Unimodular bilinear lattices, automorphism groups, vanishing cycles, monodromy groups, distinguished bases, braid group actions and moduli spaces from upper triangular matrices

TL;DR

This monograph explores the rich algebraic and geometric structures arising from upper triangular matrices with integer entries, connecting concepts like lattices, monodromy, and braid group actions across algebraic geometry and singularity theory.

Contribution

It systematically develops tools to study structures associated with upper triangular matrices, especially in rank 2 and 3, revealing complex phenomena and their relations to moduli spaces and singularities.

Findings

Unimodular bilinear lattices and Seifert forms are constructed from matrices.

Braid group actions generate orbits of distinguished bases and matrices.

Connections to complex manifolds and hypersurface singularities are established.

Abstract

This monograph starts with an upper triangular matrix with integer entries and 1's on the diagonal. It develops from this a spectrum of structures, which appear in different contexts, in algebraic geometry, representation theory and the theory of irregular meromorphic connections. It provides general tools to study these structures, and it studies sytematically the cases of rank 2 and 3. The rank 3 cases lead already to a rich variety of phenomena and give an idea of the general landscape. The first structure associated to the matrix is a Z-lattice with unimodular bilinear form (called Seifert form) and a triangular basis. It leads immediately to an even and an odd intersection form, reflections and transvections, an even and an odd monodromy group, even and odd vanishing cycles. Braid group actions lead to braid group orbits of distinguished bases and of upper triangular matrices. Finally, complex manifolds, which consist of correctly glued Stokes regions, are associated to these braid group orbits. The last chapter is a report on the case of isolated hypersurface singularities.