# 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:

$P_\lambda=\{ g\in S_n : g \text{ preserves each row of } \lambda \}$

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

$b_\lambda=\sum_{g\in Q_\lambda} \sgn(g) e_g$

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

$c_\lambda := a_\lambda b_\lambda = \sum_{g\in P_\lambda,h\in Q_\lambda} \sgn(h) e_{gh}$

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 $\mathbb{C}S_n \rightarrow \text{End} (V^{\otimes n})$ 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

$\text{Im}(a_\lambda) := a_\lambda V^{\otimes n} \cong \text{Sym}^{\lambda_1}\; V \otimes \text{Sym}^{\lambda_2}\; V \otimes \cdots \otimes \text{Sym}^{\lambda_j}\; V.$

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 $\text{Sym}^2\; V\otimes \text{Sym}^2\; V$.

The image of $b_\lambda$ is

$\text{Im}(b_\lambda) \cong \bigwedge^{\mu_1} V \otimes \bigwedge^{\mu_2} V \otimes \cdots \otimes \bigwedge^{\mu_k} V$

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

The image $\mathbb{C}S_nc_\lambda$ of $c_\lambda = a_\lambda \cdot b_\lambda$ in $\mathbb{C}S_n$ is an irreducible representation[1] of Sn, called a Specht module. We write

$\text{Im}(c_\lambda) = V_\lambda$

for the irreducible representation.

Some scalar multiple of $c_\lambda$ is idempotent, that is $c^2_\lambda = \alpha_\lambda c_\lambda$ for some rational number $\alpha_\lambda\in\mathbb{Q}$. Specifically, one finds $\alpha_\lambda=n! / \text{dim } V_\lambda$. 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).