# 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?

Note: This issue refers to TODO (linear forms for hyperplanes) at

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1

Give us a concrete example of what you tried, please.

( 2022-02-15 12:15:32 +0200 )edit

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.