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 S_n and specific Young tableau λ corresponding to a numbered partition of n, 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 }}}\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 S_n 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 ${\mathbb {C}}S_{n}c_{\lambda }$ of $c_{\lambda }=a_{\lambda }\cdot b_{\lambda }$ in ${\mathbb {C}}S_{n}$ is an irreducible representation of S_n, called a Specht module. We write

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

for the irreducible representation.

Some scalar multiple of $c_{\lambda }$ is idempotent,^[1] that is $c_{\lambda }^{2}=\alpha _{\lambda }c_{\lambda }$ 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).

Notes

↑ See (Fulton & Harris 1991, Theorem 4.3, p. 46)

References

William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Lecture 4 of Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249, ISBN 978-0-387-97527-6.
Bruce E. Sagan. The Symmetric Group. Springer, 2001.

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Young symmetrizer

Definition

Construction

See also

Notes

References