On the Multiplicities of the Primitive Idempotents of a Q-Polynomial Distance-regular Graph

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. m Let Γ denote a distance-regular graph with diameter D ≥ 3. Suppose Γ is Q -polynomial with respect to the orderingE0 , E1,⋯ , EDof the primitive idempotents. For 0 ≤ i ≤ D, let midenote the multiplicity ofEi . Then (i)mi − 1 ≤ mi (1 ≤ i ≤ D / 2),(ii)mi ≤ mD − i (0 ≤ i ≤ D / 2). ving the above theorem we resolve a conjecture of Dennis Stanton.