On four-weight spin models and their gauge transformations
A four-weight spin model on a finite non-empty set \(X\) is a 5-tuple \((W_1,W_2,W_3,W_4,D)\), where \(D^2=|X|\) and \(W_i\) are complex matrices with rows and columns indexed by X which satisfy: \(\sum_{x\in X} W_1(a,x)W_3(x,b)=|X|\delta(a,b)\), \(W_1(a,b)W_3(b,a)=1\), \(\sum_{x\in X}W_2(a,x)W_4(x,b)=|X|\delta(a,b)\), \(W_2(a,b)\;W_4(b,a)=1\), \[ \begin{aligned} \sum_{x\in X}W_2(a,x)W_2(b,x)W_4(x,c) &= DW_1(b,a)W_3(a,c)W_3(c,b)\\ =\sum_{x\in X}W_2(x,a)W_2(x,b)W_4(c,x) &=DW_1(a,b)W_3(b,c)W_3(c,a). \end{aligned} \] There exists a non-zero complex number \(\mu\), called the modulus of the spin model such that for all \(a\in X\): \(\mu=W_1(a,a)\) and \(\sum_{x\in X}W_4(a,x)=\sum_{x\in X}W_4(x,a)=D\mu\). The associated link invariant is defined for every connected oriented diagram \(L\) by \(\mu^{-T(L)}D^{-|B(L)|}Z(L)\), where \[ Z(L)=\sum_{\sigma:B(L)\to X}\prod_{v\in V(L)}\langle v,\sigma\rangle. \] Proposition 11. Let \((W_1,W_2,W_3,W_4,D)\) be a four-weight spin model. Let \(P\) be a permutation matrix in \(M_X\) such that \(W_2^{-1}PW_2\) is also a permutation matrix. \(\Delta\) be an invertible diagonal matrix in \(M_X\), and \(\lambda\) be a non-zero complex number. Then \[ (\lambda\Delta W_1 \Delta^{-1},\lambda^{-1}PW_2,\lambda^{-1}\Delta W_3\Delta^{-1},\lambda PW_4P^tD) \] is also a four-weight spin model which has the same associated link invariant as the four-weight spin model \((W_1,W_2,W_3,W_4,D)\). Proposition 13. There are exactly \(2|X|\) two-weight spin models which satisfy the modular invariance property with respect to a Bose-Mesner algebra \(\mathbb{B}\) with a duality \(\Psi\). They are mutually gauge equivalent. One of them is symmetric.
scientific article; zbMATH DE number 1495747