There are finitely many Q-polynomial association schemes with given first multiplicity at least three

  • Authors:

  • William J. Martin;Jason S. Williford

  • Affiliations:

  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States;Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA, United States

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2009

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Abstract

In this paper, we will prove a result which is formally dual to the long-standing conjecture of Bannai and Ito which claims that there are only finitely many distance-regular graphs of valency k for each k2. That is, we prove that, for any fixed m"12, there are only finitely many cometric association schemes (X,R) with the property that the first idempotent in a Q-polynomial ordering has rank m"1. As a key preliminary result, we show that the splitting field of any such association scheme is at most a degree two extension of the rationals. All of the tools involved in the proof are fairly elementary yet have wide applicability. To indicate this, a more general theorem is proved here with the result alluded to in the title appearing as a corollary to this theorem.