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[[File:Uniform triangular tiling 121212.png|thumb|upright=1.2|alt=Tiling of the plane by regular triangles|The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group {{math|''S̃''<sub>3</sub>}}]]
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 [[permutation]]s (rearrangements) of the integers ({{math|..., &minus;2, &minus;1, 0, 1, 2, ...}}) that are periodic in a certain sense, or in purely algebraic terms as a [[group (mathematics)|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 [[group extension|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 [[Permutation#Descents|descents]] and [[Inversion (discrete mathematics)|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 pattern]]s and certain [[complex reflection group]]s. Many of their combinatorial and geometric properties extend to the broader family of [[affine Coxeter group]]s.

==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.{{sfnp|Shi|1986|p = 66}}

=== Algebraic definition ===
[[File:Dynkin diagram for the affine symmetric group.png|thumb|upright=1.2|alt=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 diagram]]s for the affine symmetric groups on 2 and more than 2 generators]]
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.{{efn|More precisely, every relation between generators can be explained by the given relations, so that the group is the largest among all groups whose generators satisfy the given relations. The formal version of this definition is given in terms of [[quotient group|quotients]] of [[free group]]s.}}{{sfnp|Gallian|2013|loc = Chapter 26}} In this way, the affine symmetric group <math>\widetilde{S}_n</math> is generated by a set
<math display="block"> s_0, s_1, \ldots, s_{n - 1}</math>
of {{mvar|n}} elements that satisfy the following relations: when <math> n \geq 3 </math>,
# <math>s_i^2 = 1</math> (the generators are [[involution (mathematics)|involution]]s),
# <math> s_is_j = s_js_i </math> if {{mvar|j}} is not one of <math>i - 1, i, i + 1</math>, indicating that for these pairs of generators, the group operation is [[Commutative property|commutative]], and
# <math> s_is_{i + 1}s_i = s_{i + 1}s_is_{i + 1} </math>.
In the relations above, indices are taken [[modular arithmetic|modulo {{mvar|n}}]], so that the third relation includes as a particular case <math> s_0s_{n - 1}s_0 = s_{n - 1}s_0s_{n - 1} </math>. (The second and third relation are sometimes called the [[braid relation]]s.{{sfnp|Stembridge|1996|p=355}}) When <math> n = 2</math>, the affine symmetric group <math>\widetilde{S}_2</math> is the [[infinite dihedral group]] generated by two elements <math>s_0, s_1</math> subject only to the relations <math>s_0^2 = s_1^2 = 1</math>.{{sfnp|Björner|Brenti|2005|pp = 5–6}}

These relations can be rewritten in the special form that defines the [[Coxeter group]]s, so the affine symmetric groups are Coxeter groups, with the <math>s_i</math> as their Coxeter generating sets.{{sfnp|Björner|Brenti|2005|pp = 5–6}} Each Coxeter group may be represented by a [[Coxeter–Dynkin diagram]], in which vertices correspond to generators and edges encode the relations between them.{{sfnp|Humphreys|1990|p = 31}} For <math> n \geq 3</math>, the Coxeter–Dynkin diagram of <math>\widetilde{S}_n</math> is the [[Cycle graph|{{mvar|n}}-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 <math> n = 2</math> it consists of two nodes joined by an edge labeled <math>\infty</math>.{{sfnp|Björner|Brenti|2005|p = 2}}{{sfnp|Björner|Brenti|2005|pp = 5–6}}

=== Geometric definition ===
[[File:A 2 lattice and hyperplanes.png|thumb|upright=1.2|alt=The plane divided into equilateral triangles by three sets of parallel lines. Certain intersections of the lines (vertices of the triangles) are circled.|When {{math|1=''n'' = 3}}, the space {{mvar|V}} is a two-dimensional plane and the reflections are across lines. The points of the root lattice {{math|Λ}} are circled.]]
In the [[Euclidean space]] <math>\R^{n}</math> with coordinates <math>(x_1, \ldots, x_n)</math>, the set {{mvar|V}} of points for which <math>x_1 + x_2 + \cdots + x_n = 0</math> forms a [[hyperplane|(hyper)plane]], an {{math|(''n'' − 1)}}-dimensional subspace. For every pair of distinct elements {{mvar|i}} and {{mvar|j}} of <math>\{1, \ldots, n\}</math> and every integer {{mvar|k}}, the set of points in {{mvar|V}} that satisfy <math>x_i - x_j = k</math> forms an {{math|(''n'' − 2)}}-dimensional subspace within {{mvar|V}}, and there is a unique [[Reflection (mathematics)|reflection]] of {{mvar|V}} that fixes this subspace. Then the affine symmetric group <math>\widetilde{S}_n</math> can be realized geometrically as a collection of maps from {{mvar|V}} to itself, the compositions of these reflections.{{sfnp|Humphreys|1990|pp = 87–89, 95–6}}

Inside {{mvar|V}}, the subset of points with integer coordinates forms the ''[[root lattice]]'', {{math|Λ}}. It is the set of all the [[integer]] vectors <math>(a_1, \ldots, a_n)</math> such that <math>a_1 + \cdots + a_n = 0</math>.{{sfnp|Humphreys|1990|p = 41}} Each reflection preserves this lattice, and so the lattice is preserved by the whole group.{{sfnp|Humphreys|1990|p = 87}}

The fixed subspaces of these reflections divide {{mvar|V}} into congruent [[simplex|simplices]], called ''alcoves''.{{sfnp|Humphreys|1990|loc = Section 4.3}} The situation when <math>n = 3</math> is shown in the figure; in this case, the root lattice is a triangular lattice, the reflecting lines divide {{mvar|V}} into equilateral triangle alcoves, and the roots are the centers of nonoverlapping hexagons made up of six triangular alcoves.{{sfnp|Petersen|2015|loc = Chapter 14}}{{sfnp|Coxeter|1973|loc = Chapter 5}}
[[File:Affine A 2 hyperplanes.png|thumb|upright=1.4|alt=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.]]
To translate between the geometric and algebraic definitions, one fixes an alcove and consider the {{mvar|n}} 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 <math>(x_1, \ldots, x_n)</math> such that <math> x_1 \geq x_2 \geq \cdots \geq x_n \geq x_1 - 1</math>, which is bounded by the hyperplanes <math> x_1 - x_2 = 0,</math> <math>x_2 - x_3 = 0,</math> ..., and <math> x_1 - x_n = 1,</math> illustrated in the case <math>n = 3</math>. For <math>i = 1, \ldots, n- 1</math>, one may identify the reflection through <math>x_i - x_{i + 1} = 0</math> with the Coxeter generator <math>s_i</math>, and also identify the reflection through <math> x_1 - x_n = 1</math> with the generator <math> s_0 = s_n</math>.{{sfnp|Humphreys|1990|loc = Section 4.3}}

=== Combinatorial definition ===
The elements of the affine symmetric group may be realized as a group of periodic permutations of the integers. In particular, say that a function <math>u \colon \Z \to \Z</math> is an ''affine permutation'' if
*it is a [[bijection]] (each integer appears as the value of <math>u(x)</math> for exactly one <math>x</math>),
*<math>u(x + n) = u(x) + n</math> for all integers {{mvar|x}} (the function is [[Equivariant map|equivariant]] under shifting by <math>n</math>), and
*<math> u(1) + u(2) + \cdots + u(n) = 1 + 2 + \cdots + n = \frac{n(n + 1)}{2}</math>, the <math>n</math>th [[triangular number]].
For every affine permutation, and more generally every shift-equivariant bijection, the numbers <math>u(1), \ldots, u(n)</math> must all be distinct modulo {{mvar|n}}. An affine permutation is uniquely determined by its ''window notation'' <math>[u(1), \ldots, u(n)]</math>, because all other values of <math>u</math> can be found by shifting these values. Thus, affine permutations may also be identified with [[tuple]]s <math>[u(1), \ldots, u(n)]</math> of integers that contain one element from each congruence class modulo {{mvar|n}} and sum to <math>1 + 2 + \cdots + n</math>.{{sfnp|Björner|Brenti|2005|loc = Chapter 8.3}}

To translate between the combinatorial and algebraic definitions, for <math>i = 1, \ldots, n- 1</math> one may identify the Coxeter generator <math>s_i</math> with the affine permutation that has window notation <math>[1, 2, \ldots, i - 1, i + 1, i, i + 2, \ldots, n ] </math>, and also identify the generator <math> s_0 = s_n</math> with the affine permutation <math>[0, 2, 3, \ldots, n - 2, n - 1, n + 1] </math>. More generally, every reflection (that is, a conjugate of one of the Coxeter generators) can be described uniquely as follows: for distinct integers {{mvar|i}}, {{mvar|j}} in <math>\{1, \ldots, n\}</math> and arbitrary integer {{mvar|k}}, it maps {{mvar|i}} to {{math|''j'' − ''kn''}}, maps {{mvar|j}} to {{math|''i'' + ''kn''}}, and fixes all inputs not congruent to {{mvar|i}} or {{mvar|j}} modulo {{mvar|n}}.{{sfnp|Björner|Brenti|2005|loc = Proposition 8.3.5}}

=== Representation as matrices ===
[[File:Matrix representation of an affine permutation.png|thumb|alt=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.]]
Affine permutations can be represented as infinite periodic [[permutation matrix|permutation matrices]].{{sfnp|Chmutov|Pylyavskyy|Yudovina|2018|loc = Section 1.6}} If <math>u : \mathbb{Z} \to \mathbb{Z}</math> is an affine permutation, the corresponding matrix has entry 1 at position <math>(i, u(i))</math> in the infinite grid <math> \mathbb{Z} \times \mathbb{Z}</math> for each integer {{mvar|i}}, and all other entries are equal to 0. Since {{mvar|u}} is a bijection, the resulting matrix contains exactly one 1 in every row and column. The periodicity condition on the map {{mvar|u}} ensures that the entry at position <math>(a, b)</math> is equal to the entry at position <math>(a + n, b + n)</math> for every pair of integers <math>(a, b)</math>.{{sfnp|Chmutov|Pylyavskyy|Yudovina|2018|loc = Section 1.6}} For example, a portion of the matrix for the affine permutation <math>[2, 0, 4] \in \widetilde{S}_3</math> is shown in the figure. In row 1, there is a 1 in column 2; in row 2, there is a 1 in column 0; and in row 3, there is a 1 in column 4. The rest of the entries in those rows and columns are all 0, and all the other entries in the matrix are fixed by the periodicity condition.

== Relationship to the finite symmetric group ==
The affine symmetric group <math>\widetilde{S}_n</math> contains the finite symmetric group <math>S_n</math> of permutations on <math>n</math> elements as both a [[subgroup]] and a [[quotient group]].{{sfnp|Lewis|McCammond|Petersen|Schwer|2019}} These connections allow a direct translation between the combinatorial and geometric definitions of the affine symmetric group.

=== As a subgroup ===
There is a [[wikt:canonical#Adjective|canonical]] way to choose a subgroup of <math>\widetilde{S}_n</math> that is isomorphic to the finite symmetric group <math>S_n</math>.
In terms of the algebraic definition, this is the subgroup of <math>\widetilde{S}_n</math> generated by <math>s_1, \ldots, s_{n - 1}</math> (excluding the simple reflection <math>s_0 = s_n</math>). Geometrically, this corresponds to the subgroup of transformations that fix the origin, while combinatorially it corresponds to the window notations for which <math>\{u(1), \ldots, u(n) \} = \{1, 2, \ldots, n \}</math> (that is, in which the window notation is the [[one-line notation]] of a finite permutation).{{sfnp|Björner|Brenti|2005|p = 260}}{{sfnp|Kane|2001|loc=Section 11-3}}

If <math> u = [u(1), u(2), \ldots, u(n)]</math> is the window notation of an element of this standard copy of <math>S_n \subset \widetilde{S}_n</math>, its action on the hyperplane {{mvar|V}} in <math>\R^n</math> is given by permutation of coordinates: <math> (x_1, x_2, \ldots, x_n) \cdot u = (x_{u(1)}, x_{u(2)}, \ldots, x_{u(n)})</math>.{{sfnp|Lewis|McCammond|Petersen|Schwer|2019|p = 4118}} (In this article, the geometric action of permutations and affine permutations is on the right; thus, if {{mvar|u}} and {{mvar|v}} are two affine permutations, the action of {{math|''uv''}} on a point is given by first applying {{mvar|u}}, then applying {{mvar|v}}.)

There are also many nonstandard copies of <math>S_n</math> contained in <math>\widetilde{S}_n</math>. A geometric construction is to pick any point {{mvar|a}} in {{math|Λ}} (that is, an integer vector whose coordinates sum to 0); the subgroup <math>(\widetilde{S}_n)_a</math> of <math>\widetilde{S}_n</math> of [[isometry|isometries]] that fix {{mvar|a}} is isomorphic to <math>S_n</math>.{{sfnp|Lewis|McCammond|Petersen|Schwer|2019|loc = Corollary 2.5}}

=== As a quotient ===

There is a simple map (technically, a [[surjection|surjective]] [[group homomorphism]]) {{mvar|π}} from <math>\widetilde{S}_n</math> onto the finite symmetric group <math>S_n</math>. In terms of the combinatorial definition, an affine permutation can be mapped to a permutation by reducing the window entries modulo {{mvar|n}} to elements of <math>\{1, 2, \ldots, n\}</math>, leaving the one-line notation of a permutation.{{sfnp|Shi|1986|pp = 85&ndash;6}} In this article, the image <math>\pi(u)</math> of an affine permutation {{mvar|u}} is called the ''underlying permutation'' of {{mvar|u}}.

The map {{mvar|π}} sends the Coxeter generator <math> s_0 = [0, 2, 3, 4, \ldots, n - 2, n - 1, n + 1]</math> to the permutation whose one-line notation and [[cycle notation]] are <math> [n , 2 , 3 , 4, \ldots , n - 2 , n - 1 , 1]</math> and <math>(1 \; n)</math>, respectively.{{sfnp|Petersen|2015|loc = Section 14.4.1}}{{sfnp|Shi|1986|pp = 85&ndash;6}}

The [[Kernel (algebra)|kernel]] of {{mvar|π}} is by definition the set of affine permutations whose underlying permutation is the [[identity element|identity]]. The window notations of such affine permutations are of the form <math>[1 - a_1 \cdot n, 2 - a_2 \cdot n, \ldots, n - a_n \cdot n]</math>, where <math>(a_1, a_2, \ldots, a_n)</math> is an integer vector such that <math>a_1 + a_2 + \ldots + a_n = 0</math>, that is, where <math>(a_1, \ldots, a_n) \in \Lambda</math>. Geometrically, this kernel consists of the [[translation (geometry)|translations]], the isometries that shift the entire space {{mvar|V}} without rotating or reflecting it.{{sfnp|Kane|2001|loc=Section 11-1}} In an [[abuse of notation]], the symbol {{math|Λ}} is used in this article for all three of these sets (integer vectors in {{mvar|V}}, affine permutations with underlying permutation the identity, and translations); in all three settings, the natural group operation turns {{math|Λ}} into an [[abelian group]], generated [[Free abelian group|freely]] by the {{math|''n'' − 1}} vectors <math>\{(1, -1, 0, \ldots, 0), (0, 1, -1, \ldots, 0), \ldots, (0, \ldots, 0, 1, -1)\}</math>.{{sfnp|Humphreys|1990|loc = Section 2.10}}

=== Connection between the geometric and combinatorial definitions ===
[[File:Alcoves labeled by affine permutations.png|thumb|upright=1.2|alt=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 <math>\widetilde{S}_3</math> labeled by affine permutations. An alcove {{mvar|A}} is labeled by the window notation for a permutation {{mvar|u}} if {{mvar|u}} sends the fundamental alcove (shaded) to {{mvar|A}}. Negative numbers are denoted by overbars.]]

The affine symmetric group <math>\widetilde{S}_n</math> has {{math|Λ}} as a [[normal subgroup]], and is isomorphic to the [[semidirect product]]
<math display = block> \widetilde{S}_n \cong S_n \ltimes \Lambda </math>
of this subgroup with the finite symmetric group <math>S_n</math>, where the action of <math>S_n</math> on {{math|Λ}} is by permutation of coordinates. Consequently, every element {{mvar|u}} of <math>\widetilde{S}_n</math> has a unique realization as a product 
<math> u = r \cdot t </math> where <math>r</math> is a permutation in the standard copy of <math>S_n</math> in <math>\widetilde{S}_n</math> and <math>t</math> is a translation in {{math|Λ}}.{{sfnp|Lewis|McCammond|Petersen|Schwer|2019|loc=Section 4.1}}

This point of view allows for a direct translation between the combinatorial and geometric definitions of <math>\widetilde{S}_n</math>: if one writes <math> [u(1), \ldots, u(n)] = [r_1 - a_1 \cdot n, \ldots, r_n - a_n \cdot n]</math> where <math>r = [r_1, \ldots, r_n] = \pi(u)</math> and <math>(a_1, a_2, \ldots, a_n) \in \Lambda</math> then the affine permutation {{mvar|u}} corresponds to the rigid motion of {{mvar|V}} defined by{{sfnp|Lewis|McCammond|Petersen|Schwer|2019|loc=Section 4.1}}
<math display = block> (x_1, \ldots, x_n) \cdot u = \left(x_{r(1)} + a_1, \ldots, x_{r(n)} + a_n \right).</math>

Furthermore, as with every affine Coxeter group, the affine symmetric group [[transitive group action|acts transitively]] and [[free group action|freely]] on the set of alcoves: for each two alcoves, a unique group element takes one alcove to the other.{{sfnp|Humphreys|1990|loc = Chapter 4.5}} Hence, making an arbitrary choice of alcove <math>A_0</math> places the group in one-to-one correspondence with the alcoves: the identity element corresponds to <math>A_0</math>, and every other group element {{mvar|g}} corresponds to the alcove <math>A = A_0 \cdot g</math> that is the image of <math>A_0</math> under the action of {{mvar|g}}.{{sfnp|Humphreys|1990|loc = Chapter 4}}

=== Example: {{math|''n'' {{=}} 2}} ===
[[File:Infinite dihedral group.png|thumb|alt=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 <math>\widetilde{S}_2</math> acts on the line {{mvar|V}} in the Euclidean plane. The reflections are through the dashed lines. The vectors of the root lattice {{math|Λ}} are marked.]]
Algebraically, <math>\widetilde{S}_2</math> is the infinite dihedral group, generated by two generators <math>s_0, s_1</math> subject to the relations <math>s_0^2 = s_1^2 = 1</math>.{{sfnp|Björner|Brenti|2005|pp = 5–6}} Every other element of the group can be written as an alternating product of copies of <math>s_0</math> and <math>s_1</math>.{{sfnp|Gallian|2013|p = 454}}

Combinatorially, the affine permutation <math>s_1</math> has window notation <math>[2, 1]</math>, corresponding to the bijection <math>2k \mapsto 2k - 1, 2k - 1 \mapsto 2k</math> for every integer {{mvar|k}}. The affine permutation <math>s_0</math> has window notation <math>[0, 3]</math>, corresponding to the bijection <math>2k \mapsto 2k + 1, 2k + 1 \mapsto 2k</math> for every integer {{mvar|k}}. Other elements have the following window notations:

: <math>
\begin{align}
\overbrace{s_0 s_1 \cdots s_0 s_1}^{2k \text{ factors}} & = [1 + 2k, 2 - 2k ], \\[5pt]
\overbrace{s_1 s_0 \cdots s_1 s_0}^{2k \text{ factors}} & = [1 - 2k, 2 + 2k ], \\[5pt]
\overbrace{s_0 s_1 \cdots s_0}^{2k + 1 \text{ factors}} & = [2 + 2k, 1 - 2k ], \\[5pt]
\overbrace{s_1 s_0 \cdots s_1}^{2k + 1 \text{ factors}} & = [2 - 2(k + 1), 1 + 2(k + 1) ].
\end{align}
</math>

Geometrically, the space {{mvar|V}} on which <math>\widetilde{S}_2</math> acts is a line, with infinitely many equally spaced reflections.{{sfnp|Gallian|2013|p = 455}} It is natural to identify the line {{mvar|V}} with the real line <math> \R^1</math>, <math>s_0</math> with reflection around the point {{math|0}}, and <math>s_1</math> with reflection around the point {{math|1}}. In this case, the reflection <math>(s_0 s_1)^k s_0</math> reflects across the point {{math|–''k''}} for any integer {{mvar|k}}, the composition <math>s_0 s_1</math> translates the line by {{math|–2}}, and the composition <math>s_1 s_0</math> translates the line by {{math|2}}.{{sfnp|Lewis|Reiner|2016|loc = Section 4.1}}{{sfnp|Gallian|2013|p = 455}}

==Permutation statistics and permutation patterns==
Many [[Random permutation statistics|permutation statistics]] and other features of the combinatorics of finite permutations can be extended to the affine case.{{sfnp|Björner|Brenti|2005|p = 245}}

===Descents, length, and inversions===
The ''length'' <math>\ell(g)</math> of an element {{mvar|g}} of a Coxeter group {{mvar|G}} is the smallest number {{mvar|k}} such that {{mvar|g}} can be written as a product <math> g= s_{i_1} \cdots s_{i_k}</math> of {{mvar|k}} Coxeter generators of {{mvar|G}}.{{sfnp|Björner|Brenti|2005|p = 15}}
Geometrically, the length of an element {{mvar|g}} in <math>\widetilde{S}_n</math> is the number of reflecting hyperplanes that separate <math>A_0</math> and <math> A_0 \cdot g</math>, where <math>A_0</math> is the fundamental alcove (the simplex bounded by the reflecting hyperplanes of the Coxeter generators <math>s_0, s_1, \ldots, s_{n - 1}</math>).{{efn|In fact, the same is true for any affine Coxeter group.}}{{sfnp|Humphreys|1990|p = 93}}
Combinatorially, the length of an affine permutation is encoded in terms of an appropriate notion of [[Inversion (discrete mathematics)|inversions]]: for an affine permutation {{mvar|u}}, the length is{{sfnp|Björner|Brenti|2005|p = 261}}
<math display = block> \ell(u) = \# \left\{ (i, j) \colon i \in \{1, \ldots, n\}, j \in \mathbb{Z}, i < j, \text{ and } u(i) > u(j) \right\}.</math>
Alternatively, it is the number of equivalence classes of pairs <math> (i, j) \in \mathbb{Z} \times \mathbb{Z} </math> such that <math> i < j</math> and <math> u(i) > u(j)</math> under the equivalence relation <math> (i, j) \equiv (i', j') </math> if <math>(i - i', j - j') = (kn, kn)</math> for some integer {{mvar|k}}.
The [[generating function]] for length in <math>\widetilde{S}_n</math> is{{sfnp|Björner|Brenti|2005|p = 208}}{{sfnp|Björner|Brenti|1996|loc = Cor. 4.7}}
<math display = block> \sum_{g \in \widetilde{S}_n} q^{\ell(g)} = \frac{1 - q^n}{(1 - q)^n}.</math>

Similarly, there is an affine analogue of [[Permutation#Descents|descents]] in permutations: an affine permutation {{mvar|u}} has a descent in position {{mvar|i}} if <math>u(i) > u(i + 1)</math>. (By periodicity, {{mvar|u}} has a descent in position {{mvar|i}} if and only if it has a descent in position <math>i + kn</math> for all integers {{mvar|k}}.)
Algebraically, the descents corresponds to the ''right descents'' in the sense of Coxeter groups; that is, {{mvar|i}} is a descent of {{mvar|u}} if and only if <math> \ell(u \cdot s_i) < \ell (u)</math>.{{sfnp|Björner|Brenti|2005|p = 263}} The left descents (that is, those indices {{mvar|i}} such that <math> \ell(s_i \cdot u) < \ell (u)</math>) are the descents of the inverse affine permutation <math>u^{-1}</math>; equivalently, they are the values {{mvar|i}} such that {{mvar|i}} occurs before {{math| ''i'' − 1}} in the sequence <math> \ldots, u(-2), u(-1), u(0), u(1), u(2), \ldots</math>.{{sfnp|Chmutov|Lewis|Pylyavskyy|2022|loc = Section 3.2}}
Geometrically, {{mvar|i}} is a descent of {{mvar|u}} if and only if the fixed hyperplane of <math>s_i</math> separates the alcoves <math>A_0</math> and <math> A_0 \cdot u.</math>{{sfnp|Shi|1987|p=55}}

Because there are only finitely many possibilities for the number of descents of an affine permutation, but infinitely many affine permutations, it is not possible to [[wikt:Naive#Adjective|naively]] form a generating function for affine permutations by number of descents (an affine analogue of [[Eulerian polynomial]]s).{{sfnp|Reiner|1995|p = 2}} One possible resolution is to consider affine descents (equivalently, cyclic descents) in the finite symmetric group <math>S_n</math>.{{sfnp|Petersen|2015|loc=Chapter 14}} Another is to consider simultaneously the length and number of descents of an affine permutation. The [[multivariate generating function]] for these statistics over <math>\widetilde{S}_n</math> simultaneously for all {{mvar|n}} is
<math display = block> \sum_{n \geq 1} \frac{x^n}{1 - q^n} \sum_{w \in \widetilde{S}_n} t^{\operatorname{des}(w)} q^{\ell(w)}
=
\left[
\frac{x \cdot \frac{\partial}{\partial{x}} \log(\exp(x; q))}{1 - t \exp(x; q)}
\right]_{x \mapsto x \frac{1 - t}{1 - q}}
</math>
where {{math|des(''w'')}} is the number of descents of the affine permutation {{mvar|w}} and <math>\exp(x; q) = \sum_{n \geq 0} \frac{x^n (1 - q)^n}{(1 - q)(1 - q^2) \cdots (1 - q^n)}</math> is the [[q-exponential|{{mvar|q}}-exponential function]].{{sfnp|Reiner|1995|loc = Theorem 6}}

=== Cycle type and reflection length ===

Any bijection <math>u : \mathbb{Z} \to \mathbb{Z}</math> partitions the integers into a (possibly infinite) list of (possibly infinite) cycles: for each integer {{mvar|i}}, the cycle containing {{mvar|i}} is the sequence <math> ( \ldots, u^{-2}(i), u^{-1}(i), i, u(i), u^2(i), \ldots )</math> where exponentiation represents functional composition. <!--For example, the affine permutation in <math>\widetilde{S}_5</math> with window notation <math>[ 6, 3, 2, 0, 4]</math> contains the two infinite cycles
<math> (\ldots, -9, -4, 1, 6, 11, \ldots)</math> and 
<math> (\ldots, 10, 9, 5, 4, 0, -1, -5 \ldots)</math>
as well as infinitely many finite cycles <math> (5k + 2, 5k + 3)</math> for each <math>k \in \mathbb{Z}</math>. Cycles of an affine permutation correspond to cycles of the underlying permutation in an obvious way: in the example above, with underlying permutation <math> [1, 3, 2, 5, 4] = (1)(23)(45)</math>, the first infinite cycle corresponds to the cycle (1), the second corresponds to the cycle (45), and the finite cycles all correspond to the cycle (23).-->
For an affine permutation {{mvar|u}}, the following conditions are equivalent: all cycles of {{mvar|u}} are finite, {{mvar|u}} has finite [[order (group theory)|order]], and the geometric action of {{mvar|u}} on the space {{mvar|V}} has at least one fixed point.{{sfnp|Lewis|McCammond|Petersen|Schwer|2019|loc=Propositions 1.31 and 4.24}}

The ''reflection length'' <math>\ell_R(u)</math> of an element {{mvar|u}} of <math>\widetilde{S}_n</math> is the smallest number {{mvar|k}} such that there exist reflections <math>r_1, \ldots, r_k</math> such that <math>u = r_1 \cdots r_k</math>. (In the symmetric group, reflections are transpositions, and the reflection length of a permutation {{mvar|u}} is <math>n - c(u)</math>, where <math>c(u)</math> is the number of cycles of {{mvar|u}}.{{sfnp|Lewis|McCammond|Petersen|Schwer|2019}}) In {{harv|Lewis|McCammond|Petersen|Schwer|2019}}, the following formula was proved for the reflection length of an affine permutation {{mvar|u}}: for each cycle of {{mvar|u}}, define the ''weight'' to be the integer ''k'' such that consecutive entries congruent modulo {{mvar|n}} differ by exactly {{math|''kn''}}. <!--(For example, in the permutation <math>[ 5, 2, 0, 3]</math> above, the first infinite cycle has weight 1 and the second infinite cycle has weight −1; all finite cycles have weight 0.)--> Form a tuple of cycle weights of {{mvar|u}} (counting translates of the same cycle by multiples of {{mvar|n}} only once), and define the ''nullity'' <math>\nu(u)</math> to be the size of the smallest [[set partition]] of this tuple so that each part sums to 0. <!--(In the example above, the tuple is <math>(1, -1, 0)</math> and the nullity is 2, with corresponding partition <math>(1, -1), (0)</math>.)--> Then the reflection length of {{mvar|u}} is 
<math display = block> \ell_R(u) = n - 2 \nu(u) + c(\pi(u)),</math>
where <math>\pi(u)</math> is the underlying permutation of {{mvar|u}}.{{sfnp|Lewis|McCammond|Petersen|Schwer|2019|loc = Theorem 4.25}}

For every affine permutation {{mvar|u}}, there is a choice of subgroup {{mvar|W}} of <math>\widetilde{S}_n</math> such that <math>W \cong S_n</math>, <math> \widetilde{S}_n = W \ltimes \Lambda</math>, and for the standard form <math> u = w \cdot t </math> implied by this semidirect product, the reflection lengths are additive, that is, <math> \ell_R(u) = \ell_R(w) + \ell_R(t)</math>.{{sfnp|Lewis|McCammond|Petersen|Schwer|2019|loc = Corollary 2.5}}

===Fully commutative elements and pattern avoidance===
A ''reduced word'' for an element {{mvar|g}} of a Coxeter group is a tuple <math>(s_{i_1}, \ldots, s_{i_{\ell(g)}})</math> of Coxeter generators of minimum possible length such that <math> g = s_{i_1} \cdots s_{i_{\ell(g)}}</math>.{{sfnp|Björner|Brenti|2005|p = 15}} The element {{mvar|g}} is called ''fully commutative'' if any reduced word can be transformed into any other by sequentially swapping pairs of factors that commute.{{sfnp|Stembridge|1996|p = 353}} For example, in the finite symmetric group <math>S_4</math>, the element <math> 2143 = (12)(34)</math> is fully commutative, since its two reduced words <math>(s_1, s_3)</math> and <math>(s_3, s_1)</math> can be connected by swapping commuting factors, but <math> 4132 = (142)(3)</math> is not fully commutative because there is no way to reach the reduced word <math>(s_3, s_2, s_3, s_1)</math> starting from the reduced word <math>(s_2, s_3, s_2, s_1)</math> by commutations.{{sfnp|Billey|Jockusch|Stanley|1993|p=358}}

{{harvtxt|Billey|Jockusch|Stanley|1993}} proved that in the finite symmetric group <math>S_n</math>, a permutation is fully commutative if and only if it avoids the [[permutation pattern]] 321, that is, if and only if its one-line notation contains no three-term decreasing subsequence. In {{harv|Green|2002}}, this result was extended to affine permutations: an affine permutation {{mvar|u}} is fully commutative if and only if there do not exist integers <math> i < j < k</math> such that <math> u(i) > u(j) > u(k)</math>.{{efn|The three positions {{mvar|i}}, {{mvar|j}}, and {{mvar|k}} need not lie in a single window. For example, the affine permutation {{mvar|w}} in <math>\widetilde{S}_4</math> with window notation <math>[-4, -1, 1, 14]</math> is not fully commutative, because <math> w(0) = 10</math>, <math>w(3) = 1</math>, and <math>w(5) = 0</math>, even though no four consecutive positions contain a decreasing subsequence of length three.{{sfnp|Hanusa|Jones|2010|p = 1345}}}}

The number of affine permutations avoiding a single pattern {{mvar|p}} is finite if and only if {{mvar|p}} avoids the pattern 321,{{sfnp|Crites|2010|loc = Theorem 1}} so in particular there are infinitely many fully commutative affine permutations. These were enumerated by length in {{harv|Hanusa|Jones|2010}}.

==Parabolic subgroups and other structures==
The parabolic subgroups of <math>\widetilde{S}_n</math> and their [[coset]] representatives offer a rich combinatorial structure. Other aspects of affine symmetric groups, such as their [[Bruhat order]] and [[representation theory]], may also be understood via combinatorial models.{{sfnp|Björner|Brenti|2005|p = 245}}

===Parabolic subgroups, coset representatives===
[[File:Abacus diagram for affine S 4.png|thumb|right|upright|alt=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]]]

A ''standard [[parabolic subgroup of a reflection group|parabolic subgroup]]'' of a Coxeter group is a subgroup generated by a subset of its Coxeter generating set.{{sfnp|Björner|Brenti|2005|p = 38}} The maximal parabolic subgroups are those that come from omitting a single Coxeter generator. In <math>\widetilde{S}_n</math>, all maximal parabolic subgroups are isomorphic to the finite symmetric group <math>S_n</math>. The subgroup generated by the subset <math> \{s_0, \ldots, s_{n - 1} \} \smallsetminus \{s_i\}</math> consists of those affine permutations that stabilize the interval <math>[i + 1, i + n]</math>, that is, that map every element of this interval to another element of the interval.{{sfnp|Björner|Brenti|2005|p = 263}}

For a fixed element {{mvar|i}} of <math>\{0, \ldots, n - 1\}</math>, let <math> J = \{s_0, \ldots, s_{n - 1} \} \smallsetminus \{s_i\}</math> be the maximal proper subset of Coxeter generators omitting <math>s_i</math>, and let <math>(\widetilde{S}_n)_J</math> denote the parabolic subgroup generated by {{mvar|J}}. Every [[coset]] <math> g \cdot (\widetilde{S}_n)_J </math> has a unique element of minimum length. The collection of such representatives, denoted <math>(\widetilde{S}_n)^J</math>, consists of the following affine permutations:{{sfnp|Björner|Brenti|2005|p = 263}}
<math display = block> (\widetilde{S}_n)^J = \left \{ u \in \widetilde{S}_n \colon u(i - n + 1) < u(i - n + 2) < \cdots < u(i - 1) < u(i) \right \}.</math>

In the particular case that <math> J = \{s_1, \ldots, s_{n - 1} \}</math>, so that <math>(\widetilde{S}_n)_J \cong S_n</math> is the standard copy of <math>S_n</math> inside <math>\widetilde{S}_n</math>, the elements of <math>(\widetilde{S}_n)^J \cong \widetilde{S}_n/S_n</math> may naturally be represented by ''abacus diagrams'': the integers are arranged in an infinite strip of width {{mvar|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. For example, the minimal coset representative <math> u = [-5, 0, 6, 9]</math> is represented by the abacus diagram at right. To compute the length of the representative from the abacus diagram, one adds up the number of uncircled numbers that are smaller than the last circled entry in each column. (In the example shown, this gives <math> 5 + 3 + 0 + 1 = 9</math>.){{sfnp|Hanusa|Jones|2010|loc = Section 2.2}}

Other combinatorial models of minimum-length coset representatives for <math>\widetilde{S}_n/S_n</math> can be given in terms of core partitions ([[integer partition]]s in which no [[Young tableau#Arm and leg length|hook length]] is divisible by {{mvar|n}}) or bounded partitions (integer partitions in which no part is larger than {{math|''n'' − 1}}). Under these correspondences, it can be shown that the [[weak Bruhat order]] on <math>\widetilde{S}_n / S_n</math> is isomorphic to a certain subposet of [[Young's lattice]].{{sfnp|Lapointe|Morse|2005}}{{sfnp|Berg|Jones|Vazirani|2009}}

===Bruhat order===
The [[Bruhat order]] on <math>\widetilde{S}_n</math> has the following combinatorial realization. If {{mvar|u}} is an affine permutation and {{mvar|i}} and {{mvar|j}} are integers, define
<math> u [i, j] </math>
to be the number of integers {{mvar|a}} such that <math> a \leq i </math> and <math> u(a) \geq j</math>. (For example, with <math> u = [2, 0, 4] \in \widetilde{S}_3</math>, one has <math>u [ 3, 1 ] = 3</math>: the three relevant values are <math> a = 0, 1, 3 </math>, which are respectively mapped by {{mvar|u}} to 1, 2, and 4.) Then for two affine permutations {{mvar|u}}, {{mvar|v}}, one has that <math>u \leq v</math> in Bruhat order if and only if <math> u[i, j] \leq v[i, j] </math> for all integers {{mvar|i}}, {{mvar|j}}.{{sfnp|Björner|Brenti|2005|p = 264}}

=== Representation theory and an affine Robinson–Schensted correspondence ===
In the finite symmetric group, the [[Robinson–Schensted correspondence]] gives a bijection between the group and pairs <math>(P, Q)</math> of [[standard Young tableau]]x of the same shape. This bijection plays a central role in the combinatorics and the [[representation theory of the symmetric group]]. For example, in the language of [[Kazhdan–Lusztig theory]], two permutations lie in the same left cell if and only if their images under Robinson–Schensted have the same tableau {{mvar|Q}}, and in the same right cell if and only if their images have the same tableau {{mvar|P}}. In {{harv|Shi|1986}}, Jian-Yi Shi showed that left cells for <math>\widetilde{S}_n</math> are indexed instead by ''tabloids'',{{efn|A tabloid is a filling of the Young diagram with distinct entries, where two fillings are equivalent if they differ in the order of elements in rows. They are equinumerous with row-strict tableaux, in which entries are required to increase along rows (whereas standard Young tableau have entries that increase along rows and down columns).{{sfnp|Chmutov|Frieden|Kim|Lewis|2022|loc = Section 2.2.2}}}} and in {{harv|Shi|1991}} he gave an algorithm to compute the tabloid analogous to the tableau {{mvar|P}} for an affine permutation. In {{harv|Chmutov|Pylyavskyy|Yudovina|2018}}, the authors extended Shi's work to give a bijective map between <math>\widetilde{S}_n</math> and triples <math>(P, Q, \rho)</math> 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 [[Viennot's geometric construction|shadow construction]], introduced in {{harv|Viennot|1977}}.

==Inverse realizations==
[[File:Alcoves labeled by affine permutations-inverse.png|thumb|upright=1.2|alt=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 <math>\widetilde{S}_3</math> labeled by affine permutations, inverse to the labeling above]]
In some situations, one may wish to consider the action of the affine symmetric group on <math>\Z</math> or on alcoves that is inverse to the one given above.{{efn|In other words, one might be interested in switching from a [[left action|left group action]] to a right action or vice versa.{{sfnp|Shi|1986|p=68}}}} These alternate realizations are described below.

In the combinatorial action of <math>\widetilde{S}_n</math> on <math>\Z</math>, the generator <math>s_i</math> acts by switching the ''values'' {{mvar|i}} and {{math| ''i'' + 1}}. In the inverse action, it instead switches the entries in ''positions'' {{mvar|i}} and {{math| ''i'' + 1}}. Similarly, the action of a general reflection will be to switch the entries at ''positions'' {{math|''j'' − ''kn''}} and {{math|''i'' + ''kn''}} for each {{mvar|k}}, fixing all inputs at positions not congruent to {{mvar|i}} or {{mvar|j}} modulo {{mvar|n}}.{{sfnp|Knutson|Lam|Speyer|2013|loc=Section 2.1}}{{efn|In the finite symmetric group <math>S_n</math>, the analogous distinction is between the ''active'' and ''passive'' forms of a permutation.<ref>As in {{harv|Cameron|1994|loc=Section 3.5}}.</ref>}}

In the geometric action of <math>\widetilde{S}_n</math>, the generator <math>s_i</math> acts on an alcove {{mvar|A}} by reflecting it across one of the bounding planes of the fundamental alcove {{math|''A''<sub>0</sub>}}. In the inverse action, it instead reflects {{mvar|A}} across one of ''its own'' bounding planes. From this perspective, a reduced word corresponds to an ''alcove walk'' on the tessellated space {{mvar|V}}.<ref>As in, for example, {{harv|Beazley|Nichols|Park|Shi|2015}}, {{harv|Lam|2015}}.</ref>

== Relationship to other mathematical objects ==
The affine symmetric groups are closely related to a variety of other mathematical objects.

=== Juggling patterns ===
{{Wide image|Juggling-cropped.png|800px|alt=An alternating sequence of black and white circles runs from left to right. Interweaving paths in three colors are drawn; each circle lies on exactly one path, and the paths each connect two consecutive balls, then skip over three, then skip over three, and repeat this pattern. |The juggling pattern [[441 (juggling)|441]] visualized as an arc diagram: the height of each throw corresponds to the length of an arc; the two colors of nodes are the left and right hands of the juggler. This pattern has four crossings, which repeat periodically.}}
[[File:Juggling_441.gif|thumb|upright=0.5|alt=A stick-figure person juggling three balls|The juggling pattern 441]]
In {{harv|Ehrenborg|Readdy|1996}}, a correspondence is given between affine permutations and [[juggling pattern]]s encoded in a version of [[siteswap notation]].{{sfnp|Polster|2003|p=42}} Here, a juggling pattern of period {{mvar|n}} is a sequence <math>(a_1, \ldots, a_n)</math> of nonnegative integers (with certain restrictions) that captures the behavior of balls thrown by a juggler, where the number <math>a_i</math> indicates the length of time the {{mvar|i}}th throw spends in the air (equivalently, the height of the throw).{{efn|Not every sequence of {{mvar|n}} nonnegative integers is a juggling sequence. In particular, a sequence corresponds to a "simple juggling pattern", with one ball caught and thrown at a time, if and only if the function <math>i \mapsto i + a_i \mod{n}</math> is a permutation of <math>\{1, \ldots, n\}</math>.{{sfnp|Polster|2003|p=22}}}} The number {{mvar|b}} of balls in the pattern is the average <math>b = \frac{a_1 + \cdots + a_n}{n}</math>.{{sfnp|Polster|2003|p=15}} The Ehrenborg&ndash;Readdy correspondence associates to each juggling pattern <math>{\bf a} = (a_1, \ldots, a_n)</math> of period {{mvar|n}} the function <math>w_{\bf a} \colon \Z \to \Z</math> defined by
<math display = block> w_{\bf a}(i) = i + a_i - b,</math>
where indices of the sequence '''a''' are taken modulo {{mvar|n}}. Then <math>w_{\bf a}</math> is an affine permutation in <math>\widetilde{S}_n</math>, and moreover every affine permutation arises from a juggling pattern in this way.{{sfnp|Polster|2003|p=42}} Under this bijection, the length of the affine permutation is encoded by a natural statistic in the juggling pattern:
<math display = block>\ell(w_{\bf a}) = (b - 1)n - \operatorname{cross}({\bf a}),</math>
where <math>\operatorname{cross}({\bf a})</math> is the number of crossings (up to periodicity) in the arc diagram of '''a'''. This allows an elementary proof of the generating function for affine permutations by length.{{sfnp|Polster|2003|p=43}}

For example, the juggling pattern [[441 (juggling)|441]] has <math>n = 3</math> and <math> b = \frac{4 + 4 + 1}{3} = 3</math>. Therefore, it corresponds to the affine permutation <math>w_{441} = [1 + 4 - 3, 2 + 4 - 3, 3 + 1 - 3] = [2, 3, 1]</math>. The juggling pattern has four crossings, and the affine permutation has length <math>\ell(w_{441}) = (3 - 1) \cdot 3 - 4 = 2</math>.{{sfnp|Polster|2003|loc=Section 2.7}}

Similar techniques can be used to derive the generating function for minimal coset representatives of <math>\widetilde{S}_n/S_n</math> by length.{{sfnp|Clark|Ehrenborg|2011|loc = Theorem 2.2}}

=== Complex reflection groups ===
In a finite-dimensional real [[inner product space]], a ''reflection'' is a [[linear transformation]] that fixes a linear hyperplane pointwise and negates the vector orthogonal to the plane. This notion may be extended to vector spaces over other [[field (mathematics)|fields]]. In particular, in a complex inner product space, a ''reflection'' is a [[unitary transformation]] {{mvar|T}} of finite order that fixes a hyperplane.{{efn|In some sources, unitary reflections are called ''pseudoreflections''.{{sfnp|Lehrer|Taylor|2009|p=9}}}} This implies that the vectors orthogonal to the hyperplane are eigenvectors of {{mvar|T}}, and the associated eigenvalue is a complex [[root of unity]]. A ''[[complex reflection group]]'' is a finite group of linear transformations on a complex vector space generated by reflections.{{sfnp|Lehrer|Taylor|2009|p=10}}

The complex reflection groups were fully classified by {{harvtxt|Shephard|Todd|1954}}: each complex reflection group is isomorphic to a product of irreducible complex reflection groups, and every irreducible either belongs to an infinite family <math>G(m, p, n)</math> (where {{mvar|m}}, {{mvar|p}}, and {{mvar|n}} are positive integers such that {{mvar|p}} divides {{mvar|m}}) or is one of 34 other (so-called "exceptional") examples. The group <math>G(m, 1, n)</math> is the [[generalized symmetric group]]: algebraically, it is the [[wreath product]] <math> (\Z / m \Z) \wr S_n</math> of the [[cyclic group]] <math>\Z / m \Z</math> with the symmetric group <math>S_n</math>. Concretely, the elements of the group may be represented by [[monomial matrix|monomial matrices]] (matrices having one nonzero entry in every row and column) whose nonzero entries are all {{mvar|m}}th roots of unity. The groups <math>G(m, p, n)</math> are subgroups of <math>G(m, 1, n)</math>, and in particular the group <math>G(m, m, n)</math> consists of those matrices in which the product of the nonzero entries is equal to 1.{{sfnp|Lehrer|Taylor|2009|loc=Chapter 2}}

In {{harv|Shi|2002}}, Shi showed that the affine symmetric group is a ''generic cover'' of the family <math>\left\{G(m, m, n) \colon m \geq 1 \right\}</math>, in the following sense: for every positive integer {{mvar|m}}, there is a surjection <math>\pi_m</math> from <math>\widetilde{S}_n</math> to <math>G(m, m, n)</math>, and these maps are compatible with the natural surjections <math>G(m, m, n) \twoheadrightarrow G(p, p, n)</math> when <math>p \mid m</math> that come from raising each entry to the {{math|''m''/''p''}}th power. Moreover, these projections respect the reflection group structure, in that the image of every reflection in <math>\widetilde{S}_n</math> under <math>\pi_m</math> is a reflection in <math>G(m, m, n)</math>; and similarly when <math>m > 1</math> the image of the standard [[Coxeter element]] <math>s_0 \cdot s_1 \cdots s_{n - 1}</math> in <math>\widetilde{S}_n</math> is a Coxeter element in <math> G(m, m, n)</math>.{{sfnp|Lewis|2020|loc = Section 3.2}}

=== Affine Lie algebras ===
Each affine Coxeter group is associated to an [[affine Lie algebra]], a certain [[Infinite-dimensional space|infinite-dimensional]] [[non-associative algebra]] with unusually nice representation-theoretic properties.{{efn|For example, like finite-dimensional [[semisimple Lie algebra]]s, they admit an explicit parameterization of their [[integrable representation|integrable]] [[highest weight module]]s; whereas there is no corresponding general theory for general infinite-dimensional Lie algebras.{{sfnp|Kac|1990|loc = Introduction}} }} In this association, the Coxeter group arises as a group of symmetries of the root space of the Lie algebra (the dual of the [[Cartan subalgebra]]).{{sfnp|Kac|1990|loc = Chapter 3}} In the classification of affine Lie algebras, the one associated to <math>\widetilde{S}_n</math> is of (untwisted) type <math>A_{n - 1}^{(1)}</math>, with [[Cartan matrix#Lie algebras|Cartan matrix]] <math> \left[ \begin{array}{rr} 2 & - 2 \\ - 2& 2 \end{array} \right] </math> for <math>n = 2</math> and
<math display = block> \left[ \begin{array}{rrrrrr} 2 & -1 & 0 & \cdots & 0 & -1 \\
-1 & 2 & -1 & \cdots & 0 & 0 \\
0 & -1 & 2 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 2 & -1 \\
-1 & 0 & 0 & \cdots & -1 & 2
\end{array} \right]
</math>
(a [[circulant matrix]]) for <math> n > 2 </math>.{{sfnp|Kac|1990|loc = Chapter 4}}

Like other [[Kac–Moody algebra]]s, affine Lie algebras satisfy the [[Weyl–Kac character formula]], which expresses the [[character theory|characters]] of the algebra in terms of their [[highest weight]]s.{{sfnp|Kac|1990|loc = Chapter 10}} In the case of affine Lie algebras, the resulting identities are equivalent to the [[Macdonald identities]]. In particular, for the affine Lie algebra of type <math>A_1^{(1)}</math>, associated to the affine symmetric group <math>\widetilde{S}_2</math>, the corresponding Macdonald identity is equivalent to the [[Jacobi triple product]].{{sfnp|Kac|1990|loc = Chapter 12}}

=== Braid group and group-theoretic properties ===

Coxeter groups have a number of special properties not shared by all groups. These include that their [[word problem (mathematics)|word problem]] is [[Undecidable problem|decidable]] (that is, there exists an [[algorithm]] that can determine whether or not any given product of the generators is equal to the identity element) and that they are [[linear group]]s (that is, they can be represented by a group of invertible matrices over a field).{{sfnp|Björner|Brenti|2005|pp = 75, 92}}{{sfnp|McCammond|2017|p=6}}

Each Coxeter group {{mvar|W}} is associated to an [[Artin&ndash;Tits group]] <math>B_W</math>, which is defined by a similar presentation that omits relations of the form <math>s^2 = 1</math> for each generator {{mvar|s}}.{{sfnp|McCammond|2017|loc=Section 1.1}} In particular, the Artin&ndash;Tits group associated to <math>\widetilde{S}_n</math> is generated by {{mvar|n}} elements <math> \sigma_0, \sigma_1, \ldots, \sigma_{n - 1}</math> subject to the relations <math>\sigma_i \sigma_{i + 1} \sigma_i = \sigma_{i + 1}\sigma_i \sigma_{i + 1}</math> for <math>i = 0, \ldots, n - 1</math> (and no others), where as before the indices are taken modulo {{mvar|n}} (so <math> \sigma_n = \sigma_0</math>).{{sfnp|Kent, IV|Peifer|2002}} Artin&ndash;Tits groups of Coxeter groups are conjectured to have many nice properties: for example, they are conjectured to be [[torsion-free group|torsion-free]], to have trivial [[center (group theory)|center]], to have solvable word problem, and to satisfy the <math>K(\pi, 1)</math> conjecture. These conjectures are not known to hold for all Artin&ndash;Tits groups, but in {{harv|Charney|Peifer|2003}} it was shown that <math>B_{\widetilde{S}_n}</math> has these properties. (Subsequently, they have been proved for the Artin&ndash;Tits groups associated to affine Coxeter groups.){{sfnp|McCammond|Sulway|2017}}{{sfnp|McCammond|2017|pp = 14–17}}{{sfnp|Paolini|Salvetti|2021}} In the case of the affine symmetric group, these proofs make use of an associated [[Garside group|Garside structure]] on the Artin&ndash;Tits group.{{sfnp|McCammond|2017|p=17}}

[[File:Artin-Tits braid group of affine type A generators.svg|thumb|alt=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 {{math|''n'' {{=}} 4}}) and as braids drawn on a cylinder (for {{math|''n'' {{=}} 3}})]]

Artin&ndash;Tits groups are sometimes also known as ''generalized braid groups'', because the Artin&ndash;Tits group <math>B_{S_n}</math> of the (finite) symmetric group is the [[braid group]] on {{mvar|n}} strands.{{sfnp|McCammond|2017|p=11}} Not all Artin&ndash;Tits groups have a natural representation in terms of geometric braids. However, the Artin&ndash;Tits group of the [[hyperoctahedral group]] <math>S^{\pm}_n</math> (geometrically, the symmetry group of the ''n''-dimensional [[hypercube]]; combinatorially, the group of [[signed permutation]]s of size ''n'') does have such a representation: it is given by the subgroup of the braid group on <math>n + 1</math> strands consisting of those braids for which a particular strand ends in the same position it started in, or equivalently as the braid group of {{mvar|n}} strands in an [[annulus (mathematics)|annular]] region.{{sfnp|Kent, IV|Peifer|2002}}{{sfnp|Charney|Peifer|2003|pp=587–8}} Moreover, the Artin&ndash;Tits group of the hyperoctahedral group <math>S^{\pm}_n</math> can be written as a semidirect product of <math>B_{\widetilde{S}_n}</math> with an infinite cyclic group.{{sfnp|Charney|Peifer|2003|p=588}} It follows that <math>B_{\widetilde{S}_n}</math> may be interpreted as a certain subgroup consisting of geometric braids, and also that it is a [[linear group]].{{sfnp|Allcock|2002}}{{sfnp|Kent, IV|Peifer|2002}}{{sfnp|Charney|Peifer|2003|pp=586}}

=== Extended affine symmetric group ===
The affine symmetric group is a subgroup of the ''extended affine symmetric group''. The extended group is isomorphic to the [[wreath product]] <math> \Z \wr S_n</math>. Its elements are ''extended affine permutations'': bijections <math>u \colon \Z \to \Z</math> such that <math>u(x + n) = u(x) + n</math> for all integers {{mvar|x}}. Unlike the affine symmetric group, the extended affine symmetric group is not a Coxeter group. But it has a natural generating set that extends the Coxeter generating set for <math>\widetilde{S}_n</math>: the ''shift operator'' <math>\tau</math> whose window notation is <math>\tau = [2, 3, \ldots, n, n + 1]</math> generates the extended group with the simple reflections, subject to the additional relations <math> \tau s_i \tau^{-1} = s_{i + 1}</math>.{{sfnp|Chmutov|Pylyavskyy|Yudovina|2018|loc = Section 1.6}}

=== Combinatorics of other affine Coxeter groups ===
The geometric action of the affine symmetric group <math>\widetilde{S}_n</math> places it naturally in the family of [[affine Coxeter group]]s, each of which has a similar geometric action on an affine space. The combinatorial description of the <math>\widetilde{S}_n</math> may also be extended to many of these groups: in {{harvtxt|Eriksson|Eriksson|1998}}, an axiomatic description is given of certain permutation groups acting on <math>\Z</math> (the "George groups", in honor of [[George Lusztig]]), and it is shown that they are exactly the "classical" Coxeter groups of finite and affine types A, B, C, and D. (In the classification of affine Coxeter groups, the affine symmetric group is type A.) Thus, the combinatorial interpretations of descents, inversions, etc., carry over in these cases.{{sfnp|Björner|Brenti|2005|loc = Chapter 8}} Abacus models of minimum-length coset representatives for parabolic quotients have also been extended to this context.{{sfnp|Hanusa|Jones|2012}}

==History==
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. The modern systematic study (connecting the algebraic and geometric definitions of finite and affine Coxeter groups) began in work of Coxeter in the 1930s.{{sfnp|Björner|Brenti|2005|p = 24}} The combinatorial description of the affine symmetric group first appears in work of {{harvtxt|Lusztig|1983}}, and was expanded upon by {{harvtxt|Shi|1986}}; both authors used the combinatorial description to study the Kazhdan–Lusztig cells of <math>\widetilde{S}_n</math>.{{sfnp|Björner|Brenti|2005|p = 293}}{{sfnp|Green|2002}} The proof that the combinatorial definition agrees with the algebraic definition was given by {{harvtxt|Eriksson|Eriksson|1998}}.{{sfnp|Green|2002}}

== References ==
{{Academic peer reviewed|Q100400684|doi-access=free|m|mode=cs2}}
{{reflist|25em}}

===Notes===
{{notelist}}

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[[Category:Coxeter groups]]
[[Category:Reflection groups]]
[[Category:Permutation groups]]
[[Category:Symmetry]]
[[Category:Representation theory]]