Spin models of index 2 and Hadamard models
Let \(X\) be a finite nonempty set of size \(n\), \(W\) be a matrix of \(M_{\mathbb C}(X)\) with non-zero entries. For each pair \((a,b)\in X\times X\), we define a vector \(Y_{a,b}^W\) by \[ Y_{a,b}^W(x)=\frac {W(x,a)}{W(x,b)}. \] \(W\) is said to be a type II matrix, if \(\sum _{x\in X} Y_{a,b}^W(x)=n\delta_{a,b}\) holds for any \(a\) and \(b\) in \(X\). Let \(D\) denote one of the square roots of \(|X|\). A spin model on \(X\) with loop variable \(D\) is a type II matrix \(W\) which satisfies the type III condition: \[ \sum _{x\in X}\frac {W(\alpha,x)W(\beta,x)}{W(\gamma,x)}= D\frac {W(\alpha,\beta)}{W(\alpha,\gamma)W(\gamma,\beta)}. \] If \(W\) is a spin model, then \(^tW\) is obtained from \(W\) by permutation of rows, the order of this permutation is called the index of \(W\). Every spin model of index 2 has the following form: \[ W=\left ( \begin{matrix} A & A & B & -B \\ A & A & -B & B \\ -^tB & ^tB & C & C \\ ^tB & -^tB & C & C \end{matrix} \right), \] where \(A,B,C\) are square matrices of equal sizes. Spin models of index 2 with \(A\) having constant non-diagonal entries are classified in this paper (Theorem 1.1).
scientific article; zbMATH DE number 1901445