Affine symmetric group

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 (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations.

As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation.

Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups.

[4] Each Coxeter group may be represented by a Coxeter–Dynkin diagram, in which vertices correspond to generators and edges encode the relations between them.

[7] Inside V, the subset of points with integer coordinates forms the root lattice, Λ.

is shown in the figure; in this case, the root lattice is a triangular lattice, the reflecting lines divide V into equilateral triangle alcoves, and the roots are the centers of nonoverlapping hexagons made up of six triangular alcoves.

[11][12] To translate between the geometric and algebraic definitions, one fixes an alcove and consider the n hyperplanes that form its boundary.

More generally, every reflection (that is, a conjugate of one of the Coxeter generators) can be described uniquely as follows: for distinct integers i, j in

[16] These connections allow a direct translation between the combinatorial and geometric definitions of the affine symmetric group.

A geometric construction is to pick any point a in Λ (that is, an integer vector whose coordinates sum to 0); the subgroup

Geometrically, this kernel consists of the translations, the isometries that shift the entire space V without rotating or reflecting it.

[23] In an abuse of notation, the symbol Λ is used in this article for all three of these sets (integer vectors in V, affine permutations with underlying permutation the identity, and translations); in all three settings, the natural group operation turns Λ into an abelian group, generated freely by the n − 1 vectors

[16]) In (Lewis et al. 2019), the following formula was proved for the reflection length of an affine permutation u: for each cycle of u, define the weight to be the integer k such that consecutive entries congruent modulo n differ by exactly kn.

Other aspects of affine symmetric groups, such as their Bruhat order and representation theory, may also be understood via combinatorial models.

may naturally be represented by abacus diagrams: the integers are arranged in an infinite strip of width n, increasing sequentially along rows and then from top to bottom; integers are circled if they lie directly above one of the window entries of the minimal coset representative.

This bijection plays a central role in the combinatorics and the representation theory of the symmetric group.

In (Chmutov, Pylyavskyy & Yudovina 2018), the authors extended Shi's work to give a bijective map between

consisting of two tabloids of the same shape and an integer vector whose entries satisfy certain inequalities.

Their procedure uses the matrix representation of affine permutations and generalizes the shadow construction, introduced in (Viennot 1977).

From this perspective, a reduced word corresponds to an alcove walk on the tessellated space V.[57] The affine symmetric groups are closely related to a variety of other mathematical objects.

In (Ehrenborg & Readdy 1996), a correspondence is given between affine permutations and juggling patterns encoded in a version of siteswap notation.

of nonnegative integers (with certain restrictions) that captures the behavior of balls thrown by a juggler, where the number

[58] Under this bijection, the length of the affine permutation is encoded by a natural statistic in the juggling pattern:

[62] Similar techniques can be used to derive the generating function for minimal coset representatives of

In particular, in a complex inner product space, a reflection is a unitary transformation T of finite order that fixes a hyperplane.

[h] This implies that the vectors orthogonal to the hyperplane are eigenvectors of T, and the associated eigenvalue is a complex root of unity.

[86] Abacus models of minimum-length coset representatives for parabolic quotients have also been extended to this context.

[87] The study of Coxeter groups in general could be said to first arise in the classification of regular polyhedra (the Platonic solids) in ancient Greece.

[90] This article was adapted from the following source under a CC BY 4.0 license (2021) (reviewer reports): Joel B. Lewis (21 April 2021), "Affine symmetric group" (PDF), WikiJournal of Science, 4 (1): 3, doi:10.15347/WJS/2021.003, ISSN 2470-6345, Wikidata Q100400684

Tiling of the plane by regular triangles
The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group S̃ 3
The first part of the figure is labeled "S̃ sub n for n > 2". It consists of a cycle of circular nodes, labeled s sub 1, s sub 2, ..., s sub n - 1, and one circle labeled "s sub 0 = s sub n". Adjacent nodes in the cycle are connected by straight lines, non-adjacent nodes are not connected. The second part of the figure is labeled "S̃ sub 2". It consists of two circular nodes, labeled s sub 0 and s sub 1. They are connected by a straight line segment, which is labeled "infinity".
Dynkin diagrams for the affine symmetric groups on 2 and more than 2 generators
The plane divided into equilateral triangles by three sets of parallel lines. Certain intersections of the lines (vertices of the triangles) are circled.
When n = 3 , the space V is a two-dimensional plane and the reflections are across lines. The points of the root lattice Λ are circled.
The plane divided into triangles by three sets of parallel lines. One triangle is shaded; the lines that form its edges are thickened and labeled by the equations y - z = 0, x - y = 0, and x - z = 0.
Reflections and alcoves for the affine symmetric group. The fundamental alcove is shaded.
A grid is drawn. The columns are labeled "..., −1, 0, 1, 2, 3, 4, 5, 6, ..." from left to right, and the rows are labeled "..., −2, −1, 0, 1, 2, 3, 4, 5, ..." from top to bottom. Heavy lines are drawn between columns 0 and 1, columns 3 and 4, rows 0 and 1, and rows 3 and 4. The cells in row-column pairs (−2, −1), (0, 1), (1, 2), (2, 0), (3, 4), (4, 5), and (5, 3) are marked with a filled circle.
The matrix representation of the affine permutation [2, 0, 4], with the conventions that 1s are replaced by • and 0s are omitted. Row and column labelings are shown.
The plane is divided into equilateral triangles by three sets of parallel lines. Each triangle is labeled by a triple of three numbers. One triangle, labeled by [1, 2, 3], is shaded. One of its vertices is the origin. The other five triangles that share this vertex are labeled (in clockwise order) by [2, 1, 3], [3, 1, 2], [3, 2, 1], [2, 3, 1], and [1, 3, 2]. The third triangle adjacent to [2, 1, 3] is labeled [2, 0, 4].
Alcoves for labeled by affine permutations. An alcove A is labeled by the window notation for a permutation u if u sends the fundamental alcove (shaded) to A . Negative numbers are denoted by overbars.
Coordinate x- and y-axes in the plane. A thick line labeled V runs from upper left to lower right, passing through the origin. It is crossed by several equally spaced dashed lines that are perpendicular to it. At every other intersection point, a node is drawn. The dashed line through the origin is labeled s_1, and the dashed line nearest to it is labeled s_0.
The affine symmetric group acts on the line V in the Euclidean plane. The reflections are through the dashed lines. The vectors of the root lattice Λ are marked.
The numbers from -7 to 16, arranged in order in a rectangular grid with four numbers per row. The numbers 9, 6, -5, and 0 are circled, as well as all of the numbers above them.
Abacus diagram of the affine permutation [−5, 0, 6, 9]
The plane is divided into equilateral triangles by three sets of parallel lines. Each triangle is labeled by a triple of three numbers. One triangle, labeled by [1, 2, 3], is shaded. One of its vertices is the origin. The other five triangles that share this vertex are labeled (in clockwise order) by [2, 1, 3], [2, 3, 1], [3, 2, 1], [3, 1, 2], and [1, 3, 2]. The third triangle adjacent to [2, 1, 3] is labeled [0, 1, 5].
Alcoves for labeled by affine permutations, inverse to the labeling above
A stick-figure person juggling three balls
The juggling pattern 441
Above, four pictures, each of five vertical strands of thread. In the first, labeled "sigma sub 1", the first strand crosses over the second, while the other three strands go from top to bottom without crossing any other strand. The second and third (labeled "sigma sub 2" and "sigma sub 3") are similar, but with the second strand crossing over the third or the third strand crossing over the fourth, respectively. In the fourth picture, the second, third, and fifth strands go in a straight line from top to bottom; the first strand crosses behind all other strands before wrapping in front of the fifth strand and then under the fourth strand, ending in the fourth position; after crossing over the first strand, the fourth strand crosses over the fifth strand, then behind all other strands, ending in the first position. Below, three pictures, each of which show three strands drawn on a cylinder. In the first picture, the first strand crosses over the second, while the third goes from top to bottom without crossing anything; in the second picture, the second strand crosses over the third, while the first goes from top to bottom without crossing anything; in the final picture, the first and third strands wrap around the back of the cylinder with the third crossing over the first, while the second goes from top to bottom without crossing anything.
Generators of the Artin-Tits group associated with the affine symmetric group, represented as braids with one fixed strand (for n = 4 ) and as braids drawn on a cylinder (for n = 3 )