On thin irreducible T-modules with endpoint 1

• Main Author: Baustista, Paolo Lorenzo Y.
• Format: text
• Language: English
• Published: Animo Repository 2011
• Subjects: Irreducible polynomials; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_masteral/6883
• Institution: De La Salle University

Consider a distance-regular graph Γ = (X, R) with D ≥ 3 and adjacency matrix A. The subalgebra of M atX(C) generated by A is called the Bose-Mesner algebra M of Γ. Fix a vertex x ∈ X. Let E ∗ 0 , . . . , E∗ D denote the dual primitive idempotents of Γ with respect to x. The subalgebra of M atX(C) generated by A, E∗ 0 , . . . , E∗ D is called the subconstituent algebra or Terwilliger algebra of Γ with respect to x and denoted by T. Let V = C X 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 6 = v ∈ E ∗ 1V such that hv, s1i = 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.