Gems of Algebra:Gems of Algebra:The secret life of theThe secret life of the

symmetric groupsymmetric group

Some things that you may not have known about permutations.

Permutation of

= the set of permutations of

Two line notation

Example

One line notation (word of a permutation)

Example two line notation:

One line notation (word of a permutation)

Example one line notation:

Cycle notationA cycle

Cycle notationA cycle

means

Cycle notationA cycle

means

where each of the cycles contain disjoint sets of integers

This representation is not unique, the cycles may be writtenin any order and any number in the cycle can be listed first.

Two line notation:

One line notation:

Cycle notation:

Two line notation:

One line notation:

Cycle notation:

Two line notation:

One line notation:

Cycle notation:

Two line notation:

One line notation:

Cycle notation:

Multiplication of permutations

Composition of two permutations as functions gives another permutation

Multiplication of permutations

Composition of two permutations as functions gives another permutation

will be another permutation

Multiplication of permutations

Composition of two permutations as functions gives another permutation

will be another permutation

The definition of this permutation will be

Multiplication of permutations

Composition of two permutations as functions gives another permutation

will be another permutation

The definition of this permutation will be

will sometimes be written as

Multiplication using two line notation

Multiplication using two line notation

Multiplication using two line notation

Under the operation of multiplication forms a group.

Closed under the operation of multiplication

if then

Under the operation of multiplication forms a group.

Closed under the operation of multiplication

The set has an identity element.

if then

Under the operation of multiplication forms a group.

Closed under the operation of multiplication

The set has an identity element.

if then

Every element in the set has an inverse

Under the operation of multiplication forms a group.

Closed under the operation of multiplication

The set has an identity element.

if then

Every element in the set has an inverse

Under the operation of multiplication forms a group.

Closed under the operation of multiplication

The set has an identity element.

if then

Every element in the set has an inverse

The symmetric group contains every groupof order n as a subgroup

The group of permutations is called the symmetric group

are the elements of a group of order n

The symmetric group contains every groupof order n as a subgroup

The group of permutations is called the symmetric group

are the elements of a group of order n

The symmetric group contains every groupof order n as a subgroup

The group of permutations is called the symmetric group

are the elements of a group of order n

Then these corresponding elements will multiply in thesymmetric group just as they do in their own group.

Generators and relations

Generators and relations

generateThe elements

where these elements are characterized by the relations

(Coxeter) The set of permutations can be realizedas compositions of reflections across hyperplanesin which divide the space into chambers.

fundamentalchamber

Hyperplanes of representing

From this perspective we have the notion of the lengthof a permutation.

The length of a permutation is the length of the smallestword of elements that can be used to represent .

Consider:

Consider:

Example:

where

and

is called an Eulerian ‘statistic’

The length of a permutation is equal to thethe number of inversions in the permutation.

number of inversions left of: 0 0 0 0 3 5 6 51

Weak order

We may ‘draw’ this with a graph (vertices and edges) sothat the vertices are permutations and there is an edge betweentwo permutations if

To create the following images we also put someadditional restrictions1. the level will depend on the length of the permutation2. the color of the edge will determine the position that is changing so that every permutation will have one edge of each color.

Graph of Weakorder for

Graph of Weakorder for

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Note that all faces of the permuta-hedrons are made up of two types of facets.

A permutation asa set of points

A permutation asa set of points

Contains the permutation 123

A permutation asa set of points

Contains the permutation 132

A permutation asa set of points

Contains the permutation 213

A permutation asa set of points

Contains the permutation 231

A permutation asa set of points

Contains the permutation 312

A permutation asa set of points

Contains the permutation 321

Define

to be the set of partitions

which do not contain the pattern

contains 123contains 132contains 213contains 231

avoids 312avoids 321

123132213231312

The following permutations of size n=3, 4

1234124313241342142321342143

2314234124133124314234124123

A new way of looking at permutations?