Group Association Schemes Research Articles

Amide I absorption band: description of the urethane group association scheme in polyether urethane elastomers

On the strength of analysis of the amide I absorption band contour of the i.r. spectra of polyether urethane elastomers based on poly(tetramethylene oxide), poly(propylene oxide) and 4,4′-diphenylmethane diisocyanate, a scheme of association of urethane groups has been suggested. The amide I band has been shown to include five individual components at 1740, 1730, 1725, 1713 and 1702 cm −1. The isolated bands are considered to be characteristic for urethane groups forming part of various associations.

Association Schemes and Fusion Algebras (An Introduction)

We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition (ST)3 = S 2. We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H(d, q) has the modular invariance property.

Subschemes of some association schemes

By using the character tables of known commutative association schemes, we can, in many instances, construct some new commutative association schemes as their subschemes. In particular, we obtain many new non-group-case association schemes from the group association schemes of finite simple groups. Although examples worked out in this paper are limited to some special cases, it might be possible that this phenomenon occurs universally for all the group association schemes of finite Chevalley groups. The subschemes obtained in this way could be considered as an association scheme version of Weyl groups for Chevalley groups.

Minimal subschemes of the group association schemes of Mathieu groups

Minimal subschemes of the group association schemes of Mathieu groupsM n (n = 11, 12, 22, 23, 24) are determined. It is proved that for eachM n (n = 11, 12, 22, 23, 24), there is a unique minimal subscheme of . The class numbers of these minimal subschemes are 7, 11, 9, 11 and 20 respectively. A general computer program to determine subschemes of group association schemes of relatively small class numbers is discussd.