Special Symplectic Subgroup over Integers Arising as a Factor of The Braid Group

TL;DR

This paper investigates a special symplectic subgroup over integers derived from the braid group, classifies its orbits, and calculates subgroup indices, contributing to the understanding of braid group actions on matrices.

Contribution

It introduces a new extremal case involving rank two matrices, classifies the associated orbits, and determines the subgroup indices within symplectic groups.

Findings

The linear group is a proper subgroup of finite index in Sp(2n,Z) or its semidirect product.

Orbits of the action are classified.

Indices of the subgroups are explicitly calculated.

Abstract

There is a natural action of the braid group on the symmetric matrices with units on the diagonal, appearing in various fields as Singularity Theory, Frobenius Manifolds or Isomonodromic deformations of certain classes of linear differential systems. The problem of finding the finite orbits of this action for the case of non-degenerate three-dimensional matrices was solved and it was conjectured that the finite orbits on any non-degenerate matrices correspond to finite Coxeter groups and conjugacy classes of quasicoxeter elements in them. In the present work it is considered another extremal case of rank two matrices of arbitrary dimension. The problem of classification of the orbits in this case involves an examination of a linear representation of a factor of the braid group. It is shown that this linear group is a proper subgroup of finite index in the the symplectic group $Sp(2n,\Z)$ or the semidirect product $Sp(2n,\Z)\ltimes\Z^{2n}$. The orbits are classified and the indices of these subgroups are calculated.