Representations of Deligne-Mostow lattices into PGL(3, C)

TL;DR

This paper classifies certain Deligne-Mostow lattice representations into PGL(3,C), demonstrating local rigidity in specific cases and identifying conditions for deformations, using computational tools like SAGE and Maple.

Contribution

It provides a detailed classification and rigidity results for Deligne-Mostow lattices into PGL(3,C), including computational methods and new insights into their deformation space.

Findings

Proved local rigidity for lattices with 3-fold symmetry and type one generators.

Established local rigidity without generator type constraints for six lattices.

Identified cases where local deformations of representations exist.

Abstract

We classify representations of a class of Deligne-Mostow lattices into PGL(3;C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are of the same type as the generators of Deligne-Mostow lattices. We also show local rigidity without constraints on the type of generators for six of them and we show the existence of local deformations for a number of representations in three of them. We use formal computations in SAGE and Maple to obtain the results. The code files are available on GitHub.