Bose-Mesner Algebras Related to Type II Matrices and Spin Models

  • Authors:

  • François Jaeger;Makoto Matsumoto;Kazumasa Nomura

  • Affiliations:

  • Laboratoire Leibniz, Institut IMAG,BP 53 38041 Grenoble cedex 9, France;Faculty of Science and Technology, Keio University, Hiyoshi, Kohoku-ku, Yokohama, 223 Japan. E-mail: matsumoto@math.keio.ac.jp;College of Liberal Arts and Sciences, Tokyo Medical and Dental University, Kohnodai, Ichikawa, 272 Japan. E-mail: nomura@tmd.ac.jp

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 1998

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Abstract

A type II matrix is a square matrixW with non-zero complex entries such that the entrywisequotient of any two distinct rows of W sums to zero. Hadamardmatrices and character tables of abelian groups are easy examples, andother examples called spin models and satisfying anadditional condition can be used as basic data to construct invariants oflinks in 3-space. Our main result is the construction, for every type IImatrix W, of a Bose-Mesner algebraN(W), which is a commutative algebra of matrices containing theidentity I, the all-one matrix J, closed undertransposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). Thetransposition of matrices W corresponds to a classical notionof duality for the corresponding Bose-Mesner algebrasN(W). Every Bose-Mesner algebra encodes a highly regularcombinatorial structure called an association scheme,and we give an explicit construction of this structure. This allows us tocompute N(W) for a number of examples.