Nomura algebras of nonsymmetric Hadamard models

Spin models for link invariants were introduced by Jones [7], and their connection to combinatorics was revealed first in [4]. Jaeger and Nomura [6] constructed nonsymmetric spin models for link invariants from Hadamard matrices, and showed that these models give link invariants which depend nontrivially on link orientation. These models are a modification of the Hadamard model originally constructed by Nomura [9]. Jaeger and Nomura also pointed out a similarity between the association scheme of a Hadamard graph and the association scheme containing their new nonsymmetric spin model. Nomura [10], and later Jaeger, Matsumoto and Nomura [5] introduced an algebra called the Nomura algebra of a type II matrix W , and showed that this algebra coincides with the Bose–Mesner algebra of some self-dual association schemes when W is a spin model. By [5] the Nomura algebra of the Hadamard model coincides with the Bose– Mesner algebra of the corresponding Hadamard graph. The purpose of this paper is to determine the Nomura algebra of a nonsymmetric Hadamard model to be the Bose–Mesner algebra of the corresponding directed Hadamard graph. We also show that the directed Hadamard graph can be constructed from the ordinary Hadamard graphs by means of a general method given by Klin, Muzychuk, Pech, Woldar and Zieschang [8].