Representation theory of the symmetric group

Remarks

Example: the sign map $\sgn \colon S_n \to \set{\pm 1} \subseteq GL_1(\C)$ is a representation of degree $1$, because $\sgn(\sigma\tau) = \sgn(\sigma)\sgn(\tau)$. Non-example: let $\rho \colon S_3 \to GL_1(\C)$ send $1$ to $1$, each transposition to $-1$ and each $3$-cycle to $\omega = e^{2\pi i/3}$. It is not a representation: the product of the transpositions $(1\,2)$ and $(2\,3)$ is a $3$-cycle, while $(-1)(-1) = 1 \ne \omega$. Over $\C$, the number of irreducible representations of $S_n$ is the number of conjugacy classes of $S_n$. Conjugacy classes correspond to cycle types, so this is the number $p(n)$ of partitions of $n$. The Specht modules $S^\lambda$, for $\lambda$ a partition of $n$, are a complete list of irreducible complex representations, without repetition. Over a field of characteristic $p$ with $p \le n$, the theory is the modular representation theory of $S_n$, and is much less understood.