Changes

Jump to: navigation, search

Linear representation theory of symmetric group:S3

874 bytes added, 15:27, 16 July 2011
Character table
{| class="sortable" border="1"
!Representation/Conjugacy class representative!! <math>()</math> (identity element) -- size 1 || <math>(1,2,3)</math> (23-transpositioncycle) -- size 3 2 || <math>(1,2,3)</math> (32-cycletransposition) -- size 2
|-
| [[Trivial representation]] || 1 || 1 || 1
|-
| [[Sign representation]] || 1 || -1 || -1
|-
| [[Standard representation of symmetric group:S3|Standard representation]] || 2 || 0 -1 || -10
|}
The same character table applies in any characteristic not equal to 2 or 3, where 0,-1,1,2 are interpreted, not as integers, but as elements of that field.
 
There is a canonical bijection between conjugacy classes and representations (as is the case with all symmetric groups). Under the bijection, the trivial representation corresponds to the identity element, the standard representation corresponds to the 2-transposition, and the sign representation corresponds to the 3-cycle.
Here are the size-degree weighted characters (i.e., the product of the character value by the size of the conjugacy class divided by the degree of the representation).
{| class="sortable" border="1"
!Rep/Conj class !! <math>()</math> (identity element) || <math>(1,2,3)</math> (23-transpositioncycle) -- size 2 || <math>(1,2,3)</math> (32-cycletransposition)-- size 3
|-
| [[Trivial representation]] || 1 || 3 2 || 23
|-
| [[Sign representation]] || 1 || 2 || -3 || 2
|-
| [[Standard representation of symmetric group:S3|Standard representation]] || 1 || 0 -1 || -10
|}
Here is the orthogonal matrix obtained by multiplying each character value by the square root of the quotient of the size of its conjugacy class by the order of the group. Note that this is an orthogonal matrix due to the orthogonality relations between the characters.
<math>\begin{pmatrix} 1/\sqrt{6} & 1/\sqrt{23} & 1/\sqrt{32} \\ 1/\sqrt{6} & -1/\sqrt{23} & -1/\sqrt{32} \\ 2/\sqrt{6} & 0 & -1/\sqrt{3} & 0 \end{pmatrix}</math> ==Table of matrix entries== ===Using real orthogonal matrices as dihedral group=== This table satisfies the [[grand orthogonality theorem]] -- in particular, any two rows are orthogonal and each row has norm <math>1/n</math> where <math>n</math> is the degree of the representation. Note that unlike the character table, this table is not canonical and depends on the specific choice of matrices used for the two-dimensional representation. {| class="sortable" border="1"! Representation/element !! <math>()</math> !! <math>(1,2,3)</math> !! <math>(1,3,2)</math> !! <math>(1,2)</math> !! <math>(2,3)</math> !! <math>(1,3)</math>|-| trivial || 1 || 1 || 1 || 1 || 1 || 1|-| sign || 1 || 1 || 1 || -1 || -1 || -1|-| standard -- top left entry || 1 || -1/2 || -1/2 || -1/2 || 1 || -1/2|-| standard -- top right entry || 0 || <math>-\sqrt{3}/2</math> || <math>\sqrt{3}/2</math> || <math>\sqrt{3}/2</math> || 0 || <math>-\sqrt{3}/2</math>|-| standard -- bottom left entry || 0 || <math>\sqrt{3}/2</math> || <math>-\sqrt{3}/2</math> || <math>\sqrt{3}/2</math> || 0 || <math>-\sqrt{3}/2</math>|-| standard -- bottom right entry || 1 || -1/2 || -1/2 || 1/2 || -1 || 1/2|}
==Realizability information==
Bureaucrats, emailconfirmed, Administrators
38,910
edits

Navigation menu