Journal of Algebraic Combinatorics

, Volume 6, Issue 1, pp 53–58

An Algebra Associated with a Spin Model

  • Kazumasa Nomura

DOI: 10.1023/A:1008644201287

Cite this article as:
Nomura, K. Journal of Algebraic Combinatorics (1997) 6: 53. doi:10.1023/A:1008644201287


To each symmetric n × n matrix W with non-zero complex entries, we associate a vector space N, consisting of certain symmetric n × n matrices. If W satisfies\(\sum\limits_{x = 1}^n {\frac{{W_{a,x} }}{{W_{b,x} }} = n{\delta }_{a,b} } (a,b = 1,...,n),\) then N becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that N is the Bose-Mesner algebra of some association scheme. If W satisfies the star-triangle equation:\(\frac{1}{{\sqrt n }}\sum\limits_{x = 1}^n {\frac{{W_{a,x} W_{b,x} }}{{W_{c,x} }} = \frac{{W_{a,b} }}{{W_{a,c} W_{b,c} }}} (a,b,c = 1,...,n),\) then W belongs to N. This gives an algebraic proof of Jaeger's result which asserts that every spin model which defines a link invariant comes from some association scheme.

spin model association scheme Bose-Mesner algebra 

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Kazumasa Nomura
    • 1
  1. 1.Tokyo Ikashika UniversityKounodai, IchikawaJapan

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