Affine symmetric group

Summary

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects. Each one is an infinite extension of a finite symmetric group, the group of permutations (rearrangements) of a finite set. In addition to their geometric description, the affine symmetric groups may be defined as collections 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 generators and relations. They are studied as part of the fields of combinatorics and representation theory.

The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group 3

Many important combinatorial properties of the finite symmetric groups can be extended to affine symmetric groups. Permutation statistics such as descents and inversions can be defined in the affine case. As in the finite case, the natural combinatorial definitions for these statistics also have a geometric interpretation.

The affine symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups.

DefinitionsEdit

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.[1]

Algebraic definitionEdit

 
Dynkin diagrams 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 satisfied by those elements that imply all of the other equations they satisfy. In this way, the affine symmetric group   is generated by a set

 
of n elements that satisfy the following relations: when  ,
  1.   (the generators are involutions),
  2.   if j is not one of  , indicating that for these pairs of generators, the group operation is commutative, and
  3.  .

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

These relations can be rewritten in the special form that defines the Coxeter groups, so the affine symmetric groups are Coxeter groups, with the   as their Coxeter generating sets. For  , the Coxeter–Dynkin diagram of   is the n-cycle, while for   it consists of two nodes joined by an edge labeled  .[3] In these diagrams, the vertices represent the generators, which for Coxeter groups must be involutions. The edges of the cycle correspond to the relations between pairs of consecutive generators, while the absence of an edge between other pairs of generators indicates that they commute.[4]

Geometric definitionEdit

 
When n = 3, the space V is a two-dimensional plane and the reflections are across lines. The points of the root lattice Λ are circled.

In the Euclidean space   with coordinates  , the set V of points for which   forms a (hyper)plane, an (n − 1)-dimensional subspace. For every pair of distinct elements i and j of   and every integer k, the set of points in V that satisfy   forms an (n − 2)-dimensional subspace within V, and there is a unique reflection of V that fixes this subspace. Then the affine symmetric group can be realized geometrically as a collection of maps from V to itself, the compositions of these reflections.[5]

Inside V, the subset of points with integer coordinates forms the root lattice, Λ. It is the set of all the integer vectors   such that  . Each reflection preserves this lattice, and so the lattice is preserved by the whole group.[5]

The fixed subspaces of these reflections divide V into congruent simplices, called alcoves.[6] The situation when   is shown in the figure; in this case, the root lattice is a triangular lattice, with reflecting lines dividing it into equilateral triangle alcoves. However for higher dimensions, the alcoves are not regular simplices.

 
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 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   such that  , which is bounded by the hyperplanes  ,  , ..., and  , illustrated in the case  . For  , one may identify the reflection through   with the Coxeter generator  , and also identify the reflection through   with the generator  .[6]

Combinatorial definitionEdit

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   is an affine permutation if

  • it is a bijection (each integer appears as the value of   for exactly one  ),
  •   for all integers x (the function is equivariant under shifting by  ), and
  •  , the  th triangular number.

For every affine permutation, and more generally every shift-equivariant bijection, the numbers   must all be distinct modulo n. An affine permutation is uniquely determined by its window notation  , because all other values of   can be found by shifting these values. Thus, affine permutations may also be identified with tuples   of integers that contain one element from each congruence class modulo n and sum to  .[7]

To translate between the combinatorial and algebraic definitions, for   one may identify the Coxeter generator   with the affine permutation that has window notation  , and also identify the generator   with the affine permutation  . More generally, every reflection (that is, a conjugate of one of the Coxeter generators) can be described uniquely as follows: for distinct integers i, j in   and arbitrary integer k, it maps i to jkn, maps j to i + kn, and fixes all inputs not congruent to i or j modulo n.[8]

Representation as matricesEdit

 
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 matrices.[9] If   is an affine permutation, the corresponding matrix has entry 1 at position   in the infinite grid   for each integer i, and all other entries are equal to 0. Since u is a bijection, the resulting matrix contains exactly one 1 in every row and column. The periodicity condition on the map u ensures that the entry at position   is equal to the entry at position   for every pair of integers  .[9] For example, a portion of the matrix for the affine permutation   is shown in the figure.

Relationship to the finite symmetric groupEdit

The affine symmetric group   contains the finite symmetric group   of permutations on   elements as both a subgroup and a quotient group.[10]

As a subgroupEdit

There is a canonical way to choose a subgroup of   that is isomorphic to the finite symmetric group  . In terms of the algebraic definition, this is the subgroup of   generated by   (excluding the simple reflection  ). Geometrically, this corresponds to the subgroup of transformations that fix the origin, while combinatorially it corresponds to the window notations for which   (that is, in which the window notation is the one-line notation of a finite permutation).[11][5]

If   is the window notation of an element of this standard copy of  , its action on the hyperplane V in   is given by permutation of coordinates:  .[12] (In this article, the geometric action of permutations and affine permutations is on the right; thus, if u and v are two affine permutations, the action of uv on a point is given by first applying u, then applying v.)

There are also many nonstandard copies of   contained in  . A geometric construction is to pick any point a in Λ (that is, an integer vector whose coordinates sum to 0); the subgroup   of   of isometries that fix a is isomorphic to  .[13]

As a quotientEdit

There is a simple map (technically, a surjective group homomorphism) π from   onto the finite symmetric group  . In terms of the combinatorial definition, an affine permutation can be mapped to a permutation by reducing the window entries modulo n to elements of  , leaving the one-line notation of a permutation.[14] In this article, the image   of an affine permutation u is called the underlying permutation of u.

The map π sends the Coxeter generator   to the permutation whose one-line notation and cycle notation are   and  , respectively.[15][14]

The kernel of π is by definition the set of affine permutations whose underlying permutation is the identity. The window notations of such affine permutations are of the form  , where   is an integer vector such that  , that is, where  . Geometrically, this kernel consists of the translations, the isometries that shift the entire space V without rotating or reflecting it.[16] In an abuse of notation, the symbol Λ is used in this article for all three of these sets (integer vectors in V, affine permutations with underlying permutation the identity, and translations); in all three settings, the natural group operation turns Λ into an abelian group, generated freely by the n − 1 vectors  .[17]

Connection between the geometric and combinatorial definitionsEdit

 
Alcoves for   labeled by affine permutations. An alcove A is labeled by the window notation for a permutation u if u sends the fundamental alcove (shaded) to A. Negative numbers are denoted by overbars.

The affine symmetric group   has Λ as a normal subgroup, and is isomorphic to the semidirect product

 
of this subgroup with the finite symmetric group  , where the action of   on Λ is by permutation of coordinates. Consequently, every element u of   has a unique realization as a product   where   is a permutation in the standard copy of   in   and   is a translation in  .[18]

This point of view allows for a direct translation between the combinatorial and geometric definitions of  : if one writes   where   and   then the affine permutation u corresponds to the rigid motion of V defined by[18]

 

Furthermore, as with every affine Coxeter group, the affine symmetric group acts transitively and freely on the set of alcoves: for each two alcoves, a unique group element takes one alcove to the other. Hence, making an arbitrary choice of alcove   places the group in one-to-one correspondence with the alcoves: the identity element corresponds to  , and every other group element g corresponds to the alcove   that is the image of   under the action of g.[5]

Example: n = 2Edit

 
The affine symmetric group   acts on the line V in the Euclidean plane. The reflections are through the dashed lines. The vectors of the root lattice Λ are marked.

Algebraically,   is the infinite dihedral group, generated by two generators   subject to the relations  .[2] Every other element of the group can be written as an alternating product of copies of   and  .[19]

Combinatorially, the affine permutation   has window notation  , corresponding to the bijection   for every integer k. The affine permutation   has window notation  , corresponding to the bijection   for every integer k. Other elements have the following window notations:

  •  
  •  
  •  
  •  

Geometrically, the space V on which   acts is a line, with infinitely many equally spaced reflections.[20] It is natural to identify the line V with the real line  ,   with reflection around the point 0, and   with reflection around the point 1. In this case, the reflection   reflects across the point k for any integer k, the composition   translates the line by –2, and the composition   translates the line by 2.[21][20]

Permutation statistics and permutation patternsEdit

Many permutation statistics and other features of the combinatorics of finite permutations can be extended to the affine case.[22]

Descents, length, and inversionsEdit

The length   of an element g of a Coxeter group G is the smallest number k such that g can be written as a product   of k Coxeter generators of G.[23] Geometrically, the length of an element g in   is the number of reflecting hyperplanes that separate   and  , where   is the fundamental alcove (the simplex bounded by the reflecting hyperplanes of the Coxeter generators  ).[a][24] Combinatorially, the length of an affine permutation is encoded in terms of an appropriate notion of inversions: for an affine permutation u, the length is[25]

 
Alternatively, it is the number of equivalence classes of pairs   such that   and   under the equivalence relation   if   for some integer k. The generating function for length in   is[26][27]
 

Similarly, there is an affine analogue of descents in permutations: an affine permutation u has a descent in position i if  . (By periodicity, u has a descent in position i if and only if it has a descent in position   for all integers k.)[28] Algebraically, the descents corresponds to the right descents in the sense of Coxeter groups; that is, i is a descent of u if and only if  .[28] The left descents (that is, those indices i such that   are the descents of the inverse affine permutation  ; equivalently, they are the values i such that i occurs before i − 1 in the sequence  .[29] Geometrically, i is a descent of u if and only if the fixed hyperplane of   separates the alcoves   and  [30]

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 naively form a generating function for affine permutations by number of descents (an affine analogue of Eulerian polynomials).[31] One possible resolution is to consider affine descents (equivalently, cyclic descents) in the finite symmetric group  .[32] Another is to consider simultaneously the length and number of descents of an affine permutation. The generating function for these statistics over   simultaneously for all n is

 
where des(w) is the number of descents of the affine permutation w and   is the q-exponential function.[33]

Cycle type and reflection lengthEdit

Any bijection   partitions the integers into a (possibly infinite) list of (possibly infinite) cycles: for each integer i, the cycle containing i is the sequence   where exponentiation represents functional composition. For an affine permutation u, the following conditions are equivalent: all cycles of u are finite, u has finite order, and the geometric action of u on the space V has at least one fixed point.[34]

The reflection length   of an element u of   is the smallest number k such that there exist reflections   such that  . (In the symmetric group, reflections are transpositions, and the reflection length of a permutation u is  , where   is the number of cycles of u.[35]) In (Lewis et al. 2019), the following formula was proved for the reflection length of an affine permutation u: for each cycle of u, define the weight to be the integer k such that consecutive entries congruent modulo n differ by exactly kn. Form a tuple of cycle weights of u (counting translates of the same cycle by multiples of n only once), and define the nullity   to be the size of the smallest set partition of this tuple so that each part sums to 0. Then the reflection length of u is

 
where   is the underlying permutation of u.[36]

For every affine permutation u, there is a choice of subgroup W of   such that  ,  , and for the standard form   implied by this semidirect product, the reflection lengths are additive, that is,  .[13]

Fully commutative elements and pattern avoidanceEdit

A reduced word for an element g of a Coxeter group is a tuple   of Coxeter generators of minimum possible length such that  .[23] The element g is called fully commutative if any reduced word can be transformed into any other by sequentially swapping pairs of factors that commute.[37] For example, in the finite symmetric group  , the element   is fully commutative, since its two reduced words   and   can be connected by swapping commuting factors, but   is not fully commutative because there is no way to reach the reduced word   starting from the reduced word   by commutations.[38]

Billey, Jockusch & Stanley (1993) proved that in the finite symmetric group  , 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 (Green 2002), this result was extended to affine permutations: an affine permutation u is fully commutative if and only if there do not exist integers   such that  .[b]

The number of affine permutations avoiding a single pattern p is finite if and only if p avoids the pattern 321,[40] so in particular there are infinitely many fully commutative affine permutations. These were enumerated by length in (Hanusa & Jones 2010).

Parabolic subgroups and other structuresEdit

The parabolic subgroups of   and their coset representatives offer a rich combinatorial structure. Other aspects of the affine symmetric group, such as its Bruhat order and representation theory, may also be understood via combinatorial models.[22]

Parabolic subgroups, coset representativesEdit

 
Abacus diagram of the affine permutation [−5, 0, 6, 9].

A standard parabolic subgroup of a Coxeter group is a subgroup generated by a subset of its Coxeter generating set. The maximal parabolic subgroups are those that come from omitting a single Coxeter generator. In  , all maximal parabolic subgroups are isomorphic to the finite symmetric group  . The subgroup generated by the subset   consists of those affine permutations that stabilize the interval  , that is, that map every element of this interval to another element of the interval.[28]

For a fixed element i of  , let   be the maximal proper subset of Coxeter generators omitting  , and let   denote the parabolic subgroup generated by J. Every coset   has a unique element of minimum length. The collection of such representatives, denoted  , consists of the following affine permutations:[28]

 

In the particular case that  , so that   is the standard copy of   inside  , the elements of   may naturally be represented by abacus diagrams: the integers are arranged in an infinite strip of width n, increasing sequentially along rows and then from top to bottom; integers are circled if they lie directly above one of the window entries of the minimal coset representative. For example, the minimal coset representative   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  .)[41]

Other combinatorial models of minimum-length coset representatives for   can be given in terms of core partitions (integer partitions in which no hook length is divisible by n) or bounded partitions (integer partitions in which no part is larger than n − 1). Under these correspondences, it can be shown that the weak Bruhat order on   is isomorphic to a certain subposet of Young's lattice.[42][43]

Bruhat orderEdit

The Bruhat order on   has the following combinatorial realization. If u is an affine permutation and i and j are integers, define   to be the number of integers a such that   and  . (For example, with  , one has  : the three relevant values are  , which are respectively mapped by u to 1, 2, and 4.) Then for two affine permutations u, v, one has that   in Bruhat order if and only if   for all integers i, j.[44]

Representation theory and an affine Robinson–Schensted correspondenceEdit

In the finite symmetric group, the Robinson–Schensted correspondence gives a bijection between the group and pairs   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 Kazhdan–Lusztig theory, two permutations lie in the same left cell if and only if their images under Robinson–Schensted have the same tableau Q, and in the same right cell if and only if their images have the same tableau P. In (Shi 1986), Jian-Yi Shi showed that left cells for   are indexed instead by tabloids,[c] and in (Shi 1991) he gave an algorithm to compute the tabloid analogous to the tableau P for an affine permutation. In (Chmutov, Pylyavskyy & Yudovina 2018), the authors extended Shi's work to give a bijective map between   and triples   consisting of two tabloids of the same shape and an integer vector whose entries satisfy certain inequalities. Their procedure uses the matrix representation of affine permutations and generalizes the shadow construction, introduced in (Viennot 1977).

Inverse realizationsEdit

 
Alcoves for   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   or on alcoves that is inverse to the one given above.[d] We describe these alternate realizations now.

In the combinatorial action of   on  , the generator   acts by switching the values i and i + 1. In the inverse action, it instead switches the entries in positions i and i + 1. Similarly, the action of a general reflection will be to switch the entries at positions jkn and i + kn for each k, fixing all inputs at positions not congruent to i or j modulo n.[47][e]

In the geometric action of  , the generator   acts on an alcove A by reflecting it across one of the bounding planes of the fundamental alcove A0. In the inverse action, it instead reflects A across one of its own bounding planes. From this perspective, a reduced word corresponds to an alcove walk on the tessellated space V.[49]

Relationship to other mathematical objectsEdit

The affine symmetric group is closely related to a variety of other mathematical objects.

Juggling patternsEdit

 
The juggling pattern 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.
 
The juggling pattern 441

In (Ehrenborg & Readdy 1996), a correspondence is given between affine permutations and juggling patterns encoded in a version of siteswap notation.[50] Here, a juggling pattern of period n is a sequence   of nonnegative integers (with certain restrictions) that captures the behavior of balls thrown by a juggler, where the number   indicates the length of time the ith throw spends in the air (equivalently, the height of the throw).[f] The number b of balls in the pattern is the average  .[52] The Ehrenborg–Readdy correspondence associates to each juggling pattern   of period n the function   defined by

 
where indices of the sequence a are taken modulo n. Then   is an affine permutation in  , and moreover every affine permutation arises from a juggling pattern in this way.[50] Under this bijection, the length of the affine permutation is encoded by a natural statistic in the juggling pattern:
 
where   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.[53]

For example, the juggling pattern 441 has   and  . Therefore, it corresponds to the affine permutation  . The juggling pattern has four crossings, and the affine permutation has length  .[54]

Similar techniques can be used to derive the generating function for minimal coset representatives of   by length.[55]

Complex reflection groupsEdit

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 fields. In particular, in a complex inner product space, a reflection is a unitary transformation T of finite order that fixes a hyperplane.[g] This implies that the vectors orthogonal to the hyperplane are eigenvectors of 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.[57]

The complex reflection groups were fully classified by 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   (where m, p, and n are positive integers such that p divides m) or is one of 34 other (so-called "exceptional") examples. The group   is the generalized symmetric group: algebraically, it is the wreath product   of the cyclic group   with the symmetric group  . Concretely, the elements of the group may be represented by monomial matrices (matrices having one nonzero entry in every row and column) whose nonzero entries are all mth roots of unity. The groups   are subgroups of  , and in particular the group   consists of those matrices in which the product of the nonzero entries is equal to 1.[58]

In (Shi 2002), Shi showed that the affine symmetric group is a generic cover of the family  , in the following sense: for every positive integer m, there is a surjection   from   to  , and these maps are compatible with the natural surjections   when   that come from raising each entry to the m/pth power. Moreover, these projections respect the reflection group structure, in that the image of every reflection in   under   is a reflection in  ; and similarly when   the image of the standard Coxeter element   in   is a Coxeter element in  .[59]

Affine Lie algebrasEdit

Each affine Coxeter group is associated to an affine Lie algebra, a certain infinite-dimensional non-associative algebra with unusually nice representation-theoretic properties. 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).[60] In the classification of affine Lie algebras, the one associated to   is of (untwisted) type  , with Cartan matrix   for   and

 
(a circulant matrix) for  .[61]

Like other Kac–Moody algebras, affine Lie algebras satisfy the Weyl–Kac character formula, which expresses the characters of the algebra in terms of their highest weights.[62] 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  , associated to the affine symmetric group  , the corresponding Macdonald identity is equivalent to the Jacobi triple product.[63]

Extended affine symmetric groupEdit

The affine symmetric group is a subgroup of the extended affine symmetric group. The extended group is isomorphic to the wreath product  . Its elements are extended affine permutations: bijections   such that   for all integers 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  : the shift operator   whose window notation is   generates the extended group with the simple reflections, subject to the additional relations  .[9]

Combinatorics of other affine Coxeter groupsEdit

The geometric action of the affine symmetric group   places it naturally in the family of affine Coxeter groups, each of which has a similar geometric action on an affine space. The combinatorial description of the   may also be extended to many of these groups: in Eriksson & Eriksson (1998), an axiomatic description is given of certain permutation groups acting on   (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.[64] Abacus models of minimum-length coset representatives for parabolic quotients have also been extended to this context.[65]

HistoryEdit

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.[66] The combinatorial description of the affine symmetric group first appears in work of Lusztig (1983), and was expanded upon by Shi (1986).[67]

ReferencesEdit

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

  1. ^ Shi (1986), p. 66.
  2. ^ a b Björner & Brenti (2005), pp. 5–6.
  3. ^ Humphreys (1990), p. 17.
  4. ^ Björner & Brenti (2005), p. 2.
  5. ^ a b c d Humphreys (1990), Chapter 4.
  6. ^ a b Humphreys (1990), Section 4.3.
  7. ^ Björner & Brenti (2005), Chapter 8.3.
  8. ^ Björner & Brenti (2005), Proposition 8.3.5.
  9. ^ a b c Chmutov, Pylyavskyy & Yudovina (2018), Section 1.6.
  10. ^ Shi (1986).
  11. ^ Björner & Brenti (2005), p. 260.
  12. ^ Lewis et al. (2019), p. 4118.
  13. ^ a b Lewis et al. (2019), Corollary 2.5.
  14. ^ a b Shi (1986), pp. 85–6.
  15. ^ Petersen (2015), Section 14.4.1.
  16. ^ Kane (2001), Chapter 11.1.
  17. ^ Humphreys (1990), Section 2.10.
  18. ^ a b Lewis et al. (2019), Section 4.1.
  19. ^ Gallian (2013), p. 454.
  20. ^ a b Gallian (2013), p. 455.
  21. ^ Lewis & Reiner (2016), Section 4.1.
  22. ^ a b Björner & Brenti (2005), p. 245.
  23. ^ a b Björner & Brenti (2005), p. 15.
  24. ^ Humphreys (1990), p. 93.
  25. ^ Björner & Brenti (2005), p. 261.
  26. ^ Björner & Brenti (2005), p. 208.
  27. ^ Björner & Brenti (1996), Cor. 4.7.
  28. ^ a b c d Björner & Brenti (2005), p. 263.
  29. ^ Chmutov, Lewis & Pylyavskyy (2017), Section 3.2.
  30. ^ Shi (1986), p. 91.
  31. ^ Reiner (1995), p. 2.
  32. ^ Petersen (2015), Chapter 14.
  33. ^ Reiner (1995), Theorem 6.
  34. ^ Lewis et al. (2019), Propositions 1.31 and 4.24.
  35. ^ Lewis et al. (2019).
  36. ^ Lewis et al. (2019), Theorem 4.25.
  37. ^ Stembridge (1996), p. 353.
  38. ^ Billey, Jockusch & Stanley (1993), p. 358.
  39. ^ Hanusa & Jones (2010), p. 1345.
  40. ^ Crites (2010), Theorem 1.
  41. ^ Hanusa & Jones (2010), Section 2.2.
  42. ^ Lapointe & Morse (2005).
  43. ^ Berg, Jones & Vazirani (2009).
  44. ^ Björner & Brenti (2005), p. 264.
  45. ^ Chmutov et al. (2018), Section 2.2.2.
  46. ^ Shi (1986), p. 68.
  47. ^ Knutson, Lam & Speyer (2013), Section 2.1.
  48. ^ As in (Cameron 1994, Section 3.5).
  49. ^ As in, for example, (Beazley et al. 2015), (Lam 2015).
  50. ^ a b Polster (2003), p. 42.
  51. ^ Polster (2003), p. 22.
  52. ^ Polster (2003), p. 15.
  53. ^ Polster (2003), p. 43.
  54. ^ Polster (2003), Section 2.7.
  55. ^ Clark & Ehrenborg (2011), Theorem 2.2.
  56. ^ Lehrer & Taylor (2009), p. 9.
  57. ^ Lehrer & Taylor (2009), p. 10.
  58. ^ Lehrer & Taylor (2009).
  59. ^ Lewis (2020), Section 3.2.
  60. ^ Kac (1990), Chapter 3.
  61. ^ Kac (1990), Chapter 4.
  62. ^ Kac (1990), Chapter 10.
  63. ^ Kac (1990), Chapter 12.
  64. ^ Björner & Brenti (2005), Chapter 8.
  65. ^ Hanusa & Jones (2012).
  66. ^ Björner & Brenti (2005), p. 24.
  67. ^ Björner & Brenti (2005), p. 293.

NotesEdit

  1. ^ In fact, the same is true for any affine Coxeter group.
  2. ^ The three positions i, j, and k need not lie in a single window. For example, the affine permutation w in   with window notation   is not fully commutative, because  ,  , and  , even though no four consecutive positions contain a decreasing subsequence of length three.[39]
  3. ^ 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).[45]
  4. ^ In other words, one might be interested in switching from a left group action to a right action or vice-versa.[46]
  5. ^ In the finite symmetric group  , the analogous distinction is between the active and passive forms of a permutation.[48]
  6. ^ Not every sequence of 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   is a permutation of  .[51]
  7. ^ In some sources, unitary reflections are called pseudoreflections.[56]

Works citedEdit

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