Group-type subfactors and Hadamard matrices

Abstract

A hyperfinite I I 1 \mathrm {II}_1 subfactor may be obtained from a symmetric commuting square via iteration of the basic construction. For certain commuting squares constructed from Hadamard matrices, we describe this subfactor as a group-type inclusion R H ⊂ R ⋊ K R^H \subset R \rtimes K , where H H and K K are finite groups with outer actions on the hyperfinite I I 1 \mathrm {II}_1 factor R R . We find the group of outer automorphisms generated by H H and K K and use the method of Bisch and Haagerup to determine the principal and dual principal graphs. In some cases a complete classification is obtained by examining the element of H 3 ( H ∗ K / I n t R ) H^3(H \ast K / \mathrm {Int} R) associated with the action.