Isolation and Deformation Results for Commuting Squares of Finite Dimensional Matrix Algebras

Commuting squares arise as algebraic-combinatorial invariants in Jones' theory of subfactors. They can also be used to construct subfactors via iterating the basic construction, and a lot of the known examples of subfactors are obtained this way. In this thesis, we use deformation techniques to investigate the structure of the moduli space of commuting squares, and of a larger class of generalized commuting squares.

In the first part of the thesis, we consider generalizations of commuting squares, called twisted commuting squares, obtained by having the commuting square orthogonality condition hold with respect to the inner product given by a faithful state on a finite dimensional matrix algebra. We present various examples of twisted commuting squares, most of which are computationally easy to work with, and we prove an isolation result. We also give an application to the theory of associative deformations of the matrix multiplication.

In the second part of the thesis, we investigate commuting squares arising from finite groups. We define the undephased defect d(G) and the dephased defect d’(G) for a finite group G, which generalize the existing notions of defect for Fourier matrices. The undephased and dephased defects give upper bounds on the number of independent directions in which the commuting square associated to G can be deformed by any (possibly isomorphic) commuting squares, respectively by non-isomorphic commuting squares. We find a canonical basis of independent directions in which the commuting square associated to G can be deformed, and we explicitly construct parametric families of commuting squares in each of the d(G) directions of this basis. In particular, we obtain parametric families of complex Hadamard matrices stemming from the Fourier matrix of non-prime dimension.