View source for Young symmetrizer

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

<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>

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" />).

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" />.



====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.