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| In [[mathematics]], a '''Young symmetrizer''' is an element of the [[group ring|group algebra]] of the [[symmetric group]], constructed in such a way that the image of the element corresponds to an [[irreducible representation]] of the symmetric group over the [[complex number]]s. A similar construction works over any field, and the resulting representations are called '''[[Specht module]]s'''. The Young symmetrizer is named after British mathematician [[Alfred Young]].
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| ==Definition==
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| Given a finite symmetric group ''S''<sub>''n''</sub> and specific [[Young tableau]] λ corresponding to a numbered partition of ''n'', define two [[permutation group|permutation subgroups]] <math>P_\lambda</math> and <math>Q_\lambda</math> of ''S''<sub>''n''</sub> as follows:
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| :<math>P_\lambda=\{ g\in S_n : g \text{ preserves each row of } \lambda \}</math>
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| | | <li>[http://nmjnfda.gov.cn/plus/feedback.php?aid=4 http://nmjnfda.gov.cn/plus/feedback.php?aid=4]</li> |
| and
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| | | <li>[http://www.cnbike.net/plus/feedback.php?aid=106 http://www.cnbike.net/plus/feedback.php?aid=106]</li> |
| :<math>Q_\lambda=\{ g\in S_n : g \text{ preserves each column of } \lambda \}.</math>
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| | | <li>[http://sunline-racing.com/blog/question/indextest.cgi http://sunline-racing.com/blog/question/indextest.cgi]</li> |
| Corresponding to these two subgroups, define two vectors in the [[group algebra]] <math>\mathbb{C}S_n</math> as
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| | | </ul> |
| :<math>a_\lambda=\sum_{g\in P_\lambda} e_g</math>
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| and
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| :<math>b_\lambda=\sum_{g\in Q_\lambda} \sgn(g) e_g</math>
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| where <math>e_g</math> is the unit vector corresponding to ''g'', and <math>\sgn(g)</math> is the signature of the permutation. The product
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| :<math>c_\lambda := a_\lambda b_\lambda = \sum_{g\in P_\lambda,h\in Q_\lambda} \sgn(h) e_{gh}</math>
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| is the '''Young symmetrizer''' corresponding to the [[Young tableau]] λ. Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer. (If we replace the [[complex number]]s by more general [[field (mathematics)|field]]s the corresponding representations will not be irreducible in general.)
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| ==Construction==
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| Let ''V'' be any [[vector space]] over the [[complex number]]s. Consider then the [[tensor product]] vector space <math>V^{\otimes n}=V \otimes V \otimes \cdots \otimes V</math> (''n'' times). Let ''S''<sub>n</sub> act on this tensor product space by permuting the indices. One then has a natural [[group ring|group algebra]] representation <math>\mathbb{C}S_n \rightarrow \text{End} (V^{\otimes n})</math> on <math>V^{\otimes n}</math>.
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| Given a partition λ of ''n'', so that <math>n=\lambda_1+\lambda_2+ \cdots +\lambda_j</math>, then the [[image (mathematics)|image]] of <math>a_\lambda</math> is
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| :<math>\text{Im}(a_\lambda) := a_\lambda V^{\otimes n} \cong
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| \text{Sym}^{\lambda_1}\; V \otimes
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| \text{Sym}^{\lambda_2}\; V \otimes \cdots \otimes
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| \text{Sym}^{\lambda_j}\; V.
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| </math>
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| For instance, if <math>n =4</math>, and <math>\lambda = (2,2)</math>, with the canonical Young tableau <math>\{\{1,2\},\{3,4\}\}</math>. Then the corresponding <math>a_\lambda</math> is given by <math> a_\lambda = e_{\text{id}} + e_{(1,2)} + e_{(3,4)} + e_{(1,2)(3,4)}</math>. Let an element in <math>V^{\otimes 4}</math> be given by <math>v_{1,2,3,4}:=v_1 \otimes v_2 \otimes v_3 \otimes v_4</math>. Then
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| :<math> a_\lambda v_{1,2,3,4} = v_{1,2,3,4} + v_{2,1,3,4} + v_{1,2,4,3} + v_{2,1,4,3} = (v_1 \otimes v_2 + v_2 \otimes v_1) \otimes (v_3 \otimes v_4 + v_4 \otimes v_3). </math>
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| The latter clearly span <math> \text{Sym}^2\; V\otimes \text{Sym}^2\; V</math>.
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| The image of <math>b_\lambda</math> is
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| :<math>\text{Im}(b_\lambda) \cong | |
| \bigwedge^{\mu_1} V \otimes
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| \bigwedge^{\mu_2} V \otimes \cdots \otimes
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| \bigwedge^{\mu_k} V
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| </math>
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| where μ is the conjugate partition to λ. Here, <math>\text{Sym}^i V </math> and <math>\bigwedge^j V</math> are the [[symmetric algebra|symmetric]] and [[exterior algebra|alternating tensor product spaces]].
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| The image <math>\mathbb{C}S_nc_\lambda</math> of <math>c_\lambda = a_\lambda \cdot b_\lambda</math> in <math>\mathbb{C}S_n</math> is an irreducible representation<ref>See {{harv|Fulton|Harris|1991|loc=Theorem 4.3, p. 46}}</ref> of ''S''<sub>n</sub>, called a [[Specht module]]. We write
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| :<math>\text{Im}(c_\lambda) = V_\lambda</math>
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| for the irreducible representation.
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| Some scalar multiple of <math>c_\lambda</math> is idempotent, that is <math>c^2_\lambda = \alpha_\lambda c_\lambda</math> for some rational number <math>\alpha_\lambda\in\mathbb{Q}</math>. Specifically, one finds <math>\alpha_\lambda=n! / \text{dim } V_\lambda</math>. In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra <math>\mathbb{Q}S_n</math>.
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| Consider, for example, ''S''<sub>3</sub> and the partition (2,1). Then one has <math>c_{(2,1)} = e_{123}+e_{213}-e_{321}-e_{312}</math>
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| If ''V'' is a complex vector space, then
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| the images of <math>c_\lambda</math> on spaces <math>V^{\otimes d}</math> provides essentially all the finite-dimensional irreducible representations of GL(V).
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| ==See also==
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| * [[Representation theory of the symmetric group]]
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| ==Notes==
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| <references/>
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| ==References==
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| * William Fulton. ''Young Tableaux, with Applications to Representation Theory and Geometry''. Cambridge University Press, 1997.
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| * Lecture 4 of {{Fulton-Harris}}
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| * [[Bruce Sagan|Bruce E. Sagan]]. ''The Symmetric Group''. Springer, 2001.
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| [[Category:Representation theory of finite groups]]
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| [[Category:Symmetric functions]]
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| [[Category:Permutations]]
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'Get out
Din! Really makes a fire in my belly, these old guys ルイヴィトン レディース really want ルイヴィトン限定財布 to punch the boom into slag! '
four under phase ブランド ルイヴィトン with each ルイヴィトン ケース other, have come to realize, it was a monumental temple to the pressure.
......
Yin Kong city.
'Get out, get out!' crowd roaring riding Huangshou soldiers, will give way to a lot of people rushed to one side.
'side, stick!'
This is a large group of soldiers led by pointing to the bare walls in the street next to the ordered other soldiers suddenly jumped ルイヴィトンさいふ Huangshou one, quickly down the paper will ルイヴィトンルイヴィトン be distributed at the beginning of the whole picture to go to the wall, and soon, it clearly incomparable picture, the reward will make ルイヴィトン 服 it shocking that people on the street stunned.
here belong thoroughfare, between the flow very much.
'Just a little bit with the message, on the award of one バック ルイヴィトン million ...... treasure money? million treasure money? that are much?'
'I have not personally seen the treasure money too.'
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