On the multiplicities of the primitive idempotents of a Q-polynomial distance-regular graph

Abstract

Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem. THEOREM. Let Γ denote a distance-regular graph with diameter D ≥ 3. Suppose Γ is Q-polynomial with respect to the ordering E0, E1,..., ED of the primitive idempotents. For 0 ≤ i ≤ D, let mi, denote the multiplicity of E i. Then (i) mi-1 ≤ mi (1 ≤ i ≤ D/2), (ii) mi ≤ mD-i (0 ≤ i ≤ D/2). By proving the above theorem we resolve a conjecture of Dennis Stanton. © 2002 Elsevier Science Ltd. All rights reserved.