The dihedral group is the symmetry group of the regular -sided polygon. The group consists of reflections, rotations, and the identity transformation. In this entry we will denote the group in question by . An alternate notation is ; in this approach, the subscript indicates the order of the group.
and the reflections by
The abstract group structure is given by
where the addition and subtraction is carried out modulo .
The group can also be described in terms of generators and relations as
This means that is a rank-1 Coxeter group.
Since the group acts by linear transformations
there is a corresponding action on polynomials , defined by
the latter polynomial being the real part of . It is easy to check that these two polynomials are invariant. The first polynomial describes the distance of a point from the origin, and this is unaltered by Euclidean reflections through the origin. The second polynomial is unaltered by a rotation through radians, and is also invariant with respect to complex conjugation. These two transformations generate the dihedral group. Showing that these two invariants polynomially generate the full algebra of invariants is somewhat trickier, and is best done as an application of Chevalley’s theorem regarding the invariants of a finite reflection group.
|Date of creation||2013-03-22 12:22:53|
|Last modified on||2013-03-22 12:22:53|
|Last modified by||rmilson (146)|