# The Unapologetic Mathematician

## Some Symmetric Group Representations

Tuesday, we talked about tensor powers of the standard representation of $\mathrm{GL}(V)$. Today we’ll look at some representations of the symmetric group $S_n$ and see how they’re related.

We start by looking at the $n$th tensor power $V^{\otimes n}$. Now since the category of vector spaces is symmetric, we get a representation of the symmetric group $S_n$. Indeed, we just use the element of the symmetric group to permute the tensorands. That is, given $\sigma\in S_n$ and a pure tensor $v_1\otimes...\otimes v_n\in V^{\otimes n}$ we define the representation $\pi$ by

$\left[\pi(\sigma)\right]\left(v_1\otimes...\otimes v_n\right)=v_{\sigma(1)}\otimes...\otimes v_{\sigma(n)}$

Indeed, it’s straightforward to check that $\pi(\sigma)\pi(\tau)=\pi(\sigma\tau)$, given the convention we’ve picked for symmetric group composition.

This representation of $S_n$ on $V^{\otimes n}$ is almost trivial, so why do we care? Well, it turns out that every single one of the transformations in the representation commutes with the action of $\mathrm{GL}(V)$ on $V^{\otimes n}$! Indeed, because of the way we defined the group to act on the tensor powers by doing the exact same thing to each tensorand, we can shuffle around the tensorands and get the same result. If you’re still not convinced, write out the square that needs to commute and verify both compositions.

In fact, there’s a really beautiful theorem (that I’m not about to prove here (yet)) that says the situation is even nicer. Let’s consider specifically $V=\mathbb{C}^k$ — a finite-dimensional vector space over the complex numbers. Then the representations of (the group algebras of) $S_n$ and $\mathrm{GL}_k(\mathbb{C})$ determine subalgebras (call them $A$ and $B$, respectively) of the endomorphism algebra $\mathrm{End}\left(\left(\mathbb{C}^k\right)^{\otimes n}\right)$. And each one is the “centralizer” of the other. That is, $A$ is the subalgebra consisting of all algebra elements of $\mathrm{End}\left(\left(\mathbb{C}^k\right)^{\otimes n}\right)$ which commute with every element of $B$, and vice-versa. This situation is called “Schur-Weyl duality”, and it turns out to be fantastically useful in studying representations of both the symmetric groups and the general linear groups.

December 4, 2008 - Posted by | Algebra, Representation Theory