# Changes

## Linear representation theory of symmetric group:S3

, 15:27, 27 May 2011
Representations
===Trivial representation===
The trivial or principal representation is a one-dimensional representation sending every element of the symmetric group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements.:
{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value
|-
| identity element || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](1,2,3)[/itex] || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](1,3,2)[/itex] || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](1,2)[/itex] || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](2,3)[/itex] || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](1,3)[/itex] || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|}
===Sign representation===
The sign representation is a one-dimensional representation sending every permutation to its ''sign'': the even permutations get sent to 1 and the odd permutations get sent to -1. The kernel of this representation (i.e. the permutations that get sent to one) is the alternating group: the unique cyclic subgroup of order three comprising permutations [itex](1,2,3)[/itex], [itex](1,3,2)[/itex] and the identity permutation. The three permutations of order two all get sent to -1.
This representation makes sense over any field, but when the characteristic of the field is two, it is the same as the trivial representation, because math>1 = -1[/itex] in characteristic two.
{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value
|-
| identity element || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](1,2,3)[/itex] || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](1,3,2)[/itex] || [itex]( 1 )[/itex] || [itex]x - 1[/itex] || [itex]x - 1[/itex] || 1
|-
| [itex](1,2)[/itex] || [itex]( -1 )[/itex] || [itex]x + 1[/itex] || [itex]x + 1[/itex] || -1
|-
| [itex](2,3)[/itex] || [itex]( -1 )[/itex] || [itex]x + 1[/itex] || [itex]x + 1[/itex] || -1
|-
| [itex](1,3)[/itex] || [itex]( -1 )[/itex] || [itex]x + 1[/itex] || [itex]x + 1[/itex] || -1
|}
===Standard representation===