The Unapologetic Mathematician

Mathematics for the interested outsider

The Symmetrizer and Antisymmetrizer

Today we’ll introduce two elements of the group algebra \mathbb{F}[S_n] of the symmetric group S_n which have some interesting properties. Each one, given a representation of the symmetric group, will give us a subrepresentation of that representation.

The symmetrizer gets its name from the way that it takes an arbitrary vector and turns it into one that the symmetric group will act trivially on. Specifically, we use the element

\displaystyle S=\frac{1}{n!}\sum\limits_{\pi\in S_n}\pi

That is, we take all n! permutations in the symmetric group and — in a sense — average them out. If we compose this with any permutation \hat{\pi} we find

\displaystyle\begin{aligned}\hat{\pi}S=\hat{\pi}\frac{1}{n!}\sum\limits_{\pi\in S_n}\pi\\=\frac{1}{n!}\sum\limits_{\pi\in S_n}\hat{\pi}\pi\end{aligned}

But as \pi runs over all the permutations in the group, \hat{\pi}\pi does as well. So this is just the symmetrizer S back again. The upshot is that if we have a representation \rho:S_n\rightarrow\mathrm{GL}(V) we find that

\displaystyle\begin{aligned}\left[\rho(\hat{\pi})\right]\left(\left[\rho(S)\right](v)\right)=\left[\rho(\hat{\pi}S)\right](v)\\=\left[\rho(S)\right](v)\end{aligned}

Thus every vector in the image of \rho(S) is left unchanged by the action of any permutation. That is, \mathrm{Im}\left(\rho(S)\right)\subseteq V is a subrepresentation on which S_n acts trivially.

The antisymmetrizer, on the other hand, on the other hand, will give us vectors on which the symmetric group acts by the signum representation. We use the group algebra element

\displaystyle A=\frac{1}{n!}\sum\limits_{\pi\in S_n}\mathrm{sgn}(\pi)\pi

Now if we compose this with any permutation \hat{\pi} we find

\displaystyle\begin{aligned}\hat{\pi}A=\hat{\pi}\frac{1}{n!}\sum\limits_{\pi\in S_n}\mathrm{sgn}(\pi)\pi\\=\frac{1}{n!}\sum\limits_{\pi\in S_n}\mathrm{sgn}(\pi)\hat{\pi}\pi\\=\frac{1}{n!}\sum\limits_{\pi\in S_n}\mathrm{sgn}(\hat{\pi})\mathrm{sgn}(\hat{\pi}\pi)\hat{\pi}\pi\\=\mathrm{sgn}(\hat{\pi})\frac{1}{n!}\sum\limits_{\pi\in S_n}\mathrm{sgn}(\hat{\pi}\pi)\hat{\pi}\pi\\=\mathrm{sgn}(\hat{\pi})\hat{\pi}A\end{aligned}

Now given a representation \rho:S_n\rightarrow\mathrm{GL}(V) we find that

\displaystyle\begin{aligned}\left[\rho(\hat{\pi})\right]\left(\left[\rho(A)\right](v)\right)=\left[\rho(\hat{\pi}A)\right](v)\\=\left[\rho(\mathrm{sgn}(\hat{\pi})A)\right](v)\\=\mathrm{sgn}(\hat{\pi})\left[\rho(A)\right](v)\end{aligned}

Thus every vector in the image of \rho(A) is multiplied by the signum of any permutation. That is, \mathrm{Im}\left(\rho(A)\right)\subseteq V is a subrepresentation on which S_n acts by the signum representation.

Now, one thing to be careful about: I haven’t said that the subrepresentations are nontrivial. That is, when we (anti)symmetrize a representation, the subrepresentation we get may be zero — maybe no vectors in the representation transform trivially or by the signum representation. In fact, let’s check what happens when we multiply the symmetrizer and antisymmetrizer:

\displaystyle\begin{aligned}SA=\left(\frac{1}{n!}\sum\limits_{\pi\in S_n}\pi\right)\left(\frac{1}{n!}\sum\limits_{\hat{\pi}\in S_n}\mathrm{sgn}(\hat{\pi})\hat{\pi}\right)\\=\frac{1}{n!}\frac{1}{n!}\sum\limits_{\substack{\pi\in S_n\\\hat{\pi}\in S_n}}\mathrm{sgn}(\hat{\pi})\pi\hat{\pi}\\=\frac{1}{n!}\frac{1}{n!}\sum\limits_{\substack{\pi\in S_n\\\hat{\pi}\in S_n}}\mathrm{sgn}(\pi)\mathrm{sgn}(\pi\hat{\pi})\pi\hat{\pi}\\=\frac{1}{n!}\frac{1}{n!}\left(\sum\limits_{\pi\in S_n}\mathrm{sgn}(\pi)\right)\left(\sum\limits_{\pi\hat{\pi}\in S_n}\mathrm{sgn}(\pi\hat{\pi})\pi\hat{\pi}\right)\\=\frac{1}{n!}\frac{1}{n!}\left(\sum\limits_{\pi\in S_n}\mathrm{sgn}(\pi)\right)A\\=0\end{aligned}

Where the sum comes to zero because we’re just adding up \frac{n!}{2} terms where \mathrm{sgn}(\pi)=1 and \frac{n!}{2} where \mathrm{sgn}(\pi)=-1, and so everything cancels out. That is, the symmetric part of an antisymmetrized representation is trivial. Similarly, the antisymmetric part of a symmetrized representation is trivial.

December 19, 2008 Posted by | Algebra, Representation Theory | 5 Comments

   

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