1. Gems of Algebra: The secret life of the symmetric group
Some things that you may not have known about permutations.

2. Permutation of
= the set of permutations of

3. Two line notation
Example

4. One line notation (word of a permutation)
Example two line notation:

5. One line notation (word of a permutation)
Example one line notation:

6. Cycle notation
A cycle

7. Cycle notation
A cycle
means

8. 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.

9. Two line notation:
One line notation:
Cycle notation:

10. Two line notation:
One line notation:
Cycle notation:

11. Two line notation:
One line notation:
Cycle notation:

12. Two line notation:
One line notation:
Cycle notation:

13. Multiplication of permutations
Composition of two permutations as functions gives another permutation

14. Multiplication of permutations
Composition of two permutations as functions gives another permutation
will be another permutation

15. Multiplication of permutations
Composition of two permutations as functions gives another permutation
will be another permutation
The definition of this permutation will be

16. 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

17. Multiplication using two line notation

18. Multiplication using two line notation

19. Multiplication using two line notation

20. Under the operation of multiplication forms a group.
Closed under the operation of multiplication if
then

21. Under the operation of multiplication forms a group.
Closed under the operation of multiplication if
then
The set has an identity element.

22. 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

23. 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

24. 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

28. 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

29. 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

30. 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.

31. Generators and relations

32. Generators and relations
The elements
generate
where these elements are characterized by the relations

33. (Coxeter) The set of permutations can be realized
as compositions of reflections across hyperplanes
in which divide the space into chambers.
fundamental
chamber

40. Hyperplanes of
representing

41. 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 .

42. Consider:

43. Consider:
Example:

44. where
and
is called an Eulerian ‘statistic’

45. The length of a permutation is equal to the
the number of inversions in the permutation.
number of inversions left of: 3 5 6 1 5

46. 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.

47. Graph of Weak
order for

48. Graph of Weak
order for

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

52. A permutation as
a set of points

53. A permutation as
a set of points
Contains the permutation 123

54. A permutation as
a set of points
Contains the permutation 132

55. A permutation as
a set of points
Contains the permutation 213

56. A permutation as
a set of points
Contains the permutation 231

57. A permutation as
a set of points
Contains the permutation 312

58. A permutation as
a set of points
Contains the permutation 321

59. to be the set of partitions
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

60. The following permutations of size n=3, 4
123
132
213
231
312
1234
1243
1324
1342
1423
2134
2143
2314
2341
2413
3124
3142
3412
4123

61. A new way of looking at permutations?