# 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 $V^{\otimes n}$ obtained from the action of $S_{n}$ on $V^{\otimes n}$ 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, define two permutation subgroups $P_{\lambda }$ and $Q_{\lambda }$ of Sn as follows:[clarification needed]

$P_{\lambda }=\{g\in S_{n}:g{\text{ preserves each row of }}\lambda \}$ and

$Q_{\lambda }=\{g\in S_{n}:g{\text{ preserves each column of }}\lambda \}.$ Corresponding to these two subgroups, define two vectors in the group algebra $\mathbb {C} S_{n}$ as

$a_{\lambda }=\sum _{g\in P_{\lambda }}e_{g}$ and

$b_{\lambda }=\sum _{g\in Q_{\lambda }}\operatorname {sgn}(g)e_{g}$ where $e_{g}$ is the unit vector corresponding to g, and $\operatorname {sgn}(g)$ is the sign of the permutation. The product

$c_{\lambda }:=a_{\lambda }b_{\lambda }=\sum _{g\in P_{\lambda },h\in Q_{\lambda }}\operatorname {sgn}(h)e_{gh}$ 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 $V^{\otimes n}=V\otimes V\otimes \cdots \otimes V$ (n times). Let Sn act on this tensor product space by permuting the indices. One then has a natural group algebra representation $\mathbb {C} S_{n}\to \operatorname {End} (V^{\otimes n})$ on $V^{\otimes n}$ .

Given a partition λ of n, so that $n=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{j}$ , then the image of $a_{\lambda }$ is

$\operatorname {Im} (a_{\lambda }):=a_{\lambda }V^{\otimes n}\cong \operatorname {Sym} ^{\lambda _{1}}V\otimes \operatorname {Sym} ^{\lambda _{2}}V\otimes \cdots \otimes \operatorname {Sym} ^{\lambda _{j}}V.$ For instance, if $n=4$ , and $\lambda =(2,2)$ , with the canonical Young tableau $\{\{1,2\},\{3,4\}\}$ . Then the corresponding $a_{\lambda }$ is given by

$a_{\lambda }=e_{\text{id}}+e_{(1,2)}+e_{(3,4)}+e_{(1,2)(3,4)}.$ Let an element in $V^{\otimes 4}$ be given by $v_{1,2,3,4}:=v_{1}\otimes v_{2}\otimes v_{3}\otimes v_{4}$ . Then

$a_{\lambda }v_{1,2,3,4}=v_{1,2,3,4}+v_{2,1,3,4}+v_{1,2,4,3}+v_{2,1,4,3}=(v_{1}\otimes v_{2}+v_{2}\otimes v_{1})\otimes (v_{3}\otimes v_{4}+v_{4}\otimes v_{3}).$ The latter clearly span $\operatorname {Sym} ^{2}V\otimes \operatorname {Sym} ^{2}V.$ The image of $b_{\lambda }$ is

$\operatorname {Im} (b_{\lambda })\cong \bigwedge ^{\mu _{1}}V\otimes \bigwedge ^{\mu _{2}}V\otimes \cdots \otimes \bigwedge ^{\mu _{k}}V$ where μ is the conjugate partition to λ. Here, $\operatorname {Sym} ^{i}V$ and $\bigwedge ^{j}V$ are the symmetric and alternating tensor product spaces.

The image $\mathbb {C} S_{n}c_{\lambda }$ of $c_{\lambda }=a_{\lambda }\cdot b_{\lambda }$ in $\mathbb {C} S_{n}$ is an irreducible representation of Sn, called a Specht module. We write

$\operatorname {Im} (c_{\lambda })=V_{\lambda }$ for the irreducible representation.

Some scalar multiple of $c_{\lambda }$ is idempotent, that is $c_{\lambda }^{2}=\alpha _{\lambda }c_{\lambda }$ for some rational number $\alpha _{\lambda }\in \mathbb {Q} .$ 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 $\mathbb {Q} S_{n}$ .

Consider, for example, S3 and the partition (2,1). Then one has

$c_{(2,1)}=e_{123}+e_{213}-e_{321}-e_{312}.$ If V is a complex vector space, then the images of $c_{\lambda }$ on spaces $V^{\otimes d}$ provides essentially all the finite-dimensional irreducible representations of GL(V).