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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. Each one is an infinite extension of a finite symmetric group, the group of permutations of a finite set. In addition to their geometric description, the affine symmetric groups may be defined as collections of permutations 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 as part of the fields of combinatorics and representation theory.
Many important combinatorial properties of the finite symmetric groups can be extended to 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.