In the context of representation theory and the theory of symmetric groups, the **Young symmetrizer** is an important concept used to construct representations of symmetric groups and to understand how to decompose these representations into irreducible components. ### Definition For a given partition of a positive integer \( n \), a **Young diagram** can be constructed where the shape of the diagram corresponds to the partition.
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For a Young tableau , let sum its row permutations and be the signed sum of its column permutations in the group algebra of the symmetric group. The product is a Young symmetrizer. The other order is another conventional realization. It satisfies , where is the hook product of a partition, so division by that scalar gives a primitive idempotent in characteristic zero. Acting on a tensor power constructs a Schur module.