WORK IN PROGRESS!!! \section{Intro notes on symmetric group} \subsection{Overview} This subject has been active for well over 100 years, including significant contributions from Young, Schur, Frobenius, Brauer, Weyl, Garnir, James, Jucys, Murphy and many others (see reference page). Books on (or including) the subject include Rutherford, Robinson, Hamermesh ,... It is an easy exercise to see that conjugacy classes of $S_n$ are indexed by the set $\Lambda_n$ of integer partitions of $n$ (Young diagrams are simple but rather profound pictures of integer partitions). It then follows from general theorems that the complex irreducible representations of $S_n$ are indexed in the same way. As is usual in group theory, it is not quite so obvious how to build the representation associated to partition $\lambda$, but for example the trivial module has $\lambda = (n)$, and there is another easily constructed 1-dimensional representation -- the alternating representation -- with $\lambda = (1^n)$. One immediately has an element $e_n = \sum_{g \in G} g \; \in\Z G$ (with $G = S_n$) that generates the trivial module as a left (or right) ideal, and a similar element generating the alternating module. Both elements may be normalised as idempotents in $\Q S_n$. Young's work showed how, using the natural inclusion $S_{n-1} \rightarrow S_n$, to build from these elements a certain collection of primitive orthogonal elements $e_{\lambda}$ indexed by $\Lambda_n$. (Again these are normalisable as idempotents over $\Q$.) The left ideals $\Z S_n e_{\lambda}$ have bases indexed by the set of standard Young tableaux of shape $\lambda$. It follows that upon base change to $\Q$ these are a set of inequivalent irreducible modules. From here one has various options, such as beautiful combinatorial arguments related to the Robinson-Schensted correspondence, to show that this set of irreducibles is complete. In terms of fundamental invariants this is essentially the end of the story for the complex representation theory. However there are important related questions concerning the construction of explicit representations satisfying constraints (such as orthogonality). Young settled the main problems in this area with his `forms'. The story for representation theory over algebraically closed fields in other characteristics (or over other commutative rings) is another matter. This remains almost entirely open, despite being the focus of intense research over very many years. For this reason (and others) there are a range of contemporary approaches to the representation theory of the symmetric group that seek to relate the problem to other problems -- perhaps problems with a more accessible intrinsic structure. \subsection{Notation} Here $S$ is a set. For $i$ any suitable symbol, $S^i = S \times \{ i \}$.
(The notation $S^2 = S \times S$ is also used, but our intention will always be clear by context.) $\Gamma(S)$ is the set of loop-free undirected graphs on $S$. %\newcommand{\EE}{{\mathbf E}} $\EE(S)$ is the set of partitions on $S$ (in natural bijection with the set of equivalence relations on $S$). %$\nle$ Let $\pp: \Gamma(S) \rightarrow \EE(S)$ denote the map which takes graph $g$ to the partition into connected components. A diagram for $q \in \EE(S)$ is any graph $g$ such that $\pp(g) =q$. Let $TC: S\times S \rightarrow S\times S$ be reflexive, symmetric transitive (RST) closure. Define $* : \EE(S) \times \EE(T) \rightarrow \EE(S \cup T)$ by $\rho * \nu = TC( \rho \cup \nu )$. Define $\sigma : \Z \rightarrow \Z$ by $\sigma(i)=i+1$. Define $\sigma_- : \Z \rightarrow \Z$ by $\sigma_-(2)=1$ and $\sigma_-(i)=i$ otherwise.