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WikiJournal of Science, 2021, 4(1):3 doi: 10.15347/wjs/2021.003 Encyclopedic Review Article

Affine symmetric Joel Brewster Lewis¹*

Abstract The affine is a mathematical structure that describes the symmetries of the line and the regular triangular tesselation of the , as well as related higher dimensional objects. It is an infinite extension of the symmetric group, which consists of all (rearrangements) of a finite . In to its geo- metric description, the may be defined as the collection of permutations of the (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain gen- erators and relations. These different definitions allow for the extension of many important properties of the finite symmetric group to the infinite setting, and are studied as part of the fields of and .

Definitions Non-technical summary

The affine symmetric group, 푆̃푛, may be equivalently Flat, straight-edged shapes (like triangles) or 3D ones defined as an abstract group by generators and rela- (like pyramids) have only a finite number of symme- tions, or in terms of concrete geometric and combina- tries. In contrast, the affine symmetric group is a way to torial models. mathematically describe all the symmetries possible when an infinitely large flat surface is covered by trian- gular tiles. As with many subjects in mathematics, it can Algebraic definition also be thought of in a number of ways: for example, it In terms of generators and relations, 푆̃푛 is generated also describes the symmetries of the infinitely long by a set number line, or the possible arrangements of all inte- gers (..., −2, −1, 0, 1, 2, ...) with certain repetitive pat- 푠 , 푠 , … , 푠 0 1 푛−1 terns. As a result, studying the affine symmetric group of 푛 elements that satisfy the following relations: extends the study of symmetries of straight-edged when 푛 ≥ 3, shapes or of groups of permutations to the infinite case. 2 It also connects several topics in mathematics that 1. 푠푖 = 1 (the generators are involutions), were originally studied for independent reasons, rang- 2. 푠푖푠푗 = 푠푗푠푖 if 푗 is not one of 푖 − 1, 푖, 푖 + 1, and ing from complex groups to juggling se- quences. 3. 푠푖푠푖+1푠푖 = 푠푖+1푠푖푠푖+1.

In the relations above, indices are taken modulo n, so that the third relation includes as a particular case 푠0푠푛−1푠0 = 푠푛−1푠0푠푛−1. (The second and third relation are sometimes called the braid relations.) When 푛 = 2, the affine symmetric group 푆̃2 is the infinite generated by two elements 푠0, 푠1 subject only to 2 2 [1] the relations 푠0 = 푠1 = 1.

This definition endows 푆̃푛 with the structure of a Coxe- ter group, with the 푠푖 as Coxeter generating set. For 푛 ≥ 3, its Coxeter–Dynkin diagram is the 푛-cycle, while for 푛 = 2 it consists of two nodes joined by an edge labeled ∞.[2]

1 George Washington University Figure 1 | Dynkin diagrams for the affine symmetric groups on *Author correspondence: [email protected] 2 and more than 2 generators ORCID: 0000-0003-0205-8049 Licensed under: CC-BY Received 06-06-2020; accepted 21-04-2021

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Geometric definition 푛 In the ℝ with coordinates (푥1, … , 푥푛), the set 푉 of points that satisfy the equation 푥1 + 푥2 + ⋯ + 푥푛 = 0 forms a (hyper)plane (an (푛 − 1)-dimen- sional subspace). For every pair of distinct elements 푖 and 푗 of {1, … , 푛} and every 푘, the set of points in 푉 that satisfy 푥푖 − 푥푗 = 푘 forms a plane in 푉, and there is a unique reflection of 푉 that fixes this plane. Then the affine symmetric group can be realized geo- metrically as the collection of all maps from 푉 to itself that arise by composing several of these reflections.[3] Inside 푉, the type 퐴 root Λ is the subset of points with integer coordinates, that is, it is the set of all the integer vectors (푎1, … , 푎푛) such that 푎1 + ⋯ + 푎푛 = 0. Each of the reflections preserves this lattice, and so the lattice is preserved by the whole group. In fact, one may Figure 2 | When 푛 = 3, the space V is a two-dimensional ̃ define 푆푛 to be the group of rigid transformations of 푉 plane and the reflections are across lines. The points of the that preserve the lattice Λ. type A root lattice are circled. These reflecting planes divide the space 푉 into congru- ent simplicies, called alcoves.[4] The situation when 푛 = with [푢(1), … , 푢(푛)] of integers that contain one 3 is shown at right; in this case, the root lattice is a tri- element from each congruence class modulo 푛 and sum angular lattice, and the reflecting lines divide the plane to 1 + 2 + ⋯ + 푛.[5] into equilateral triangular alcoves. (For larger 푛, the al- To translate between the combinatorial and algebraic coves are not regular simplices.) definitions, for 푖 = 1, … , 푛 − 1 one may identify the To translate between the geometric and algebraic defi- Coxeter generator 푠푖 with the affine that nitions, fix an alcove and consider the 푛 hyperplanes has window notation [1, 2, … , 푖 − 1, 푖 + 1, 푖, 푖 + that form its boundary. For example, there is a unique 2, … , 푛], and also identify the generator 푠0 = 푠푛 with alcove (the fundamental alcove) consisting of points the affine permutation [0, 2, 3, … , 푛 − 2, 푛 − 1, 푛 + 1]. (푥1, … , 푥푛) such that 푥1 ≥ 푥2 ≥ ⋯ ≥ 푥푛 ≥ 푥1 − 1, More generally, every reflection (that is, a conjugate of which is bounded by the hyperplanes 푥1 − 푥2 = 0, 푥2 − 푥3 = 0, ..., and 푥1 − 푥푛 = 1. (This is illustrated in the case 푛 = 3 at right.) For 푖 = 1, … , 푛 − 1, one may iden- tify the reflection through 푥푖 − 푥푖+1 = 0 with the Coxeter generator 푠푖, and also identify the reflection through 푥1 − 푥푛 = [4] 1 with the generator 푠0 = 푠푛.

Combinatorial definition The affine symmetric group may be real- ized as a group of periodic permutations of the integers. In particular, say that a bi- jection 푢: ℤ → ℤ is an affine permutation if 푢(푥 + 푛) = 푢(푥) + 푛 for all integers 푥 and 푢(1) + 푢(2) + ⋯ + 푢(푛) = 1 + 2 + ⋯ + 푛. (It is a consequence of the first property that the 푢(1), … , 푢(푛) must all be distinct modulo 푛.) Such a is uniquely determined by its win- dow notation [푢(1), … , 푢(푛)], and so af- Figure 3 | Reflections and alcoves for the affine symmetric group. The funda- fine permutations may also be identified mental alcove is shaded.

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one of the Coxeter generators) can be described combinatorially it corresponds to the window notations uniquely as follows: for distinct integers 푖, 푗 in {1, … , 푛} for which {푢(1), … , 푢(푛)} = {1, 2, … , 푛} (that is, in and arbitrary integer 푘, it maps 푖 to 푗 − 푘푛, maps 푗 to 푖 + which the window notation is the one-line notation of a 푘푛, and fixes all inputs not congruent to 푖 or 푗 modulo finite permutation).[8][3] 푛.[6] (In terms of the geometric definition, this corre- If u = [푢(1), 푢(2), … , 푢(푛)] is the window notation of sponds to the reflection across the plane 푥 − 푥 = 푘. 푖 푗 an element of this standard copy of 푆 ⊂ 푆̃ , its action The correspondence between the geometric and com- 푛 푛 on the hyperplane 푉 in ℝ푛 is given by permutation of binatorial representations for other elements is dis- coordinates: (푥 , 푥 , … , 푥 ) ⋅ 푢 = (푥 , 푥 , … , 푥 ). cussed below.) 1 2 푛 푢(1) 푢(2) 푢(푛) (In this article, the geometric action of permutations and affine permutations is on the right; thus, if 푢 and 푣 Representation as matrices are two affine permutations, the action of 푢푣 on a point One may represent affine permutations as infinite peri- is given by first applying 푢, then applying 푣.) odic permutation matrices.[7] If 푢: ℤ → ℤ is an affine per- There are also many nonstandard copies of 푆푛 con- mutation, one places the entry 1 at position (푖, 푢(푖)) in tained in 푆̃푛. A geometric construction is to pick any the infinite grid ℤ × ℤ for each integer 푖, and all other point 푎 in Λ (that is, an integer vector whose coordi- entries are equal to 0. Since 푢 is a , the resulting nates sum to 0); the (푆̃푛) of 푆̃푛 of isometries contains exactly one 1 in every row and column. 푎 that fix 푎 is isomorphic to 푆 . The analogous combina- The periodicity condition on the map 푢 ensures that the 푛 torial construction is to choose any subset 퐴 of ℤ that entry at position (푎, 푏) is equal to the entry at position contains one element from each mod- (푎 + 푛, 푏 + 푛) for every pair of integers (푎, 푏). For ex- ulo 푛 and whose elements sum to 1 + 2 + ⋯ + 푛; the ample, a portion of matrix for the affine permutation subgroup (푆̃푛) of 푆̃푛 of affine permutations that stabi- [2, 0, 4] ∈ 푆̃3 is shown below, with the conventions that 𝐴 1s are replaced by •, 0s are omitted, rows numbers in- lize 퐴 is isomorphic to 푆푛. crease from top to bottom, column numbers increase from left to right, and the boundary of the box consist- As a ing of rows and columns 1, 2, 3 is drawn: There is a simple map (technically, a surjective ) π from 푆̃푛 onto the finite symmetric group 푆푛. In terms of the combinatorial definition, it is to reduce the window entries modulo 푛 to elements of {1, 2, … , 푛}, leaving the one-line notation of a permuta- tion. The π(푢) of an affine permutation 푢 is called the underlying permutation of 푢.

The map π sends the Coxeter generator 푠0 = [0, 2, 3, 4, … , 푛 − 2, 푛 − 1, 푛 + 1] to the permutation

whose one-line notation and cycle notation are [푛, 2, 3, 4, … , 푛 − 2, 푛 − 1, 1] and (1 푛), respectively. In Relationship to the finite symmetric terms of the Coxeter generators of 푆푛, this can be writ- ten as π(푠 ) = 푠 푠 ⋯ 푠 푠 푠 ⋯ 푠 푠 . group 0 1 2 푛−2 푛−1 푛−2 2 1 The π is the set of affine permutations whose un- derlying permutation is the identity. The window nota- Relationship to the finite symmetric group tions of such affine permutations are of the form [1 − 푎 ⋅ 푛, 2 − 푎 ⋅ 푛, … , 푛 − 푎 ⋅ 푛] The affine symmetric group 푆̃푛 contains the finite sym- 1 2 푛 , where (푎 , 푎 , … , 푎 ) 푎 + 푎 + metric group 푆푛 as both a subgroup and a quotient. 1 2 푛 is an integer vector such that 1 2 ⋯ + 푎푛 = 0, that is, where (푎1, … , 푎푛) ∈ Λ. Geometri- As a subgroup cally, this kernel consists of the translations, that is, the isometries that shift the entire space 푉 without rotating There is a canonical way to choose a subgroup of 푆̃푛 that or reflecting it. In an , the symbol Λ is is isomorphic to the finite symmetric group 푆푛. In terms used in this article for all three of these sets (integer vec- of the algebraic definition, this is the subgroup of 푆̃푛 tors in 푉, affine permutations with underlying permuta- generated by 푠1, … , 푠푛−1 (excluding the simple reflec- tion the identity, and translations); in all three settings, tion 푠0 = 푠푛). Geometrically, this corresponds to the the natural group turns Λ into an abelian subgroup of transformations that fix the origin, while

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Figure 4 | Alcoves for 푆̃3 labeled by af- fine permutations. An alcove 퐴 is la- beled by the window notation for a permutation 푢 if 푢 sends the funda- mental alcove (shaded) to 퐴. Nega- tive numbers are denoted by over- bars.

group, generated freely by the 푛 − 1 vectors under the action of 푔. This identification for 푆̃3 is illus- {(1, −1, 0, … , 0), (0, 1, −1, … , 0), … , (0, … , 0, 1, −1)}. trated in Figure 4.

Connection between the geometric and Example: n = 2 combinatorial definitions Algebraically, 푆̃2 is the infinite dihedral group, gener-

The subgroup Λ is a of 푆̃푛, and one has ated by two generators 푠0, 푠1 subject to the relations 2 2 an 푠0 = 푠1 = 1. Every other element of the group can be written as an alternating product of copies of 푠0 and 푠1. 푆̃푛 ≅ 푆푛 ⋉ Λ Combinatorially, the affine permutation 푠 has window between 푆̃ and the of the finite 1 푛 notation [2, 1], corresponding to the bijection 2푘 ↦ symmetric group 푆푛 with Λ, where the action of 푆푛 on Λ is by permutation of coordinates. Consequently, identi- fying the finite symmetric group 푆푛 as its standard copy in 푆̃푛, one has that every element 푢 of 푆̃푛 may be real- ized uniquely as a product 푢 = 푟 ⋅ 푡 where 푟 ∈ 푆푛 is a fi- nite permutation and 푡 ∈ Λ. This point of view allows for a direct be- tween the combinatorial and geometric definitions of 푆̃푛: if one writes [푢(1), … , 푢(푛)] = [푟1 − 푎1 ⋅ 푛, … , 푟푛 − 푎푛 ⋅ 푛] where 푟 = [푟1, … , 푟푛] = 휋(푢) and (푎1, 푎2, … , 푎푛) ∈ Λ then the affine permutation 푢 corre- sponds to the rigid motion of 푉 defined by

(푥1, … , 푥푛) ⋅ 푢 = (푥푟(1) + 푎1, … , 푥푟(푛) + 푎푛). Furthermore, as with every affine , the af- fine symmetric group acts transitively and freely on the set of alcoves. Hence, by making an arbitrary choice of alcove 퐴0, one may place the group in one-to-one cor- respondence with the alcoves: the cor- responds to 퐴0, and every other group element 푔 corre- Figure 5 | The affine symmetric group 푆̃2 acts on the line 푉 in sponds to the alcove 퐴 = 퐴0 ⋅ 푔 that is the image of 퐴0 the Euclidean plane. The reflections are through the dashed lines. The vectors of the root lattice Λ are marked.

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2푘 − 1, 2푘 − 1 ↦ 2푘 for every integer 푘. The affine Alternatively, it is the number of equivalence classes of permutation 푠0 has window notation [0, 3], corre- pairs (푖, 푗) ∈ ℤ × ℤ such that 푖 < 푗 and 푢(푖) > 푢(푗) un- sponding to the bijection 2푘 ↦ 2푘 + 1, 2푘 + 1 ↦ 2푘 der the (푖, 푗) ≡ (푖′, 푗′) if (푖 − 푖′, 푗 − for every integer 푘. Other elements have the following 푗′) = (푘푛, 푘푛) for some integer 푘. window notations: [12][13] The for length in 푆̃푛 is 2푘 factors 1 − 푞푛 • 푠⏞0 푠 1 ⋯ 푠 0푠 1 = [1 + 2푘, 2 − 2푘], ∑ 푞ℓ(푔) = . (1 − 푞)푛 2푘 factors ∼ 푔∈푆푛 • 푠⏞1 푠 0 ⋯ 푠 1푠 0 = [1 − 2푘, 2 + 2푘], 2푘+1 factors Similarly, one may define an affine analogue of de- • 푠⏞0 푠1 ⋯ 푠 0 = [2 + 2푘, 1 − 2푘], scents in permutations: say that an affine permutation ( ) ( ) 2푘+1 factors 푢 has a descent in position 푖 if 푢 푖 > 푢 푖 + 1 . (By pe- • 푠⏞1 푠0 ⋯ 푠 1 = [2 − 2(푘 + 1), 1 + 2(푘 + 1)]. riodicity, 푢 has a descent in position 푖 if and only if it has a descent in position 푖 + 푘푛 for all integers 푘.)[14] Geometrically, the space 푉 is the line with equation 푥 + 푦 = 0 in the Euclidean plane ℝ2. The root lattice inside Algebraically, the descents corresponds to the right de- 푉 consists of those pairs (푎, −푎) for integral 푎. The Cox- scents in the sense of Coxeter groups; that is, 푖 is a de- [14] eter generator 푠1 acts on 푉 by reflection across the line scent of 푢 if and only if ℓ(푢 ⋅ 푠푖) < ℓ(푢). The left de- x = y (that is, across the origin); the generator 푠0 acts scents (that is, those indices 푖 such that ℓ(푠푖 ⋅ 푢) < ℓ(푢) on 푉 by reflection across the line x = y + 1 (that is, are the descents of the inverse affine permutation 푢−1; 1 1 across the point ( , − ). It is natural to identify the line equivalently, they are the values 푖 such that 푖 occurs be- 2 2 fore 푖 − 1 in the … , 푢(−2), 푢(−1), 푢(0), 푢(1), 푉 ℝ1 (푥, −푥) with the real line , by sending the point to 푢(2), … . the real number 2x. With this identification, the root lat- tice consists of the even integers; the fundamental al- Geometrically, 푖 is a descent of 푢 if and only if the fixed 푘 cove is the interval [0, 1]; the element (푠1푠0) acts by hyperplane of 푠푖 separates the alcoves 퐴0 and 퐴0 ⋅ 푢. translation by 푘 for any integer 푘; and the reflection 푘 Because there are only finitely many possibilities for the 푠1(푠0푠1) reflects across the point −푘 for any integer 푘. number of descents of an affine permutation, but infi- nitely many affine permutations, it is not possible to na- ively form a generating function for affine permutations Permutation statistics and by number of descents (an affine analogue of Eulerian permutation patterns ).[15] One possible resolution is to consider affine descents (equivalently, cyclic descents) in the fi- Many permutation statistics and other features of the [16] nite symmetric group 푆푛. Another is to consider sim- combinatorics of finite permutations can be extended ultaneously the length and number of descents of an af- to the affine case. fine permutation. The generating function for these statistics over 푆̃푛 simultaneously for all 푛 is Descents, length, and inversions 휕 푛 푥 ⋅ log(exp(푥; 푞)) 푥 ( ) ( ) 휕푥 The length ℓ(푔) of an element 푔 of a Coxeter group 퐺 is ∑ ∑ 푡푑푒푠 푤 푞ℓ 푤 = [ ] 1 − 푞푛 1 − 푡exp(푥; 푞) 푛≥1 푤∈푆̃ 1−푡 the smallest number 푘 such that 푔 can be written as a 푛 푥↦푥 1−푞 [9] product 푔 = 푠푖1 ⋯ 푠푖푘 of 푘 Coxeter generators of 퐺. where des(w) is the number of descents of the affine 푛 푛 Geometrically, the length of an element 푔 in S̃푛 is the 푥 (1−푞) permutation 푊 and exp(푥; 푞) = ∑푛≥0 is number of reflecting hyperplanes that separate 퐴 and (1−푞)(1−푞2)⋯(1−푞푛) 0 the q-exponential function.[17] 퐴0 ⋅ 푔, where 퐴0 is the fundamental alcove (the bounded by the reflecting hyperplanes of the Coxeter Cycle type and reflection length generators 푠0, 푠1, … , 푠푛−1). (In fact, the same is true for [10] any affine Coxeter group.) Any bijection 푢: ℤ → ℤ partitions the integers into a Combinatorially, the length of an affine permutation is (possibly infinite) list of (possibly infinite) cycles: for encoded in terms of an appropriate notion of inver- each integer 푖, the cycle containing 푖 is the sequence −2 −1 2 sions. In particular, one has for an affine permutation 푢 (… , 푢 (푖), 푢 (푖), 푖, 푢(푖), 푢 (푖), … ) where exponentia- that[11] tion represents functional composition. For example, the affine permutation in 푆̃ with window notation ℓ(푢) = #{(푖, 푗): 푖 ∈ {1, … , 푛}, 푖 < 푗, and 푢(푖) > 푢(푗)}. 5 [6, 3, 2, 0, 4] contains the two infinite cycles

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(… , −9, −4, 1, 6, 11, … ) and (… , 10, 9, 5, 4, 0, −1, −5 … ) words (푠1, 푠3) and (푠3, 푠1) can be connected by swap- as well as infinitely many finite cycles (5푘 + 2, 5푘 + 3) ping commuting factors, but 3214 = (13)(2)(4) is not for each 푘 ∈ ℤ. Cycles of an affine permutation corre- fully commutative because there is no way to reach the spond to cycles of the underlying permutation in an ob- reduced word (푠2, 푠1, 푠2) starting from the reduced vious way: in the example above, with underlying per- word (푠1, 푠2, 푠1) by commutations. mutation [1, 3, 2, 5, 4] = (1)(23)(45), the first infinite Billey, Jockusch & Stanley (1993) proved that in the fi- cycle corresponds to the cycle (1), the second corre- nite symmetric group 푆 , a permutation is fully commu- sponds to the cycle (45), and the finite cycles all corre- 푛 tative if and only if it avoids the spond to the cycle (23). 321, that is, if and only if its one-line notation contains For an affine permutation 푢, the following conditions no three-term decreasing subsequence. In (Green are equivalent: all cycles of 푢 are finite, 푢 has finite or- 2002), this result was extended to affine permutations: der, and the geometric action of 푢 on the space 푉 has an affine permutation 푢 is fully commutative if and only at least one fixed point.[18] if there do not exist integers 푖 < 푗 < 푘 such that 푢(푖) > 푢(푗) > 푢(푘).[a] The reflection length ℓ푅(푢) of an element 푢 of 푆̃푛 is the smallest number 푘 such that there exist reflections It has also been shown that the number of affine permu- 푟1, … , 푟푘 such that 푢 = 푟1 ⋯ 푟푘. (In the symmetric group, tations avoiding a single pattern 푝 is finite if and only if reflections are transpositions, and the reflection length 푝 avoids the pattern 321,[24] so in particular there are in- of a permutation 푢 is 푛 − 푐(푢), where 푐(푢) is the num- finitely many fully commutative affine permutations. ber of cycles of 푢.[19]) In (Lewis et al. 2019), the following These were enumerated by length in (Hanusa & Jones formula was proved for the reflection length of an affine 2010). permutation 푢: for each cycle of 푢, define the weight to be the integer 푘 such that consecutive entries congru- ent modulo 푛 differ by exactly 푘푛. (For example, in the Parabolic and other permutation [6, 3, 2, 0, 4] above, the first infinite cycle structures has weight 1 and the second infinite cycle has weight

−1; all finite cycles have weight 0.) Form a of cycle The parabolic subgroups of 푆̃푛 and their repre- weights of 푢 ( translates of the same cycle by sentatives offer a rich combinatorial structure. Other multiples of 푛 only once), and define the nullity ν(푢) to aspects of the affine symmetric group, such as its Bru- be the size of the smallest set partition of this tuple so hat and representation theory, may also be un- that each part sums to 0. (In the example above, the tu- derstood via combinatorial models. ple is (1, −1, 0) and the nullity is 2, since one can take the partition (1, −1), (0).) Then the reflection length of Parabolic subgroups, coset representatives 푢 is A standard parabolic subgroup of a Coxeter group is a ℓ (푢) = 푛 − 2휈(푢) + 푐(휋(푢)), 푅 subgroup generated by a subset of its Coxeter generat- where π(푢) is the underlying permutation of 푢.[20] ing set. The maximal parabolic subgroups are those that come from omitting a single Coxeter generator. In For every affine permutation 푢, there is a choice of sub- 푆̃ , all maximal parabolic subgroups are isomorphic to group 푊 of S̃ such that W ≅ 푆 , 푆̃ = 푊 ⋉ Λ, and for 푛 푛 푛 푛 the finite symmetric group 푆 . The subgroup generated the standard form 푢 = 푤 ⋅ 푡 implied by this semidirect 푛 by the subset {푠 , … , 푠 } ∖ {푠 } consists of those af- product, one has ℓ (푢) = ℓ (푤) + ℓ (푡). 0 푛−1 푖 푅 푅 푅 fine permutations that stabilize the interval [푖 + 1, 푖 + 푛], that is, that map every element of this interval to an- Fully commutative elements and pattern other element of the interval.[14] avoidance The non-maximal parabolic subgroups of 푆̃푛 are all iso- A reduced word for an element 푔 of a Coxeter group is a morphic to parabolic subgroups of 푆푛, that is, to a Young subgroup 푆 × ⋯ × 푆 for some positive inte- tuple (푠푖1, … , 푠푖ℓ(푔)) of Coxeter generators of minimum 푎1 푎푘 [9] gers 푎1, … , 푎푘 with sum 푛. possible length such that 푔 = 푠푖1 ⋯ 푠푖ℓ(푔). The ele- ment 푔 is called fully commutative if one can transform For a fixed element 푖 of {0, … , 푛 − 1}, let 퐽 = any reduced word into any other by sequentially swap- {푠0, … , 푠푛−1} ∖ {푠푖} be the maximal proper subset of ping pairs of factors that commute.[22] For example, in Coxeter generators omitting 푠 , and let (푆̃ ) denote 푖 푛 퐽 푆 2143 = the finite symmetric group 4, the element the parabolic subgroup generated by 퐽. Every coset 푔 ⋅ (12)(34) is fully commutative, since its two reduced

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Figure 6 | Abacus diagram values are 푎 = 0, 1, 3, which are respectively mapped of the affine permutation by 푢 to 1, 2, and 4.) Then for two affine permutations 푢, [−5, 0, 6, 9]. 푣, one has that 푢 ≤ 푣 in Bruhat order if and only if 푢[푖, 푗] ≤ 푣[푖, 푗] for all integers 푖, 푗.[28]

Representation theory and an affine Robin- son–Schensted correspondence In the finite symmetric group, the Robinson–Schensted correspondence gives a bijection between the group and pairs (푃, 푄) of standard Young tableaux 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 Ka- zhdan–Lusztig theory, two permutations lie in the same left cell if and only if their images under Robinson– (푆̃ ) has a unique element of minimum length. The 푛 퐽 Schensted have the same tableau 푄, and in the same 퐽 collection of such representatives, denoted (푆̃푛) , con- right cell if and only if their images have the same tab- sists of the following affine permutations:[14] leau 푃. In (Shi 1986), J.-Y. Shi showed that left cells for 푆̃ are indexed instead by tabloids,[b] and in (Shi 1991) ̃ 퐽 ̃ 푛 (푆푛) = {푢 ∈ 푆푛: 푢(푖 − 푛 + 1) < 푢(푖 − 푛 + 2) < ⋯ he gave an algorithm to compute the tabloid analogous < 푢(푖 − 1) < 푢(푖)}. to the tableau 푃 for an affine permutation. In (Chmutov, Pylyavskyy & Yudovina 2018), the authors extended In the particular case that 퐽 = {푠1, … , 푠푛−1}, so that Shi's work to give a bijective map between 푆̃ and tri- (푆̃푛) ≅ 푆푛 is the standard copy of 푆푛 inside 푆̃푛, the ele- 푛 퐽 ples (푃, 푄, 휌) consisting of two tabloids of the same ̃ 퐽 ̃ ments of (푆푛) ≅ 푆푛/푆푛 may naturally be represented shape and an integer vector whose entries satisfy cer- by abacus diagrams: the integers are arranged in an in- tain inequalities. Their procedure uses the matrix repre- finite strip of width 푛, increasing sequentially along sentation of affine permutations and generalizes the rows and then from top to bottom; integers are circled shadow construction of Viennot (1977). if they lie directly above one of the window entries of the minimal coset representative. For example, the minimal coset representative 푢 = [−5, 0, 6, 9] is repre- Inverse realizations sented by the abacus diagram in figure 6. To compute the length of the representative from the abacus dia- In some situations, one may wish to consider the action gram, one adds up the number of uncircled numbers of the affine symmetric group on ℤ or on alcoves that is that are smaller than the last circled entry in each col- inverse to the one given above.[c] We describe these al- umn. (In the example shown, this gives 5 + 3 + 0 + 1 = ternate realizations now. 9.)[25] In the combinatorial action of 푆̃푛 on ℤ, the generator 푠푖 Other combinatorial models of minimum-length coset acts by switching the values 푖 and 푖 + 1. In the inverse 푖 representatives for 푆̃푛/푆푛 can be given in terms of core action, it instead switches the entries in positions and partitions (integer partitions in which no hook length is 푖 + 1. Similarly, the action of a general reflection will divisible by 푛) or bounded partitions (integer partitions be to switch the entries at and 푖 + 푘푛 for each 푘, fixing in which no part is larger than 푛 − 1). Under these corre- all inputs at positions not congruent to 푖 or 푗 modulo [29] spondences, it can be shown that the weak Bruhat or- 푛. (In the finite symmetric group 푆푛, the analogous distinction is between the active and passive forms of a der on 푆̃푛/푆푛 is isomorphic to a certain subposet of [30] Young's lattice.[26][27] permutation. )

In the geometric action of 푆̃푛, the generator 푠푖 acts on Bruhat order an alcove 퐴 by reflecting it across one of the bounding 퐴 The Bruhat order on 푆̃ has the following combinatorial planes of the fundamental alcove 0. In the inverse ac- 푛 퐴 realization. If 푢 is an affine permutation and 푖 and 푗 are tion, it instead reflects across one of its own bounding integers, define 푢[푖, 푗] to be the number of integers a planes. From this perspective, a reduced word corre- 푉 [31] such that 푎 ≤ 푖 and 푢(푎) ≥ 푗. (For example, with 푢 = sponds to an alcove walk on the tesselated space . [2, 0, 4] ∈ 푆̃3, one has 푢[3, 1] = 3: the three relevant

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Figure 7 | Alcoves for 푆̃3 labeled by affine permutations, inverse to the labeling above.

where indices of the sequence a are taken modulo 푛. Then 푤퐚 is an affine permutation in 푆̃푛, and moreover Relationship to other mathematical every affine permutation arises from a juggling pattern objects in this way.[32] Under this bijection, the length of the af- fine permutation is encoded by a natural statistic in the The affine symmetric group is closely related to a vari- juggling pattern: one has ety of other mathematical objects. ℓ(푤풂) = (푏 − 1)푛 − 푐푟표푠푠(풂), Juggling patterns where cross(퐚) is the number of crossings ( perio- dicity) in the arc diagram of a. This allows an elementary In (Ehrenborg & Readdy 1996), a correspondence is proof of the generating function for affine permuta- given between affine permutations and juggling pat- tions by length.[35] terns encoded in a version of notation.[32] Here, a juggling pattern of period 푛 is a sequence For example, the juggling pattern 441 (Figures 8 & 9) has 4+4+1 (푎1, … , 푎푛) of nonnegative integers (with certain re- 푛 = 3 and 푏 = = 3. Therefore, it corresponds to 3 strictions) that captures the behavior of balls thrown by the affine permutation 푤441 = [1 + 4 − 3, 2 + 4 − a juggler, where the number 푎푖 indicates the length of 3, 3 + 1 − 3] = [2, 3, 1]. The juggling pattern has four time the 푖th throw spends in the air (equivalently, the crossings, and the affine permutation has length [d] height of the throw). The number 푏 of balls in the pat- ℓ(푤 ) = (3 − 1) ⋅ 3 − 4 = 2. 푎 +⋯+푎 441 tern is the average 푏 = 1 푛.[34] The Ehrenborg–Re- 푛 Similar techniques can addy correspondence associates to each juggling pat- be used to derive the tern 퐚 = (푎 , … , 푎 ) of period 푛 the function 푤 : ℤ → ℤ 1 푛 퐚 generating function for defined by minimal coset repre- 푤 (푖) = 푖 + 푎 − 푏, 풂 푖 sentatives of 푆̃푛/푆푛 by length.[36]

Figure 9 | The juggling pattern 441. For video, follow this link: Figure 8 | The juggling pattern 441 visualized as an arc diagram: the height of each throw https://commons.wikimedia corresponds to the length of an arc; the two colors of nodes are the left and right hands of .org/wiki/File:Juggling_441.gif the juggler. This pattern has four crossings, which repeat periodically. Nummer9, CC BY SA 3.0

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̃ (1) Complex reflection groups to 푆푛 is of (untwisted) type 퐴푛−1, with 2 −2 [ ] for 푛 = 2 and In a finite-dimensional real , a reflec- −2 2 tion is a linear transformation that fixes a linear hyper- 2 −1 0 ⋯ 0 −1 plane pointwise and negates the vector orthogonal to −1 2 −1 ⋯ 0 0 the plane. This notion may be extended to vector 0 −1 2 ⋯ 0 0 spaces over other fields. In particular, in a complex inner ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ product space, a reflection is a unitary transformation 푇 0 0 0 ⋯ 2 −1 of finite order that fixes a hyperplane.[e] This implies [−1 0 0 ⋯ −1 2 ] that the vectors orthogonal to the hyperplane are ei- (a ) for 푛 > 2.[39] genvectors of 푇, and the associated eigenvalue is a Like other Kac–Moody , affine Lie algebras sat- complex . 퐴 complex is a fi- isfy the Weyl–Kac character formula, which expresses nite group of linear transformations on a complex vec- the characters of the in terms of their highest tor space generated by reflections. weights.[40] In the case of affine Lie algebras, the result- The complex reflection groups were fully classified by ing identities are equivalent to the Macdonald identi- Shephard & Todd (1954): each complex reflection (1) ties. In particular, for the of type 퐴1 , group is isomorphic to a product of irreducible complex associated to the affine symmetric group 푆̃2, the corre- reflection groups, and every irreducible either belongs sponding Macdonald identity is equivalent to the Jacobi to an infinite family 퐺(푚, 푝, 푛) (where 푚, 푝, and 푛 are triple product.[41] positive integers such that 푝 divides 푚) or is one of 34 other (so-called "exceptional") examples. The group Extended affine symmetric group 퐺(푚, 1, 푛) is the generalized symmetric group: algebra- ically, it is the (ℤ/푚ℤ) ≀ 푆푛 of the cyclic The affine symmetric group is a subgroup of the ex- group ℤ/푚ℤ with the symmetric group 푆푛. Concretely, tended affine symmetric group. The extended group is the elements of the group may be represented by mo- isomorphic to the wreath product ℤ ≀ 푆푛. Its elements nomial matrices (matrices having one nonzero entry in are extended affine permutations: 푢: ℤ → ℤ every row and column) whose nonzero entries are all that 푢(푥 + 푛) = 푢(푥) + 푛 for all integers 푥. Unlike the mth roots of unity. The groups 퐺(푚, 푝, 푛) are subgroups affine symmetric group, the extended affine symmetric of 퐺(푚, 1, 푛), and in particular the group 퐺(푚, 푚, 푛) group is not a Coxeter group. However, it has a natural consists of those matrices in which the product of the generating set that extends the Coxeter generating set nonzero entries is equal to 1. for 푆̃푛: the shift operator τ whose window notation is τ = [2, 3, … , 푛, 푛 + 1] generates the extended group In (Shi 2002), Shi showed that the affine symmetric with the simple reflections, subject to the additional re- group is a generic cover of the family {퐺(푚, 푚, 푛): 푚 ≥ lations τ푠 τ−1 = 푠 .[7] 1}, in the following sense: for every positive integer 푚, 푖 푖+1 there is a surjection 휋푚 from 푆̃푛 to 퐺(푚, 푚, 푛), and these maps are compatible with the natural surjections Combinatorics of other affine Coxeter 퐺(푚, 푚, 푛) ↠ 퐺(푝, 푝, 푛) when 푝 ∣ 푚 that come from groups raising each entry to the 푚/푝th power. Moreover, these The geometric action of the affine symmetric group 푆̃푛 projections respect the reflection group structure, in places it naturally in the family of affine Coxeter groups, ̃ that the image of every reflection in 푆푛 under 휋푚 is a all of which have a similar geometric action. The combi- reflection in 퐺(푚, 푚, 푛); and similarly when 푚 > 1 the natorial description of the 푆̃푛 may also be extended to image of the standard 푠0 ⋅ 푠1 ⋯ 푠푛−1 in many of these groups: in (Eriksson & Eriksson 1998), an ̃ [37] 푆푛 is a Coxeter element in 퐺(푚, 푚, 푛). axiomatic description is given of certain permutation groups acting on ℤ (the "George groups", in honor of Affine Lie algebras George Lusztig), and it is shown that they are exactly Each affine Coxeter group is associated to an affine Lie the "classical" Coxeter groups of finite and affine types algebra, a certain infinite-dimensional non-associative A, B, C, and D. Thus, the combinatorial interpretations algebra with unusually nice representation-theoretic of descents, inversions, etc., carry over in these [42] properties. In this association, the Coxeter group arises cases. Abacus models of minimum-length coset rep- as a group of symmetries of the root space of the Lie resentatives for parabolic have also been ex- [43] algebra (the dual of the Cartan subalgebra).[38] In the tended to this context. classification of affine Lie algebras, the one associated

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Acknowledgements References The author thanks the three referees for their many • Beazley, Elizabeth; Nichols, Margaret; Park, Min Hae; Shi, XiaoLin; Youcis, Alexander (2015), "Bijective projections on parabolic quotients of affine helpful comments, and Max Glick for his help with the Weyl groups", J. Algeb. Comb., 41: 911–948, doi:10.1007/s10801-014-0559- less technical summary. The work of the author was 9 • Berg, Chris; Jones, Brant; Vazirani, Monica (2009), "A bijection on core supported in part by a Simons Collaboration Grant partitions and a parabolic quotient of the affine symmetric group", J. (634530). Combin. Theory Ser. A, 116 (8): 1344–1360, doi:10.1016/j.jcta.2009.03.013 • Billey, Sara C.; Jockusch, William; Stanley, Richard P. (1993), "Some Combinatorial Properties of Schubert Polynomials", J. Algeb. Comb., 2: 345–374, doi:10.1023/A:1022419800503 Notes • Björner, Anders; Brenti, Francesco (1996), "Affine permutations of type A", Electron. j. Combin., 3 (2): R18, doi:10.37236/1276 1. Björner & Brenti (2005), p. 17. • Björner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter groups, 2. Humphreys (1990), p. 17. Springer, ISBN 978-3540-442387 3. Humphreys (1990), Chapter 4. • Cameron, Peter J. (1994), Combinatorics: Topics, Techniques, Algorithms, 4. Humphreys (1990), Section 4.3. Cambridge University Press, ISBN 978-0-521-45761-3 5. Björner & Brenti (2005), Chapter 8.3. • Chmutov, Michael; Pylyavskyy, Pavlo; Yudovina, Elena (2018), "Matrix-ball 6. Björner & Brenti (2005), Proposition 8.3.5. construction of affine Robinson-Schensted correspondence", Selecta 7. Chmutov, Pylyavskyy & Yudovina (2018), Section 1.6. Math. (N.S.), 24 (2): 667–750, doi:10.1007/s00029-018-0402-6 8. Björner & Brenti (2005), p. 260. • Clark, Eric; Ehrenborg, Richard (2011), "Excedances of affine 9. Björner & Brenti (2005), p. 15. permutations", Advances in Applied Mathematics, 46: 175–191, 10. Humphreys (1990), p. 93. doi:10.1016/j.aam.2009.12.006 11. Björner & Brenti (2005), p. 261. • Crites, Andrew (2010), "Enumerating pattern avoidance for affine 12. Björner & Brenti (2005), p. 208. permutations", Electron. j. Combin., 17 (1): R127, doi:10.37236/399 13. Björner & Brenti (1996), Cor. 4.7. • Ehrenborg, Richard; Readdy, Margaret (1996), "Juggling and applications 14. Björner & Brenti (2005), p. 263. to q-analogues", Discrete Math., 157: 107–125, doi:10.1016/S0012- 15. Reiner (1995), p. 2. 365X(96)83010-X 16. Petersen (2015), Chapter 14. • Eriksson, Henrik; Eriksson, Kimmo (1998), "Affine Weyl groups as infinite 17. Reiner (1995), Theorem 6. permutations", Electron. j. Combin., 5: R18, doi:10.37236/1356 18. Lewis et al. (2019), Propositions 1.31 and 4.24. • Green, R.M. (2002), "On 321-Avoiding Permutations in Affine Weyl 19. Lewis et al. (2019). Groups", J. Algeb. Comb., 15: 241–252, doi:10.1023/A:1015012524524 20. Lewis et al. (2019), Theorem 4.25. • Hanusa, Christopher R.H.; Jones, Brant C. (2010), "The of 21. Lewis et al. (2019), Corollary 2.5. fully commutative affine permutations", Eur. j. Comb., 31 (5): 1342–1359, 22. Stembridge (1996), p. 353. doi:10.1016/j.ejc.2009.11.010 23. Hanusa & Jones (2010), p. 1345. • Hanusa, Christopher R.H.; Jones, Brant C. (2012), "Abacus models for 24. Crites (2010), Theorem 1. parabolic quotients of affine Weyl groups", J. Algebra, 361: 134–162, 25. Hanusa & Jones (2010), Section 2.2. doi:10.1016/j.jalgebra.2012.03.029 26. Lapointe & Morse (2005). • Humphreys, James E. (1990), Reflection groups and Coxeter groups, 27. Berg, Jones & Vazirani (2009). Cambridge University Press, ISBN 0-521-37510-X

28. Björner & Brenti (2005), p. 264. • Kac, Victor G. (1990), Infinite dimensional Lie algebras (3rd ed.), Cambridge 29. Knutson, Lam & Speyer (2013), Section 2.1. University Press, ISBN 0-521-46693-8

30. As in (Cameron 1994, Section 3.5). • Knutson, Allen; Lam, Thomas; Speyer, David E. (2013), "Positroid varieties:

31. As in, for example, (Beazley et al. 2015), (Lam 2015). juggling and geometry", Compositio Math., 149: 1710–1752, 32. Polster (2003), p. 42. doi:10.1112/S0010437X13007240 33. Polster (2003), p. 22. • Lam, Thomas. (2015), "The shape of a random affine element 34. Polster (2003), p. 15. and random core partitions", Ann. Probab., 43 (4): 1643–1662, 35. Polster (2003), p. 43. doi:10.1214/14-AOP915 36. Clark & Ehrenborg (2011), Theorem 2.2. • Lapointe, Luc; Morse, Jennifer (2005), "Tableaux on k + 1-cores, reduced 37. Lewis (2020), Section 3.2. words for affine permutations, and k-Schur expansions", J. Combin. Theory 38. Kac (1990), Chapter 3. Ser. A, 112 (1): 44–81, doi:10.1016/j.jcta.2005.01.003 39. Kac (1990), Chapter 4. • Lewis, Joel Brewster (2020), "A note on the Hurwitz action on reflection 40. Kac (1990), Chapter 10. factorizations of Coxeter elements in complex reflection groups", Electron. 41. Kac (1990), Chapter 12. j. Combin., 27 (2): P2.54, doi:10.37236/9351 42. Björner & Brenti (2005), Chapter 8. • Lewis, Joel Brewster; McCammond, Jon; Petersen, T. Kyle; Schwer, Petra 43. Hanusa & Jones (2012). (2019), "Computing reflection length in an affine Coxeter group", Trans. 44. The three positions 푖, 푗, and 푘 need not lie in a single window. For example, Amer. Math. Soc., 371: 4097–4127, doi:10.1090/tran/7472 the affine permutation 푊 in 푆̃ with window notation [−4, −1, 1, 14] is not 4 • fully commutative, because w(0) = 10, w(3) = 1, and w(5) = 0, even Petersen, T. Kyle (2015), Eulerian Numbers, Birkhauser, doi:10.1007/978-1- though no four consecutive positions contain a decreasing subsequence of 4939-3091-3, ISBN 978-1-4939-3090-6 length three.[23] • Polster, Burkard (2003), The Mathematics of Juggling, Springer, ISBN 0- 45. In a standard , entries increase across rows and down 387-95513-5 columns; in a tabloid, they increase across rows, but there is no column • Reiner, Victor (1995), "The distribution of descents and length in a Coxeter condition. group", Electron. j. Combin., 2: R25, doi:10.37236/1219 46. In other words, one might be interested in switching from a left group • Shephard, G. C.; Todd, J. A. (1954), "Finite unitary reflection groups", action to a right action or vice-versa. Canad. j. Math., 6: 274–304, doi:10.4153/CJM-1954-028-3 47. Not every sequence of 푛 nonnegative integers is a juggling sequence. In • Shi, Jian-Yi (1986), Kazhdan–Lusztig cells of certain affine Weyl groups, particular, a sequence corresponds to a "simple juggling pattern", with one Lecture Notes in Mathematics, 1179, Springer, ISBN 3-540-16439-1 ball caught and thrown at a time, if and only if the function 푖 ↦ 푖 + • Shi, Jian-Yi (1991), "The generalized Robinson–Schensted algorithm on [33] 푎푖 mod 푛 is a permutation of {1, … , 푛}. the affine Weyl group of type An−1", J. Algebra, 139 (2): 364–394, 48. In some sources, unitary reflections are called pseudoreflections. doi:10.1016/0021-8693(91)90300-W • Shi, Jian-Yi (2002), "Certain imprimitive reflection groups and their generic versions", Trans. Amer. Math. Soc., 354 (5): 2115–2129, doi:10.1090/S0002- 9947-02-02941-0

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• Stembridge, John (1996), "On the Fully Commutative Elements of Coxeter représentation du groupe symétrique, Lecture Notes in Mathematics, 579, Groups", J. Alg. Comb., 5: 353–385, doi:10.1007/BF00193185 Springer, pp. 29–58, doi:10.1007/BFb0090011, ISBN 978-3-540-08143-2 • Viennot, G. (1977), "Une forme géométrique de la correspondance de Robinson-Schensted", in Foata, Dominique (ed.), Combinatoire et

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