Affine symmetric group

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. They are studied in combinatorics and representation theory.

A finite symmetric group consists of all permutations of a finite set. Each affine symmetric group is an infinite extension of a finite symmetric group. Many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. Permutation statistics such as 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.

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

One way of defining groups is by generators and relations. In this type of definition, generators are a subset of group elements that, when combined, produce all other elements. The relations of the definition are a system of equations that determine when two combinations of generators are equal.

(The second and third relation are sometimes called the braid relations.)

Each Coxeter group may be represented by a Coxeter–Dynkin diagram, in which vertices correspond to generators and edges encode the relations between them.

Geometric definition

Each reflection preserves this lattice, and so the lattice is preserved by the whole group.

The reflections through these boundary hyperplanes may be identified with the Coxeter generators.

Combinatorial definition

The elements of the affine symmetric group may be realized as a group of periodic permutations of the integers.

Representation as matrices

Affine permutations can be represented as infinite periodic permutation matrices. In row 1, there is a 1 in column 2; in row 2, there is a 1 in column 0; and in row 3, there is a 1 in column 4. The rest of the entries in those rows and columns are all 0, and all the other entries in the matrix are fixed by the periodicity condition.

Relationship to the finite symmetric group

These connections allow a direct translation between the combinatorial and geometric definitions of the affine symmetric group.

As a subgroup

As a quotient

Connection between the geometric and combinatorial definitions

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.

Example: n = 2

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

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). Another is to consider simultaneously the length and number of descents of an affine permutation.

Cycle type and reflection length

Fully commutative elements and pattern avoidance

These were enumerated by length in (Hanusa & Jones 2010).

Parabolic subgroups and other structures

Other aspects of affine symmetric groups, such as their Bruhat order and representation theory, may also be understood via combinatorial models.

Parabolic subgroups, coset representatives

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.

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.

Bruhat order

Representation theory and an affine Robinson–Schensted correspondence

This bijection plays a central role in the combinatorics and the representation theory of the symmetric group. Their procedure uses the matrix representation of affine permutations and generalizes the shadow construction, introduced in (Viennot 1977).

Inverse realizations

These alternate realizations are described below.

Relationship to other mathematical objects

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

Juggling patterns

In (Ehrenborg & Readdy 1996), a correspondence is given between affine permutations and juggling patterns encoded in a version of siteswap notation.

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. A complex reflection group is a finite group of linear transformations on a complex vector space generated by reflections.

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).

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. In the case of affine Lie algebras, the resulting identities are equivalent to the Macdonald identities.

Braid group and group-theoretic properties

Coxeter groups have a number of special properties not shared by all groups. These include that their word problem is decidable (that is, there exists an algorithm that can determine whether or not any given product of the generators is equal to the identity element) and that they are linear groups (that is, they can be represented by a group of invertible matrices over a field).

(Subsequently, they have been proved for the Artin–Tits groups associated to affine Coxeter groups.) In the case of the affine symmetric group, these proofs make use of an associated Garside structure on the Artin–Tits group.

Not all Artin–Tits groups have a natural representation in terms of geometric braids.

Extended affine symmetric group

The affine symmetric group is a subgroup of the extended affine symmetric group. Unlike the affine symmetric group, the extended affine symmetric group is not a Coxeter group.

Combinatorics of other affine Coxeter groups

Abacus models of minimum-length coset representatives for parabolic quotients have also been extended to this context.

History

The study of Coxeter groups in general could be said to first arise in the classification of regular polyhedra (the Platonic solids) in ancient Greece. The modern systematic study (connecting the algebraic and geometric definitions of finite and affine Coxeter groups) began in work of Coxeter in the 1930s. The proof that the combinatorial definition agrees with the algebraic definition was given by Eriksson & Eriksson (1998).