Spin models constructed from Hadamard matrices

Abstract

A spin model (for link invariants) is a square matrix W which satisfies certain axioms. For a spin model W , it is known that W T W −1 is a permutation matrix, and its order is called the index of W. F. Jaeger and K. Nomura found spin models of index 2, by modifying the construction of symmetric spin models from Hadamard matrices. The aim of this paper is to give a construction of spin models of an arbitrary even index from any Hadamard matrix. In particular, we show that our spin models of indices a power of 2 are new.

Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of François Jaeger (a survey)

“…This also gave a motivation for Nomura to construct his spin models from Hadamard graphs (Cf. [38].) Further developments in the study of spin models will be seen in the survey papers by Jaeger [30] and Nomura [40], and so we will not discuss these topics here.…”

Untitled

“…The values of P i jk can be determined in a similar way as in [29]. The non-zero values are given as P 0213 = P 2031 = P 1302 = P 1324 = P 3120 = P 3142 = P 2413 = P 4231 = 1, P 1322 = P 3122 = P 2213 = P 2231 = 2m − 2, P 1122 = P 2211 = P 2233 = P 3322 = 2m.…”

“…It is shown in [29] (see also [30] for an alternative proof) that W + = W satisfies (with appropriate W − , a and D) the invariance Eqs. (15)- (17), and the corresponding invariant of links is determined in [18,19].…”

“…This allows us to study effectively the tensor product construction for type II matrices, and a number of examples: character tables of abelian groups, Hadamard matrices of size 16, type II matrices of size 4, spin models for the Kauffman polynomial of links (see [17]), and the spin models defined on Hadamard graphs by the third author (see [29]). …”

A Property of Solutions of Modular Invariance Equations for Distance-Regular Graphs

“…In fact, some interesting spin models can be constructed on association schemes with P-polynomial property (e.g. the HigmanSims graph [7] and the Hadamard graphs [10], see also [4] for more examples). So, in this paper, we restrict our attention to self-dual distance-regular graphs.…”