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 Gamma denote a distance-regular graph with diameter D greater than or equal to 3. Suppose Gamma is Q-polynomial with respect to the ordering E-0, E-1,..., E-D of the primitive idempotents. For 0 less than or equal to i less than or equal to D, let m(i) denote the multiplicity of E-i. Then (i) m(i-1) less than or equal to m(i) (1 < i less than or equal to D/2), (ii) m(i) less than or equal to m(D-i) (0 less than or equal to i less than or equal to D/2). By proving the above theorem we resolve a conjecture of Dennis Stanton. (C) 2002 Elsevier Science Ltd. All rights reserved.