# The Unapologetic Mathematician

## Some Review

Before we push on into our new topic, let’s look back at some of the background that we’ve already covered.

We’re talking about symmetric groups, which are, of course, groups. We have various ways of writing down an element of $S_n$, including the two-line notation and the cycle notation that are covered in our earlier description of the symmetric groups. As an example, the two-line notation $\displaystyle\left\lvert\begin{matrix}1&2&3&4\\2&1&4&3\end{matrix}\right\rvert$

and the cycle notation $(1\,2)(3\,4)$ both describe the permutation $\alpha\in S_4$ that sends $1$ to $2$, $2$ back to $1$, and similarly swaps $3$ and $4$. Similarly, the two-line notation
the composition of $\displaystyle\left\lvert\begin{matrix}1&2&3&4\\4&2&1&3\end{matrix}\right\rvert$

and the cycle notation $(1\,4\,3)(2)$ or (equivalently) $(1\,4\,3)$ describe the permutation $\beta$ that cycles the elements $1$, $4$, and $3$ (in that order) and leaves $2$ untouched.

We’re specifically concerned with complex representations of these groups. That is, we want to pick some complex vector space $V$, and for each permutation $\sigma\in S_n$ we want to come up with some linear transformation $\rho(\sigma):V\to V$ for which the composition of linear transformations and the composition of permutations are “the same” in the sense that given two permutations $\sigma$ and $\tau$, the transportation corresponding to the composite $\rho(\sigma\tau)$ is equal to the composite of the corresponding transformations $\rho(\sigma)\rho(\tau)$.

We’re primarily interested in finite-dimensional representations. That is, ones for which $V$ is a finite-dimensional complex vector space. In this case, we know that we can always just assume that $V=\mathbb{C}^k$ — the space of $k$-tuples of complex numbers — and that linear transformations are described by matrices. Composition of transformations is reflected in matrix multiplication. That is, for every permutation $\sigma\in S_n$ we want to come up with an $k\times k$ matrix $\rho(\sigma)$ so that the matrix $\rho(\sigma\tau)$ corresponding to the composition of two permutations is the product $\rho(\sigma)\rho(\tau)$ of the matrices corresponding to the two permutations. I’ll be giving some more explicit examples soon.

September 8, 2010