# Changes

## Linear representation theory of symmetric group:S3

, 04:43, 17 June 2012
Products and Schur functors
|-
| standard representation || $\mathbb{Z}[\sqrt{3}/2]$ || $\mathbb{Q}(\sqrt{3})$
|}

==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 $d$ !! [[Formula calculating effect of Schur functor on character]] !! Degree of representation times this dimension
|-
| [[trivial representation]] || 1 || 1 || 3 || [[File:Youngdiag3.png|100px]] || ${d(d+1)(d+2)/6$ || $\frac{d(d+1)(d+2)}{6}$ || $(\chi(g)^3 + 3\chi(g^2)\chi(g) + 2\chi(g^3))/6$
|-
| [[sign representation]] || 1 || 1 || 1 + 1 + 1 || [[File:Youngdiag1-1-1.png|30px]] || $d(d - 1)(d - 2)/6$ || $d(d - 1)(d - 2)/6$ || $(\chi(g)^3 - 3\chi(g^2)\chi(g) + 2\chi(g^3))/6$
|-
| [[standard representation]] || 2 || 4 || 2 + 1 || [[File:Youngdiag2-1.png|60px]] || $d(d + 1)(d - 1)/3$ || $2d(d + 1)(d - 1)/3$ ||$(\chi(g)^3 - \chi(g^3))/3$
|-
! Total !! -- !! 6 (equals 3!, order of group) !! -- !! -- !! -- !! $d^3$ (as expected) !! --
|}