SO(3,4) representations of Δ(2,3,r)
We search the character variety of ℤ/2 ∗ ℤ/3 into SO(3, 4) for representations of the (2, 3, r) triangle group — triples of matrices (A, B, C) satisfying
A² = B³ = Cr = C·B·A = I.
A 3-dimensional subset of this character variety is parameterized by a point m = (m₁, m₂, m₃) ∈ ℝ³. The solver searches that 3-parameter space numerically, finding points where the eigenvalues of C are all distinct rth roots of unity. (To do: handle the case where the eigenvalues are not necessarily distinct.)
Method. For each (r, k-triple), we run Newton from N random initial positions inside a cube [m_min, m_max]³, see which converge, and take the centroid of each cluster of converged solutions as our m. The empirical spread of the cluster gives a soft confidence bar on m.
Usage. Specify a range for r, the cube to search in m-space, and the number of starts. Click Run sweep. The program returns the triples (k₁, k₂, k₃) for which a solution exists with C's eigenvalues equal to e±i 2π kj / r, together with the point (m₁, m₂, m₃) realizing that representation. Click any solution row to expand a detail panel inline; click again to collapse.