Wedge product matrices and orbits of principal congruence subgroups

TL;DR

The paper introduces a wedge product matrix approach to define invariants of matrix orbits under principal congruence subgroups, enabling computations for various subgroups and matrices, with explicit orbit representatives constructed for 3x3 matrices over Eisenstein integers.

Contribution

It presents a novel wedge product matrix method for defining invariants of matrix orbits and constructs explicit orbit representatives using Bruhat decomposition.

Findings

Wedge product matrices provide an alternative to existing invariants.

Method applicable to other congruence subgroups and matrix sizes.

Explicit orbit representatives for 3x3 matrices over Eisenstein integers.

Abstract

The orbits in $\Gamma_{\infty}(3) \backslash \Gamma(3)$ are in bijection with sets of invariants satisfying certain relations. We explain how wedge product matrices give an alternative definition of the invariants of matrix orbits. This new method provides the possibility of performing similar computations with other congruence subgroups and arbitrary $n \times n$ matrices. Using Steinberg's refined version of the Bruhat decomposition, we construct an explicit choice of coset representative for each orbit in the orbit space $\Gamma_{\infty}(3) \backslash \Gamma(3)$ of $3 \times 3$ matrices over the PID of Eisenstein integers.