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 formula_1 obtained from the action of formula_2 on formula_1 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 "S""n" and specific Young tableau λ corresponding to a numbered partition of "n", and consider the action of formula_2 given by permuting the boxes of formula_5. Define two permutation subgroups formula_6 and formula_7 of "S""n" as follows:
and
Corresponding to these two subgroups, define two vectors in the group algebra formula_10 as
and
where formula_13 is the unit vector corresponding to "g", and formula_14 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 formula_16 ("n" times). Let "Sn" act on this tensor product space by permuting the indices. One then has a natural group algebra representation formula_17 on formula_1.
Given a partition λ of "n", so that formula_19, then the image of formula_20 is
For instance, if formula_22, and formula_23, with the canonical Young tableau formula_24. Then the corresponding formula_20 is given by 
For any product vector formula_27 of formula_28 we then have 
Thus the span of all formula_30 clearly spans formula_31 and since the formula_30 span formula_28 we obtain formula_34, where we wrote informally formula_35.
Notice also how this construction can be reduced to the construction for formula_36.
Let formula_37 be the identity operator and formula_38 the swap operator defined by formula_39, thus formula_40 and formula_41. We have that 
maps into formula_43, more precisely 
is the projector onto formula_43.
Then
which is the projector onto formula_31.
The image of formula_48 is 
where μ is the conjugate partition to λ. Here, formula_50 and formula_51 are the symmetric and alternating tensor product spaces.
The image formula_52 of formula_53 in formula_54 is an irreducible representation of "Sn", called a Specht module. We write
for the irreducible representation.
Some scalar multiple of formula_56 is idempotent, that is formula_57 for some rational number formula_58 Specifically, one finds formula_59. In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra formula_60.
Consider, for example, "S"3 and the partition (2,1). Then one has 
If "V" is a complex vector space, then the images of formula_56 on spaces formula_63 provides essentially all the finite-dimensional irreducible representations of GL(V).