Hello everyone, I hope you are well. I'm working with finite complex reflection group. And I'm having a hard time getting the equation of a given hyperplane of a reflection.
First: a reflection r is a map onto a finite-dimensional vector space V so that dim fix r = dim V -1.
Second: a reflection group is a group generated by such reflections.
third: A hyperplane is the eigenspace associated with the eigenvalue 1 of r, that is, vectors of the form r(v)=v.
Sage has a function called reflection_hyperplanes(). Which returns all hyperplanes of a given group. And it also has the reflection_hyperplane(i) function that returns the i-th hyperplane of the group in question.
But the idea is to be able to associate each reflection with its respective hyperplane.
My attempt: At first given r reflection we have to r.to_matrix() transform r into a matrix. The idea would be to have a function that returns the hyperplane of r. But r belongs to a different class from matrices.
Is it possible to create a function that makes r.to_matrix() a de facto matrix and that manages to return the equation of the respective hyperplane of r?
Thank you for your attention.
Note: This issue refers to TODO (linear forms for hyperplanes) at https:// doc . sagemath . org/html/en/reference/combinat/sage/combinat/root_system/reflection_group_complex . html
