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 E 0 , E 1 ,⋯ , E D of the primitive idempotents. For 0 ≤ i ≤ D , let m i denote the multiplicity of E i . Then (i) m i − 1 ≤ m i (1 ≤ i ≤ D / 2), (ii) m i ≤ m D − i (0 ≤ i ≤ D / 2) . By proving the above theorem we resolve a conjecture of Dennis Stanton.