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Difference between revisions of "Young symmetrizer"

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An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990901.png" /> of the group ring of the [[Symmetric group|symmetric group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990902.png" /> defined by the [[Young tableau|Young tableau]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990903.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990904.png" /> by the following rule. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990905.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990906.png" />) be the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990907.png" /> consisting of all permutations permuting the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990908.png" /> in each row (respectively, column) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y0990909.png" />. Further, put
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An element $e_d$ of the group ring of the [[Symmetric group|symmetric group]] $S_m$ defined by the [[Young tableau|Young tableau]] $d$ of order $m$ by the following rule. Let $R_d$ (respectively, $C_d$) be the subgroup of $S_m$ consisting of all permutations permuting the numbers $1,\ldots,m$ in each row (respectively, column) in $d$. Further, put
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909010.png" /></td> </tr></table>
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$$r_d=\sum_{g\in R_d}g,\quad c_d=\sum_{g\in C_d}\epsilon(g)g,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909011.png" /> is the parity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909012.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909013.png" /> (sometimes one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909014.png" />).
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where $\epsilon(g)=\pm1$ is the parity of $g$. Then $e_d=c_dr_d$ (sometimes one defines $e_d=r_dc_d$).
  
The basic property of a Young symmetrizer is that it is proportional to a primitive idempotent of the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909015.png" />. The coefficient of proportionality is equal to the product of the lengths of all hooks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909016.png" />.
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The basic property of a Young symmetrizer is that it is proportional to a primitive idempotent of the group algebra $\mathbf CS_m$. The coefficient of proportionality is equal to the product of the lengths of all hooks of $d$.
  
  
  
 
====Comments====
 
====Comments====
The ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909017.png" /> is isomorphic to the Specht module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909018.png" /> defined by the Young tableau <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099090/y09909019.png" />. Cf. also [[Young tableau|Young tableau]] for references and more details.
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The ideal $\mathbf C[S_m]e_d$ is isomorphic to the Specht module of $S_m$ defined by the Young tableau $d$. Cf. also [[Young tableau|Young tableau]] for references and more details.

Latest revision as of 13:01, 10 August 2014

An element $e_d$ of the group ring of the symmetric group $S_m$ defined by the Young tableau $d$ of order $m$ by the following rule. Let $R_d$ (respectively, $C_d$) be the subgroup of $S_m$ consisting of all permutations permuting the numbers $1,\ldots,m$ in each row (respectively, column) in $d$. Further, put

$$r_d=\sum_{g\in R_d}g,\quad c_d=\sum_{g\in C_d}\epsilon(g)g,$$

where $\epsilon(g)=\pm1$ is the parity of $g$. Then $e_d=c_dr_d$ (sometimes one defines $e_d=r_dc_d$).

The basic property of a Young symmetrizer is that it is proportional to a primitive idempotent of the group algebra $\mathbf CS_m$. The coefficient of proportionality is equal to the product of the lengths of all hooks of $d$.


Comments

The ideal $\mathbf C[S_m]e_d$ is isomorphic to the Specht module of $S_m$ defined by the Young tableau $d$. Cf. also Young tableau for references and more details.

How to Cite This Entry:
Young symmetrizer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_symmetrizer&oldid=32801
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article