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Young symmetrizer

From Encyclopedia of Mathematics
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An element of the group ring of the symmetric group defined by the Young tableau of order by the following rule. Let (respectively, ) be the subgroup of consisting of all permutations permuting the numbers in each row (respectively, column) in . Further, put

where is the parity of . Then (sometimes one defines ).

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


Comments

The ideal is isomorphic to the Specht module of defined by the Young tableau . 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