Young symmetrizer

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={ 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 sign 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 \C S_n \to \operatorname{End} (V^{\otimes n}) on V^{\otimes n} (i.e. V^{\otimes n} is a right \C S_n module).

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) := V^{\otimes n} a_\lambda \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)}.

For any product vector v_{1,2,3,4}:=v_1 \otimes v_2 \otimes v_3 \otimes v_4 of V^{\otimes 4} we then have

: v_{1,2,3,4} a_\lambda = 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).

Thus the set of all a_\lambda v_{1,2,3,4} clearly spans \operatorname{Sym}^2 V\otimes \operatorname{Sym}^2 V and since the v_{1,2,3,4} span V^{\otimes 4} we obtain V^{\otimes 4} a_\lambda= \operatorname{Sym}^2 V \otimes \operatorname{Sym}^2 V, where we wrote informally V^{\otimes 4} a_\lambda \equiv \operatorname{Im}(a_\lambda).

Notice also how this construction can be reduced to the construction for n = 2. Let \mathbb{1} \in \operatorname{End} (V^{\otimes 2}) be the identity operator and S\in \operatorname{End} (V^{\otimes 2}) the swap operator defined by S(v\otimes w) = w \otimes v, thus \mathbb{1} = e_{\text{id}} and S = e_{(1,2)} . We have that

: e_{\text{id}} + e_{(1,2)} = \mathbb{1} + S

maps into \operatorname{Sym}^2 V, more precisely

: \frac{1}{2}(\mathbb{1} + S)

is the projector onto \operatorname{Sym}^2 V. Then

: \frac{1}{4} a_\lambda = \frac{1}{4} (e_{\text{id}} + e_{(1,2)} + e_{(3,4)} + e_{(1,2)(3,4)}) = \frac{1}{4} (\mathbb{1} \otimes \mathbb{1} + S \otimes \mathbb{1} + \mathbb{1} \otimes S + S \otimes S) = \frac{1}{2}(\mathbb{1} + S) \otimes \frac{1}{2} (\mathbb{1} + S)

which is the projector onto \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 \C S_nc_\lambda of c_\lambda = a_\lambda \cdot b_\lambda in \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^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, 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).

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.

References

1. See {{harv. Fulton. Harris. 1991