Wedge product matrices and orbits of principal congruence subgroups

The orbits in ${\mathrm{\Gamma }}_{\mathrm{\infty }}(3)\mathrm{\backslash }\mathrm{\Gamma }(3)$ 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 ${\mathrm{\Gamma }}_{\mathrm{\infty }}(3)\mathrm{\backslash }\mathrm{\Gamma }(3)$ of $3\times 3$ matrices over the PID of Eisenstein integers.