Young symmetrizer
From Encyclopedia of Mathematics
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
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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
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