algebra of Γ with respect to x and denoted by T . Let V = CX be the standard module of Γ with the usual Hermitian inner product. Define s1 ∈ V to be the vector with 1’s in the entries labeled by vertices adjacent to x and 0’s elsewhere. Let 0 = v ∈ E1∗V such that v, s1 = 0. Go and Terwilliger were able to show in [Europ. J. Combinatorics, 23, (2002),793-816] that the space Mv is of dimension D − 1 or D. They then showed that Mv is a thin irreducible T -module with endpoint 1 when the dimension of Mv is D−1. In this paper, we consider the case when Mv has dimension D, and show a necessary and sufficient condition for Mv to be a thin irreducible T -module with endpoint 1.