Irreducible T-modules with endpoint r, and the Q-polynomial property

• Main Author: Bautista, Paolo Lorenzo Y.
• Format: text
• Published: Animo Repository 2022
• Subjects: Irreducible polynomials; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/faculty_research/5369
• Institution: De La Salle University

Let ɼ = (X, R) denote a distance-regular graph with diameter D ≥ 3. Let A0, ..., AD denote the distance matrices of ɼ, and let M denote the subalgebra of Matx (C) generated by A1. Recall that the distance matrices form a basis for M. Fix a vertex x ϵ X. Let T = T (x) denote the subalgebra of Matx (C) generated by A1 , E0* , ..., E*D, where Ei * denotes the projection onto the ith subconstituent of ɼ with respect to x. We call T the Terwilliger algebra of ɼ 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 I Ei * W ≠ 0}. Let V = CX and endow V with the Hermitian inner product defined by ‹u, v› = ꭒt ū for ꭒ, ꭒ ϵ V. Let s1 denote the vector in V with 1's in the entries labeled by vertices adjacent to x and 0's elsewhere. Let 0 ≠ ꭒ ϵ E1 * V such that < ꭒ, s1> = 0. We have shown that M ꭒ is a thin irreducible T-module with endpoint 1 if and only if the vectors E Ai-1ꭒ are linearly dependent for 1 ≤ i ≤ D = 1. Next we let W be an irreducible T-module that is not thin. Furthermore, suppose W has endpoint 1, dim E1* W = 1 if i ϵ {1, D/2 + 1}, and dim E1 * W = 2 if 2 ≤ D/2. Let ED, E1 ..., ED be a Q-polynomial ordering of the primitive idempotents of ɼ. In this paper, we discuss some of the preliminaries needed in order to investigate a conjecture proposed by Terwilliger, that the vectors, that the vectors Ei* Ai - 1v and Ei* Ai - 1vform a basis for Ei* W for r ≤ i ≤ r + D/2.