dihedral group

The $n^{\\text{th}}$ dihedral group is the symmetry group ofthe regular $n$ -sided polygon. The group consists of $n$ reflections, $n-1$ rotations, and the identity transformation. In this entry we will denote the group in question by $\\mathcal{D}_{n}$ .An alternate notation is $\\mathcal{D}_{2n}$ ; in this approach, the subscript indicates the order of the group.

Letting $\\omega=\\exp(2\\pi i/n)$ denote a primitive $n^{\\text{th}}$ root ofunity, and assuming the polygon is centered at the origin, therotations $R_{k},\\;k=0,\\ldots,n-1$ (Note: $R_{0}$ denotes the identity)are given by

R_{k}:z\\mapsto\\omega^{k}z,\\quad z\\in\\mathbb{C},

and the reflections $M_{k},\\;k=0,\\ldots,n-1$ by

M_{k}:z\\mapsto\\omega^{k}\\bar{z},\\quad z\\in\\mathbb{C}

The abstract group structure is given by

	$\\displaystyle R_{k}R_{l}$	$\\displaystyle=R_{k+l},$	$\\displaystyle R_{k}M_{l}$	$\\displaystyle=M_{k+l}$
	$\\displaystyle M_{k}M_{l}$	$\\displaystyle=R_{k-l},$	$\\displaystyle M_{k}R_{l}$	$\\displaystyle=M_{k-l},$

where the addition and subtraction is carried out modulo $n$ .

The group can also be described in terms of generators and relations as

\\left(M_{0}\\right)^{2}=\\left(M_{1}\\right)^{2}=(M_{1}M_{0})^{n}=\\mathrm{id}.

This means that $\\mathcal{D}_{n}$ is a rank-1 Coxeter group.

Since the group acts by linear transformations

(x,y)\\to(\\hat{x},\\hat{y}),\\quad(x,y)\\in\\mathbb{R}^{2}

there is acorresponding action on polynomials $p\\to\\hat{p}$ , defined by

\\hat{p}(\\hat{x},\\hat{y})=p(x,y),\\quad p\\in\\mathbb{R}[x,y].

The polynomialsleft invariant by all the group transformations form an algebra. Thisalgebra is freely generated by the following two basic invariants:

x^{2}+y^{2},\\quad x^{n}-\\binom{n}{2}x^{n-2}y^{2}+\\cdots,

the latterpolynomial being the real part of $(x+iy)^{n}$ . It is easy to checkthat these two polynomials are invariant. The first polynomialdescribes the distance of a point from the origin, and this isunaltered by Euclidean reflections through the origin. The secondpolynomial is unaltered by a rotation through $2\\pi/n$ radians, and isalso invariant with respect to complex conjugation. These twotransformations generate the $n^{\\text{th}}$ dihedral group. Showing thatthese two invariants polynomially generate the full algebra ofinvariants is somewhat trickier, and is best done as an application ofChevalley’s theorem regarding the invariants of a finite reflectiongroup.