Representations of Deligne-Mostow lattices into (Q6564879)

The main goal of the paper is to classify representations of certain lattices in \(\mathrm{PU}(2, 1)\) into \(\mathrm{PGL}(3,\mathbb{C})\). By the Mostow-Prasad rigidity these lattices are rigid in \(\mathrm{PU}(2, 1)\) but it turns out that they are not rigid in \(\mathrm{PGL}(3,\mathbb{C})\). The concrete examples considered in the paper can be compared with the known general rigidity and superrigidity properties (a concise review of these in presented in Section~2) and can shed new light on the general picture.\N\NThe authors prove local rigidity for the representations of Deligne-Mostow lattices with three-fold symmetry and of type one where the generators are chosen to be of the same type as the generators of Deligne-Mostow lattices. They also show local rigidity without constraints on the type of generators for six of them and show the existence of local deformations for some of the representations for three of them.\N\NThe technical side of the project is roughly as follows. Starting with the group relations the authors produce a system of equations and using SAGE they construct the corresponding polynomial ideal over the ring generated by the parameters. Then they produce a Gröebner basis for the system of equations that gives an array of polynomials that generates the same ideal. Finally, they solve these polynomial equations for each variable. The full details of the procedure are given for one of the cases. The other cases are presented in an accompanying repository on GitHub.