WikiJournal of Science/Affine symmetric group

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Joel Brewster Lewis (21 April 2021), "Affine symmetric group", WikiJournal of Science, 4 (1): 3, doi:10.15347/WJS/2021.003, ISSN 2470-6345, Wikidata Q100400684




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 geometric 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 generators 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.

Non-technical summary
Flat, straight-edged shapes (like triangles) or 3D ones (like pyramids) have only a finite number of symmetries. In contrast, the affine symmetric group is a way to mathematically describe all the symmetries possible when an infinitely large flat surface is covered by triangular tiles. As with many subjects in mathematics, it can also be thought of in a number of ways: for example, it also describes the symmetries of the infinitely long number line, or the possible arrangements of all integers (..., −2, −1, 0, 1, 2, ...) with certain repetitive patterns. As a result, studying the affine symmetric group extends the study of symmetries of straight-edged shapes or of groups of permutations to the infinite case. It also connects several topics in mathematics that were originally studied for independent reasons, ranging from complex reflection groups to juggling sequences.

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.

Algebraic definition

 
Dynkin diagrams for the affine symmetric groups on 2 and more than 2 generators

In terms of generators and relations,   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  , 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  .[1]

This definition endows   with the structure of a Coxeter group, with the   as Coxeter generating set. For  , its Coxeter–Dynkin diagram is the n-cycle, while for   it consists of two nodes joined by an edge labeled  .[2]

Geometric definition

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

In the Euclidean space   with coordinates  , the set V of points that satisfy the equation   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 a plane in V, and there is a unique reflection of V that fixes this plane. Then the affine symmetric group can be realized geometrically as the collection of all maps from V to itself that arise by composing several of these reflections.[3]

Inside V, the type A root lattice Λ is the subset of points with integer coordinates, that is, it is the set of all the integer vectors   such that  . Each of the reflections preserves this lattice, and so the lattice is preserved by the whole group. In fact, one may define   to be the group of rigid transformations of V that preserve the lattice Λ.

These reflecting planes divide the space V into congruent simplicies, called alcoves.[4] The situation when   is shown at right; in this case, the root lattice is a triangular lattice, and the reflecting lines divide the plane into equilateral triangular alcoves. (For larger n, 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, fix an alcove and consider the n hyperplanes that form its boundary. For example, there is a unique alcove (the fundamental alcove) consisting of points   such that  , which is bounded by the hyperplanes  ,  , ..., and  . (This is illustrated in the case   at right.) For  , one may identify the reflection through   with the Coxeter generator  , and also identify the reflection through   with the generator  .[4]

Combinatorial definition

The affine symmetric group may be realized as a group of periodic permutations of the integers. In particular, say that a bijection   is an affine permutation if   for all integers x and  . (It is a consequence of the first property that the numbers   must all be distinct modulo n.) Such a function is uniquely determined by its window notation  , and so affine permutations may also be identified with tuples   of integers that contain one element from each congruence class modulo n and sum to  .[5]

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.[6] (In terms of the geometric definition, this corresponds to the reflection across the plane  . The correspondence between the geometric and combinatorial representations for other elements is discussed below.)

Representation as matrices

One may represent affine permutations as infinite periodic permutation matrices.[7] If   is an affine permutation, one places the 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  . For example, a portion of matrix for the affine permutation   is shown below, with the conventions that 1s are replaced by •, 0s are omitted, rows numbers increase from top to bottom, column numbers increase from left to right, and the boundary of the box consisting of rows and columns 1, 2, 3 is drawn:

 

Relationship to the finite symmetric group

The affine symmetric group   contains the finite symmetric group   as both a subgroup and a quotient.

As a subgroup

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).[8][3]

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:  . (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  . The analogous combinatorial construction is to choose any subset A of   that contains one element from each conjugacy class modulo n and whose elements sum to  ; the subgroup   of   of affine permutations that stabilize A is isomorphic to  .

As a quotient

There is a simple map (technically, a surjective group homomorphism) π from   onto the finite symmetric group  . In terms of the combinatorial definition, it is to reduce the window entries modulo n to elements of  , leaving the one-line notation of a permutation. 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. In terms of the Coxeter generators of  , this can be written as  .

The kernel π is 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, that is, the isometries that shift the entire space V without rotating or reflecting it. 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  .

Connection between the geometric and combinatorial definitions

 
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 subgroup Λ is a normal subgroup of  , and one has an isomorphism

 
between   and the semidirect product of the finite symmetric group   with Λ, where the action of   on Λ is by permutation of coordinates. Consequently, identifying the finite symmetric group   as its standard copy in  , one has that every element u of   may be realized uniquely as a product   where   is a finite permutation and  .

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

 

Furthermore, as with every affine Coxeter group, the affine symmetric group acts transitively and freely on the set of alcoves. Hence, by making an arbitrary choice of alcove  , one may place 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. This identification for   is illustrated at right.

Example: n = 2

 
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  . Every other element of the group can be written as an alternating product of copies of   and  .

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 is the line with equation   in the Euclidean plane  . The root lattice inside V consists of those pairs   for integral a. The Coxeter generator   acts on V by reflection across the line   (that is, across the origin); the generator   acts on V by reflection across the line   (that is, across the point  . It is natural to identify the line V with the real line  , by sending the point   to the real number 2x. With this identification, the root lattice consists of the even integers; the fundamental alcove is the interval [0, 1]; the element   acts by translation by k for any integer k; and the reflection   reflects across the point k for any integer k.

Permutation statistics and permutation patterns

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

Descents, length, and inversions

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.[9]

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  ). (In fact, the same is true for any affine Coxeter group.)[10]

Combinatorially, the length of an affine permutation is encoded in terms of an appropriate notion of inversions. In particular, one has for an affine permutation u that[11]

 
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[12][13]

 

Similarly, one may define an affine analogue of descents in permutations: say that 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.)[14]

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  .[14] 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  .

Geometrically, i is a descent of u if and only if the fixed hyperplane of   separates the alcoves   and  .

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).[15] One possible resolution is to consider affine descents (equivalently, cyclic descents) in the finite symmetric group  .[16] 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.[17]

Cycle type and reflection length

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 example, the affine permutation in   with window notation   contains the two infinite cycles   and   as well as infinitely many finite cycles   for each  . Cycles of an affine permutation correspond to cycles of the underlying permutation in an obvious way: in the example above, with underlying permutation  , 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 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.[18]

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.[19]) 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. (For example, in the permutation   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 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. (In the example above, the tuple is   and the nullity is 2, since one can take the partition  .) Then the reflection length of u is

 
where   is the underlying permutation of u.[20]

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, one has  .[21]

Fully commutative elements and pattern avoidance

A reduced word for an element g of a Coxeter group is a tuple   of Coxeter generators of minimum possible length such that  .[9] The element g is called fully commutative if one can transform any reduced word into any other by sequentially swapping pairs of factors that commute.[22] 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.

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  .[a]

It has also been shown that the number of affine permutations avoiding a single pattern p is finite if and only if p avoids the pattern 321,[24] so in particular there are infinitely many fully commutative affine permutations. These were enumerated by length in (Hanusa & Jones 2010).

Parabolic subgroups and other structures

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.

Parabolic subgroups, coset representatives

 
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.[14]

The non-maximal parabolic subgroups of   are all isomorphic to parabolic subgroups of  , that is, to a Young subgroup   for some positive integers   with sum n.

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:[14]

 

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  .)[25]

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.[26][27]

Bruhat order

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.[28]

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   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), J.-Y. Shi showed that left cells for   are indexed instead by tabloids,[b] 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 of Viennot (1977).

Inverse realizations

 
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.[c] 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.[29] (In the finite symmetric group  , the analogous distinction is between the active and passive forms of a permutation.[30])

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 tesselated space V.[31]

Relationship to other mathematical objects

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

Juggling patterns

 
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.
Nummer9 CC BY SA 3

In (Ehrenborg & Readdy 1996), a correspondence is given between affine permutations and juggling patterns encoded in a version of siteswap notation.[32] 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).[d] The number b of balls in the pattern is the average  .[34] 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.[32] Under this bijection, the length of the affine permutation is encoded by a natural statistic in the juggling pattern: one has
 
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.[35]

For example, the juggling pattern 441 (illustrated at right) has   and  . Therefore, it corresponds to the affine permutation  . The juggling pattern has four crossings, and the affine permutation has length  .

Similar techniques can be used to derive the generating function for minimal coset representatives of   by length.[36]

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

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.

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  .[37]

Affine Lie algebras

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).[38] In the classification of affine Lie algebras, the one associated to   is of (untwisted) type  , with Cartan matrix   for   and

 
(a circulant matrix) for  .[39]

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.[40] 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.[41]

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  . 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. However, 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  .[7]

Combinatorics of other affine Coxeter groups

The geometric action of the affine symmetric group   places it naturally in the family of affine Coxeter groups, all of which have a similar geometric action. 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. Thus, the combinatorial interpretations of descents, inversions, etc., carry over in these cases.[42] Abacus models of minimum-length coset representatives for parabolic quotients have also been extended to this context.[43]

Acknowledgements

The author thanks the three referees for their many helpful comments, and Max Glick for his help with the less technical summary. The work of the author was supported in part by a Simons Collaboration Grant (634530).

Notes

  1. Björner & Brenti (2005), p. 17.
  2. Humphreys (1990), p. 17.
  3. 3.0 3.1 Humphreys (1990), Chapter 4.
  4. 4.0 4.1 Humphreys (1990), Section 4.3.
  5. Björner & Brenti (2005), Chapter 8.3.
  6. Björner & Brenti (2005), Proposition 8.3.5.
  7. 7.0 7.1 Chmutov, Pylyavskyy & Yudovina (2018), Section 1.6.
  8. Björner & Brenti (2005), p. 260.
  9. 9.0 9.1 Björner & Brenti (2005), p. 15.
  10. Humphreys (1990), p. 93.
  11. Björner & Brenti (2005), p. 261.
  12. Björner & Brenti (2005), p. 208.
  13. Björner & Brenti (1996), Cor. 4.7.
  14. 14.0 14.1 14.2 14.3 Björner & Brenti (2005), p. 263.
  15. Reiner (1995), p. 2.
  16. Petersen (2015), Chapter 14.
  17. Reiner (1995), Theorem 6.
  18. Lewis et al. (2019), Propositions 1.31 and 4.24.
  19. Lewis et al. (2019).
  20. Lewis et al. (2019), Theorem 4.25.
  21. Lewis et al. (2019), Corollary 2.5.
  22. Stembridge (1996), p. 353.
  23. Hanusa & Jones (2010), p. 1345.
  24. Crites (2010), Theorem 1.
  25. Hanusa & Jones (2010), Section 2.2.
  26. Lapointe & Morse (2005).
  27. Berg, Jones & Vazirani (2009).
  28. Björner & Brenti (2005), p. 264.
  29. Knutson, Lam & Speyer (2013), Section 2.1.
  30. As in (Cameron 1994, Section 3.5).
  31. As in, for example, (Beazley et al. 2015), (Lam 2015).
  32. 32.0 32.1 Polster (2003), p. 42.
  33. Polster (2003), p. 22.
  34. Polster (2003), p. 15.
  35. Polster (2003), p. 43.
  36. Clark & Ehrenborg (2011), Theorem 2.2.
  37. Lewis (2020), Section 3.2.
  38. Kac (1990), Chapter 3.
  39. Kac (1990), Chapter 4.
  40. Kac (1990), Chapter 10.
  41. Kac (1990), Chapter 12.
  42. Björner & Brenti (2005), Chapter 8.
  43. Hanusa & Jones (2012).
  1. 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.[23]
  2. In a standard Young tableau, entries increase across rows and down columns; in a tabloid, they increase across rows, but there is no column condition.
  3. In other words, one might be interested in switching from a left group action to a right action or vice-versa.
  4. 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  .[33]
  5. In some sources, unitary reflections are called pseudoreflections.

References