Equation for hyperplane of a reflection - Try to do TODO on reflection group sage page
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 $\dim \operatorname{fix} r = \dim V - 1$.
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 a reflection r we use r.to_matrix()
to transform r into a matrix. The idea would be to have a function
that returns the hyperplane of r. But r belongs
in 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
Give us a concrete example of what you tried, please.
For example:
the matrix given by A=matrix(2,[0,-1,0,-1]) is a reflection actin on C^{2}.
The order of A is 2.
The hyperplane is given by the vector (-1,1), and the equation is y+x=0.
the problem is how for a big reflection group i can associate a reflection with hyperplane equation in some algoritm of the form:
input: reflection
output: y+x, equation for a given reflection.
Thank you for your attention.
I mean "example of code that you tried", for instance displaying "r" which is not a matrix