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RELATIVE POSITION BETWEEN A PAIR OF SPIN MODEL SUBFACTORS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 10 March 2025, pp. 1-38
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- Article
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Jones [‘Two subfactors and the algebraic decomposition of bimodules over
$II_1$ factors’, Acta Math. Vietnam 33(3) (2008), 209–218] proposed the study of ‘two subfactors’ of a 
$II_1$ factor as a quantization of two closed subspaces in a Hilbert space. Motivated by this, we initiate a systematic study of a special class of two subfactors, namely a pair of spin model subfactors. We characterize which pairs of distinct complex Hadamard matrices in 
$M_n(\mathbb {C})$ give rise to distinct spin model subfactors. Then, a detailed investigation is carried out for 
$n=2$, where the spin model subfactors correspond to 
$\mathbb {Z}_2$-actions on the hyperfinite type 
$II_1$ factor R. We observe that the intersection of the pair of spin model subfactors in this case is a nonirreducible vertex model subfactor and we characterize it as a diagonal subfactor. A few key invariants for the pair of spin model subfactors are computed to understand their relative positions.
