The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. In addition to this geometric description, the affine symmetric groups may be defined in other ways: as collections of permutations (rearrangements) of the integers () that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in combinatorics and representation theory.
A finite symmetric group consists of all permutations of a finite set. Each affine symmetric group is an infinite extension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as descents and inversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation.
The affine symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups.
Definitions
The affine symmetric group may be equivalently defined as an abstract group by generators and relations, or in terms of concrete geometric and combinatorial models.
Algebraic definition
One way of defining groups is by generators and relations. In this type of definition, generators are a subset of group elements that, when combined, produce all other elements. The relations of the definition are a system of equations that determine when two combinations of generators are equal. In this way, the affine symmetric group is generated by a set
of elements that satisfy the following relations: when , (the generators are involutions), if is not one of , indicating that for these pairs of generators, the group operation is commutative, and . In the relations above, indices are taken modulo , so that the third relation includes as a particular case . (The second and third relation are sometimes called the braid relations.) When , the affine symmetric group is the infinite dihedral group generated by two elements subject only to the relations .
These relations can be rewritten in the special form that defines the Coxeter groups, so the affine symmetric groups are Coxeter groups, with the as their Coxeter generating sets. Each Coxeter group may be represented by a Coxeter–Dynkin diagram, in which vertices correspond to generators and edges encode the relations between them. For , the Coxeter–Dynkin diagram of is the -cycle (where the edges correspond to the relations between pairs of consecutive generators and the absence of an edge between other pairs of generators indicates that they commute), while for it consists of two nodes joined by an edge labeled .
Geometric definition
In the Euclidean space with coordinates , the set of points for which forms a (hyper)plane, an -dimensional subspace. For every pair of distinct elements and of and every integer , the set of points in that satisfy forms an -dimensional subspace within , and there is a unique reflection of that fixes this subspace. Then the affine symmetric group can be realized geometrically as a collection of maps from to itself, the compositions of these reflections.
Inside , the subset of points with integer coordinates forms the root lattice, . It is the set of all the integer vectors such that . Each reflection preserves this lattice, and so the lattice is preserved by the whole group.
The fixed subspaces of these reflections divide into congruent simplices, called alcoves. The situation when is shown in the figure; in this case, the root lattice is a triangular lattice, the reflecting lines divide into equilateral triangle alcoves, and the roots are the centers of nonoverlapping hexagons made up of six triangular alcoves.
To translate between the geometric and algebraic definitions, one fixes an alcove and consider the hyperplanes that form its boundary. The reflections through these boundary hyperplanes may be identified with the Coxeter generators. In particular, there is a unique alcove (the fundamental alcove) consisting of points such that , which is bounded by the hyperplanes ..., and illustrated in the case . For , one may identify the reflection through with the Coxeter generator , and also identify the reflection through with the generator .
