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Linear representation theory of symmetric group:S3

1,366 bytes added, 04:43, 17 June 2012
Products and Schur functors
|-
| standard representation || <math>\mathbb{Z}[\sqrt{3}/2]</math> || <math>\mathbb{Q}(\sqrt{3})</math>
|}
 
==Schur functors corresponding to irreducible representations==
 
Note that the discussion in this section relies ''specifically'' on the group being a symmetric group, and ''does not make sense for arbitrary finite groups.''
 
{| class="sortable" border="1"
! Common name of representation !! Degree !! Square of degree !! Corresponding [[Unordered integer partition|partition]] !! [[Young diagram]] !! Formula for dimension of corresponding [[Schur functor]] applied to a vector space of dimension <math>d</math> !! [[Formula calculating effect of Schur functor on character]] !! Degree of representation times this dimension
|-
| [[trivial representation]] || 1 || 1 || 3 || [[File:Youngdiag3.png|100px]] || <math>{d(d+1)(d+2)/6</math> || <math>\frac{d(d+1)(d+2)}{6}</math> || <math>(\chi(g)^3 + 3\chi(g^2)\chi(g) + 2\chi(g^3))/6</math>
|-
| [[sign representation]] || 1 || 1 || 1 + 1 + 1 || [[File:Youngdiag1-1-1.png|30px]] || <math>d(d - 1)(d - 2)/6</math> || <math>d(d - 1)(d - 2)/6</math> || <math>(\chi(g)^3 - 3\chi(g^2)\chi(g) + 2\chi(g^3))/6</math>
|-
| [[standard representation]] || 2 || 4 || 2 + 1 || [[File:Youngdiag2-1.png|60px]] || <math>d(d + 1)(d - 1)/3</math> || <math>2d(d + 1)(d - 1)/3</math> ||<math>(\chi(g)^3 - \chi(g^3))/3</math>
|-
! Total !! -- !! 6 (equals 3!, order of group) !! -- !! -- !! -- !! <math>d^3</math> (as expected) !! --
|}
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