WikiJournal of Science, 2021, 4(1):3 doi: 10.15347/wjs/2021.003 Encyclopedic Review Article Affine symmetric group Joel Brewster Lewis¹* Abstract The affine symmetric group is a mathematical structure that describes the symmetries of the number line and the regular triangular tesselation of the plane, as well as related higher dimensional objects. It is an infinite extension of the symmetric group, which consists of all permutations (rearrangements) of a finite set. In addition to its geo- metric description, the affine symmetric group may be defined as the collection of permutations of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain 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 combinatorics and representation theory. 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 reflection 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 dihedral group 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 1 of 11 | WikiJournal of Science WikiJournal of Science, 2021, 4(1):3 doi: 10.15347/wjs/2021.003 Encyclopedic Review Article Geometric definition 푛 In the Euclidean space 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 integer 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 lattice 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 tuples [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 permutation 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 numbers 푢(1), … , 푢(푛) must all be distinct modulo 푛.) Such a function 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. 2 of 11 | WikiJournal of Science WikiJournal of Science, 2021, 4(1):3 doi: 10.15347/wjs/2021.003 Encyclopedic Review Article 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 bijection, the resulting nates sum to 0); the subgroup (푆̃푛) of 푆̃푛 of isometries matrix 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 conjugacy class 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 1s are replaced by •, 0s are omitted, rows numbers in- lize 퐴 is isomorphic to 푆푛.