The Vieta Mapping for the Coxeter Group

This Demonstration shows the real part of the Vieta mapping, which is important in singularity theory. The domain is the plane containing the green hexagon; it has been rotated around the origin to make it horizontal for the purpose of the Demonstration. A point in the plane determines a polynomial with roots , , . The image of (the tip of the arrow from ) has coordinates . When the "show orbit" checkbox is checked, the orbit of the point under the action of the Coxeter group generated by reflections in the red lines (or symmetries of the blue equilateral triangle) is shown together with the image of all the points in the orbit (the set of all the distinct images of the point under transformations by elements of the group ).

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DETAILS

The Coxeter group acts on the plane by permuting the coordinates. The space of its complex orbits can be identified with the space of polynomials , which is a smooth manifold. The Vieta map , , …, sends the space where the reflection group acts to the space of its orbits. The critical set of the Vieta map is the union of its mirrors. The variety of irregular orbits of a reflection group (orbits with less than the maximal number of points) are called its discriminant. The discriminant of the group shown in this Demonstration is the semicubical parabola in the target plane of the Vieta map. The image of the real plane under the Vieta mapping is the interior of the semicubical parabola.
Reference
[1] V. I. Arnold, The Theory of Singularities and Its Applications, New York: Lezioni Fermiane, 1993.
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