Gems of Algebra: The secret life of the symmetric group

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Gems of Algebra The secret life of
the symmetric group

• Some things that you may not have known about
  permutations.

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Permutation of
the set of permutations of
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Two line notation
Example
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One line notation (word of a permutation)
Example two line notation
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One line notation (word of a permutation)
Example one line notation
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Cycle notation
A cycle
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Cycle notation
A cycle
means
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Cycle notation
A cycle
means
where each of the cycles contain disjoint sets of
integers
This representation is not unique, the cycles may
be written in any order and any number in the
cycle can be listed first.
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Two line notation
One line notation
Cycle notation
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Two line notation
One line notation
Cycle notation
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Two line notation
One line notation
Cycle notation
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Two line notation
One line notation
Cycle notation
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Multiplication of permutations
Composition of two permutations as functions
gives another permutation
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Multiplication of permutations
Composition of two permutations as functions
gives another permutation
will be another permutation
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Multiplication of permutations
Composition of two permutations as functions
gives another permutation
will be another permutation
The definition of this permutation will be
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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
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Multiplication using two line notation
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Multiplication using two line notation
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Multiplication using two line notation
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Under the operation of multiplication forms
a group.
Closed under the operation of multiplication
if
then
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Under the operation of multiplication forms
a group.
Closed under the operation of multiplication
if
then
The set has an identity element.
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Under the operation of multiplication forms
a group.
Closed under the operation of multiplication
if
then
The set has an identity element.
Every element in the set has an inverse
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Under the operation of multiplication forms
a group.
Closed under the operation of multiplication
if
then
The set has an identity element.
Every element in the set has an inverse
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Under the operation of multiplication forms
a group.
Closed under the operation of multiplication
if
then
The set has an identity element.
Every element in the set has an inverse
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The group of permutations is called the symmetric
group
The symmetric group contains every group of order
n as a subgroup
are the elements of a group of order n
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The group of permutations is called the symmetric
group
The symmetric group contains every group of order
n as a subgroup
are the elements of a group of order n
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The group of permutations is called the symmetric
group
The symmetric group contains every group of order
n as a subgroup
are the elements of a group of order n
Then these corresponding elements will multiply
in the symmetric group just as they do in their
own group.
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Generators and relations
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Generators and relations
generate
The elements
where these elements are characterized by the
relations
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(Coxeter) The set of permutations can be
realized as compositions of reflections across
hyperplanes in which divide the space
into chambers.
fundamental chamber
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representing
Hyperplanes of
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From this perspective we have the notion of the
length of a permutation.
The length of a permutation is the length of
the smallest word of elements that can
be used to represent .
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Consider
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Consider
Example
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where
and
is called an Eulerian statistic
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The length of a permutation is equal to the the
number of inversions in the permutation.
0
0
0
0
3
5
6
5
1
number of inversions left of
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Weak order
We may draw this with a graph (vertices and
edges) so that the vertices are permutations and
there is an edge between two permutations if

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

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Graph of Weak order for
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Graph of Weak order for
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Note that all faces of the permuta-hedrons are
made up of two types of facets.
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A permutation as a set of points
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A permutation as a set of points
Contains the permutation 123
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A permutation as a set of points
Contains the permutation 132
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A permutation as a set of points
Contains the permutation 213
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A permutation as a set of points
Contains the permutation 231
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A permutation as a set of points
Contains the permutation 312
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A permutation as a set of points
Contains the permutation 321
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Define
to be the set of partitions
which do not contain the pattern
contains 123
contains 132
contains 213
contains 231
avoids 312
avoids 321
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The following permutations of size n3, 4
1234 1243 1324 1342 1423 2134 2143
2314 2341 2413 3124 3142 3412 4123
123 132 213 231 312
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A new way of looking at permutations?