Representation theory of finite symmetric groups

Symmetrizer and antisymmetrizer elements

The symmetrizer and antisymmetrizer are essential idempotents of the complex group ring of the symmetric group , defined as follows #m/def/rep/sym

The symmetrizer generates the left ideäl carrying the trivial representation, whereas the antisymmetrizer generates that carrying the alternating character.

Proof

For the symmetrizer see Trivial irrep carrying ideal of the group ring. For the antisymmetrizer note

and for any and

where we used

Thus generates the minimal ideäl carrying . The nonequivalence of these irreps may also be shown using the Equivalence of irreps on left ideals criterion: For any ,

since there exist equal even and odd permutations.

The symmetrizer and antisymmetrizer elements fall into the more general category of Young operators, the former corresponding to the one-row diagram and the latter to the one-column diagram.

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