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Linear representation theory of symmetric group:S3

1,462 bytes added, 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 || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(1,2,3)</math> || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(1,3,2)</math> || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(1,2)</math> || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(2,3)</math> || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(1,3)</math> || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 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 <math>(1,2,3)</math>, <math>(1,3,2)</math> 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</math> in characteristic two.
{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value
|-
| identity element || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(1,2,3)</math> || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(1,3,2)</math> || <math>( 1 )</math> || <math>x - 1</math> || <math>x - 1</math> || 1
|-
| <math>(1,2)</math> || <math>( -1 )</math> || <math>x + 1</math> || <math>x + 1</math> || -1
|-
| <math>(2,3)</math> || <math>( -1 )</math> || <math>x + 1</math> || <math>x + 1</math> || -1
|-
| <math>(1,3)</math> || <math>( -1 )</math> || <math>x + 1</math> || <math>x + 1</math> || -1
|}
===Standard representation===
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