Young symmetrizer

In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space obtained from the action of on by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young. Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, and consider the action of given by permuting the boxes of . Define two permutation subgroups and of Sn as follows: and Corresponding to these two subgroups, define two vectors in the group algebra as and where is the unit vector corresponding to g, and is the sign of the permutation. The product is the Young symmetrizer corresponding to the Young tableau λ. Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer. (If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.) Let V be any vector space over the complex numbers. Consider then the tensor product vector space (n times). Let Sn act on this tensor product space by permuting the indices. One then has a natural group algebra representation on (i.e. is a right module). Given a partition λ of n, so that , then the of is For instance, if , and , with the canonical Young tableau . Then the corresponding is given by For any product vector of we then have Thus the set of all clearly spans and since the span we obtain , where we wrote informally . Notice also how this construction can be reduced to the construction for . Let be the identity operator and the swap operator defined by , thus and .