Wedge product matrices and orbits of principal congruence subgroups (Q6564114)

Let \(w = e^{2\pi i/3}\), \(\mathbb{Z}[w] = \mathfrak{o}\) and \(\Gamma(3) = \{A \in \mathrm{SL}(3, \mathfrak{o}) \mid A \equiv I_3 \bmod {3\mathfrak{o}}\}\). Given the subgroup \(\Gamma_{\infty}(3)\) of upper triangular matrices of \(\Gamma(3)\), consider \(\Gamma_{\infty}(3) \backslash \Gamma(3) = \{\Gamma_{\infty}(3) \cdot A \mid A \in \Gamma(3)\}\). For a matrix \(A = \begin{bmatrix} a & b & c\\\Nd & e & f\\\Ng & h& i \end{bmatrix} \in \Gamma(3)\) define \(\mathrm{Inv}(A) = (A_1, B_1, C_1, A_2, B_2, C_2) \in \mathfrak{o}^6\), where \(A_1 = g\), \(B_1 = h\), \(C_1 = i\), \(A_2 = dh-eg\), \(B_2 = di-fg\), \(C_2 = ei-fh\). Next, define the invariant conditions on \((A_1, B_1, C_1, A_2, B_2, C_2) \in \mathfrak{o}^6\) as \(A_1 \equiv A_2 \equiv B_1 \equiv B_2 \equiv 0 \bmod {3\mathfrak{o}}\) (I1); \(C_1 \equiv C_2 \equiv 1 \bmod {3\mathfrak{o}}\) (I2); \(\gcd(A_1, B_1, C_1) = \gcd(A_2, B_2, C_2) = 1\) (I3); \(A_1C_2 - B_1B_2+C_1A_2 = 0\) (I4). Then \(\mathrm{Inv}(A)\) satisfies the conditions (I1)--(I4) for each \(A \in \Gamma(3)\) (Proposition 2.2). Furthermore, \(\Gamma_{\infty}(3) \cdot A = \Gamma_{\infty}(3) \cdot B\) if and only if \(\mathrm{Inv}(A) = \mathrm{Inv}(B)\) (Theorem 2.3).\N\NThe main result of the paper shows how, from the set of invariants \(A_1, B_1, C_1, A_2, B_2, C_2 \in \mathfrak{o}\) satisfying conditions (I1)--(I4), one can construct an explicit matrix representative of an orbit in \(\Gamma_{\infty}(3) \backslash \Gamma(3)\) whose invariants are exactly \(A_1\), \(B_1\), \(C_1\), \(A_2\), \(B_2\), \(C_2\) (Theorem 3.3).