Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: The Multiplicities of a Dual-thin Q-polynomial Association Scheme. | The Multiplicities of a Dual-thin Q-polynomial Association Scheme Bruce E. Sagan Department of Mathematics Michigan State University East Lansing MI 44824-1027 sagan@ and John S. Caughman IV Department of Mathematical Sciences Portland State University P. O. Box 751 Portland OR 97202-0751 caughman@ Submitted June 23 2000 Accepted January 28 2001. MR Subject Classification 05E30 Abstract Let Y X Ri o i D denote a symmetric association scheme and assume that Y is Q-polynomial with respect to an ordering E0 . ED of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities mi 0 i D of Y is unimodal. Talking to Terwilliger Stanton made the related conjecture that mi mi 1 and mi mD-i for i D 2. We prove that if Y is dual-thin in the sense of Terwilliger then the Stanton conjecture is true. 1 Introduction For a general introduction to association schemes we refer to 1 2 5 or 9 . Our notation follows that found in 3 . Throughout this article Y X Ri o i D will denote a symmetric D-class association scheme. Our point of departure is the following well-known result of Taylor and Levingston. Theorem. 7 If Y is P-polynomial with respect to an ordering R0 . RD of the associate classes then the corresponding sequence of valencies ko k1 . kD THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 N4 1 is unimodal. Furthermore ki ki 1 and ki kD-i for i D 2. Indeed the sequence is log-concave as is easily derived from the inequalities bi_1 bi and ci ci 1 0 i D which are satished by the intersection numbers of any F-polynomial scheme cf. 5 p. 199 . In their book on association schemes Bannai and Ito made the dual conjecture. Conjecture. 1 p. 205 If Y is Q-polynomial with respect to an ordering E0 . ED of the primitive idempotents then the corresponding sequence of multiplicities mo mi . mD is unimodal. Bannai and Ito further remark that although unimodality of the multiplicities follows easily whenever the dual .
