The rotations are the symmetries preserving the (cyclic) ordering of vertices. Thus, a rotation g is determined by the image gv, so the subgroup N of rotations has n elements. A reflection is an order-2 symmetry reversing the ordering of vertices. Imbedding the n-gon in R, there are n axes through which the n-gon can be reflected, so there are n reflections. Since |G| = 2n, every symmetry is ei...

Abstract. There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial derivatives. This paper presents an explicit form of the action of the intertwining operator on polynomials by use of harmonic and Jacobi polynomials. The ...

Soergel Diagrammatics for Dihedral Groups

We classify and study trigonal curves in Hirzebruch surfaces admitting dihedral Galois coverings. As a consequence, we obtain certain restrictions on the fundamental group of a plane curve D with a singular point of multiplicity (deg D−3).

A subset B of a group G is called a difference basis of G if each element g ∈ G can be written as the difference g = ab−1 of some elements a, b ∈ B. The smallest cardinality |B| of a difference basis B ⊂ G is called the difference size of G and is denoted by ∆[G]. The fraction ð[G] := ∆[G]/ √ |G| is called the difference characteristic of G. We prove that for every n ∈ N the dihedral group D2n ...