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.Theorem Let Γ denote a distance-regular graph with diameter D ≥  3. Suppose Γ is Q -polynomial with respect to the orderingE₀ , E₁, , E_Dof the primitive idempotents. For 0  ≤  i ≤  D, let m_idenote the multiplicity ofE_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.