Classifying $\mathsf{GL}(n,\mathbb Z)$-orbits of points and rational subspaces

TL;DR

This paper provides a complete classification of $ ext{GL}(n, ext{Z})$-orbits of points and rational subspaces in $ ext{R}^n$ using algebraic and geometric invariants, simplifying previous proofs and extending orbit classification.

Contribution

It introduces a new classification method for $ ext{GL}(n, ext{Z})$-orbits of points and rational subspaces, based on subgroup generated by coordinates and volume of associated parallelotopes.

Findings

Orbit of a point is classified by the subgroup generated by its coordinates.

Dense orbits occur when some coordinate ratios are irrational.

Orbit of a rational subspace is classified by its dimension and associated volume.

Abstract

We first show that the subgroup of the abelian real group $\mathbb{R}$ generated by the coordinates of a point in $x = (x_1,\dots,x_n)\in\mathbb{R}^n$ completely classifies the $\mathsf{GL}(n,\mathbb Z)$-orbit of $x$. This yields a short proof of J.S.Dani's theorem: the $\mathsf{GL}(n,\mathbb Z)$-orbit of $x\in\mathbb{R}^n$ is dense iff $x_i/x_j\in \mathbb{R} \setminus \mathbb Q$ for some $i,j=1,\dots,n$. We then classify $\mathsf{GL}(n,\mathbb Z)$-orbits of rational affine subspaces $F$ of $\mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope associated to $F$ yields a complete classifier of the $\mathsf{GL}(n,\mathbb Z)$-orbit of $F$.