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 $V^$ obtained from the action of $S_n$ on $V^$ 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 $S_n$ given by permuting the boxes of $\lambda$. Define two permutation subgroups $P_\lambda$ and $Q_\lambda$ of ''S''_''n'' as follows:

:$P_\lambda=\$

and

:$Q_\lambda=\.$

Corresponding to these two subgroups, define two vectors in the group algebra $\mathbbS_n$ as

:$a_\lambda=\sum_ e_g$

and

:$b_\lambda=\sum_ \sgn(g) e_g$

where $e_g$ is the unit vector corresponding to ''g'', and $\sgn(g)$ is the sign of the permutation. The product

:$c_\lambda := a_\lambda b_\lambda = \sum_ \sgn(h) e_$

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 $V^=V \otimes V \otimes \cdots \otimes V$ (''n'' times). Let ''S_n'' act on this tensor product space by permuting the indices. One then has a natural group algebra representation $\C S_n \to \operatorname (V^)$ on $V^$.

Given a partition λ of ''n'', so that $n=\lambda_1+\lambda_2+ \cdots +\lambda_j$, then the image of $a_\lambda$ is

:$\operatorname(a_\lambda) := a_\lambda V^ \cong \operatorname^ V \otimes \operatorname^ V \otimes \cdots \otimes \operatorname^ V.$

For instance, if $n = 4$, and $\lambda = (2,2)$, with the canonical Young tableau $\$. Then the corresponding $a_\lambda$ is given by

:$a_\lambda = e_ + e_ + e_ + e_.$

For any product vector $v_:=v_1 \otimes v_2 \otimes v_3 \otimes v_4$ of $V^$ we then have

:$a_\lambda v_ = v_ + v_ + v_ + v_ = (v_1 \otimes v_2 + v_2 \otimes v_1) \otimes (v_3 \otimes v_4 + v_4 \otimes v_3).$

Thus the span of all $v_$ clearly spans $\operatorname^2 V\otimes \operatorname^2 V$ and since the $v_$ span $V^$ we obtain $a_\lambda V^ = \operatorname^2 V \otimes \operatorname^2 V$, where we wrote informally $a_\lambda V^ \equiv \operatorname(a_\lambda)$.

Notice also how this construction can be reduced to the construction for $n = 2$.
Let $\mathbb \in \operatorname (V^)$ be the identity operator and $S\in \operatorname (V^)$ the swap operator defined by $S(v\otimes w) = w \otimes v$, thus $\mathbb = e_$ and $S = e_$. We have that

:$e_ + e_ = \mathbb + S$

maps into $\operatorname^2 V$, more precisely

:$\frac(\mathbb + S)$

is the projector onto $\operatorname^2 V$.
Then

:$\frac a_\lambda = \frac (e_ + e_ + e_ + e_) = \frac (\mathbb \otimes \mathbb + S \otimes \mathbb + \mathbb \otimes S + S \otimes S) = \frac(\mathbb + S) \otimes \frac (\mathbb + S)$

which is the projector onto $\operatorname^2 V\otimes \operatorname^2 V$.

The image of $b_\lambda$ is

:$\operatorname(b_\lambda) \cong \bigwedge^ V \otimes \bigwedge^ V \otimes \cdots \otimes \bigwedge^ V$

where μ is the conjugate partition to λ. Here, $\operatorname^i V$ and $\bigwedge^j V$ are the symmetric and alternating tensor product spaces.

The image $\C S_nc_\lambda$ of $c_\lambda = a_\lambda \cdot b_\lambda$ in $\C S_n$ is an irreducible representation of ''S_n'', called a Specht module. We write

:$\operatorname(c_\lambda) = V_\lambda$

for the irreducible representation.

Some scalar multiple of $c_\lambda$ is idempotent,See that is $c^2_\lambda = \alpha_\lambda c_\lambda$ for some rational number $\alpha_\lambda\in\Q.$ Specifically, one finds $\alpha_\lambda=n! / \dim V_\lambda$. In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra $\Q S_n$.

Consider, for example, ''S''₃ and the partition (2,1). Then one has

:$c_ = e_+e_-e_-e_.$

If ''V'' is a complex vector space, then the images of $c_\lambda$ on spaces $V^$ 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 {{Fulton-Harris
* Bruce E. Sagan. ''The Symmetric Group''. Springer, 2001.

Representation theory of finite groups
Symmetric functions
Permutations