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## Linear representation theory of symmetric group:S3

, 04:47, 17 June 2012
Schur functors corresponding to irreducible representations
{| class="sortable" border="1"
! Common name of representation !! Degree !! Square of degree !! Corresponding [[Unordered integer partition|partition]] !! [[Young diagram]] !! Common name of corresponding Schur functor !! Formula for dimension of corresponding [[Schur functor]] applied to a vector space of dimension $d$ !! Degree of representation times this dimension !! [[Formula for calculating effect of Schur functor on character]] if this Schur functor is applied to any (possibly unrelated) representation whose character is $\chi$
|-
| [[trivial representation]] || 1 || 1 || 3 || [[File:Youngdiag3.png|100px]] || $\operatorname{Sym}^3$ || $d(d+1)(d+2)/6$ || $d(d+1)(d+2)/6$ || $(\chi(g)^3 + 3\chi(g^2)\chi(g) + 2\chi(g^3))/6$
|-
| [[sign representation]] || 1 || 1 || 1 + 1 + 1 || [[File:Youngdiag1-1-1.png|30px]] || $\wedge^3$ || $d(d - 1)(d - 2)/6$ || $d(d - 1)(d - 2)/6$ || $(\chi(g)^3 - 3\chi(g^2)\chi(g) + 2\chi(g^3))/6$
|-
| [[standard representation]] || 2 || 4 || 2 + 1 || [[File:Youngdiag2-1.png|60px]] || $\mathbb{S}_{2,1}$ || $d(d + 1)(d - 1)/3$ || $2d(d + 1)(d - 1)/3$ ||$(\chi(g)^3 - \chi(g^3))/3$
|-
! Total !! -- !! 6 (equals 3!, order of group) !! -- !! -- !! -- !! $d^3$ (as expected) !! --
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