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. Definition
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.) Construction
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 image 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 . We have that

maps into , more precisely

is the projector onto . Then

which is the projector onto . The image of  is

where μ is the conjugate partition to λ. Here,  and  are the symmetric and alternating tensor product spaces. The image  of  in  is an irreducible representation of Sn, called a Specht module. We write

for the irreducible representation. Some scalar multiple of  is idempotent, that is   for some rational number  Specifically, one finds . In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra . Consider, for example, S3 and the partition (2,1). Then one has

If V is a complex vector space, then the images of  on spaces  provides essentially all the finite-dimensional irreducible representations of GL(V). See also
 Representation theory of the symmetric group

Notes

References
 William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997. Lecture 4 of 
 Bruce E. Sagan. The Symmetric Group. Springer, 2001. Representation theory of finite groups
Symmetric functions
Permutations