Stanley EC2 Representation Theory Reference

Question

I was reading Stanley's Enumerative Combinatorics Volume 2, and when he starts to go into the representation theory of the symmetric group (Chapter 7.18) he states that he assumes basic knowledge of representation theory of finite groups. Is there a good reference that focuses on the theory which would be needed for this?

If possible I'd love for this post to serve as sort of a syllabus for someone who is reading EC2 and wants to read 7.18 with no previous background in representation theory.

Answer 1

To complement the other answer and the references therein, here I'll give a (not so) brief summary of everything you need to understand the (characteristic $0$) representation theory of the symmetric groups. For a certain type of representation theory of a Lie-theoretic flavor, the techniques and ideas involved here are indispensable, and this might be thought of as the first in a long line of examples of increasing complexity and interest where we have succeeded in attaining as complete an understanding as we have a right to expect (essentially the same ideas may be used, for instance, to compute the character tables of the general linear groups over finite fields).

The following statements are the facts, enumerated in such a way that treating them as exercises would be a not-unreasonable enterprise for a student reading Stanley's book.

1. The first basic fact to understand is that for any finite group $G$, the group ring $\mathbf{C} G$ of $G$ over the complex numbers is semi-simple: this can be proved by observing that each $\mathbf{C} G$-module $V$ admits a positive-definite $G$-invariant Hermitian form, which can be obtained by starting with any positive definite Hermitian form on $V$ and averaging over the group $G$ (this goes through in the same way for any compact topological group). This implies that the group ring $\mathbf{C} G$ is isomorphic to a product of matrix rings $\mathrm{Mat}_{d_V}(\mathbf{C})$, where $V$ ranges over the irreducibe $\mathbf{C}G$-modules and $d_V=\mathrm{dim}(V)$. In particular, the dimension of the center of the group ring $\mathbf{C} G$ is equal to the number of irreducible $\mathbf{C} G$-modules, implying that the number of irreducible $\mathbf{C} G$-modules is equal to the number of conjugacy classes in $G$.

2. One now attempts to capture information about representations via their characters: the character of a $\mathbf{C} G$-module $V$ is the function $\chi_V$ on $G$ defined by $\chi_V(g)=\mathrm{trace}(g,V)$ for all $g \in G$ (that is, it is the trace of the operator given by the action of $g$ on $V$). Now given $\mathbf{C} G$-modules $U$ and $V$, we obtain new $\mathbf{C}G$-modules $U \oplus V$, $U \otimes_\mathbf{C} V$, and $\mathrm{Hom}_\mathbf{C}(U,V)$ via the formulas $$g \cdot (u+v)=g\cdot u + g \cdot v, \ g \cdot (u \otimes v)=g \cdot u \otimes g \cdot v, \ \text{and} \ (g \cdot \phi)(u)=g \cdot \phi(g^{-1} \cdot u).$$ Via direct calculation with bases $$\chi_{U \oplus V}(g)=\chi_U(g)+\chi_V(g), \ \chi_{U \otimes V}(g)=\chi_U(g) \chi_V(g), \quad \text{and} \ \chi_{\mathrm{Hom}(U,V)}=\overline{\chi_U(g)} \chi_V(g).$$ Each character is a class function: a class function on $G$ is a function $c:G \to \mathbf{C}$ such that $c(h g h^{-1})=c(g)$ for all $g, h \in G$. Evidently the dimension of the space of class functions is equal to the number of conjugacy classes in $G$ (which we have seen is the same as the number of irreducible $\mathbf{C} G$-modules). We will prove below that the characters of the irreducible $\mathbf{C} G$-modules are linearly independent, and therefore are a basis of the space of class functions.

3. If $V$ a a $\mathbf{C} G$-module, the operator $$\pi(v)=\frac{1}{|G|} \sum_{g \in G} g \cdot v$$ is a projection onto the subspace $V^G$ of $G$-fixed points in $V$. Its trace is therefore the dimension of $V^G$: $$\mathrm{dim}(V^G)=\mathrm{trace}(\pi,V)=\frac{1}{|G|} \sum_{g \in G} \chi_V(g).$$ Now observing that $\mathrm{Hom}_{\mathbf{C} G}(U,V)=\mathrm{Hom}_\mathbf{C}(U,V)^G$ one obtains $$\mathrm{dim}(\mathrm{Hom}_{\mathbf{C} G}(U,V))=\frac{1}{|G|} \sum_{g \in G} \overline{\chi_U(g)} \chi_V(g).$$

4. Now observe that for functions $\chi, \psi:G \to \mathbf{C}$ the formula $$(\chi,\psi)=\frac{1}{|G|} \sum_{g \in G} \overline{\chi(g)} \psi(g)$$ defines a positive-definite Hermitian form. Moreover, for irreducible $\mathbf{C} G$-modules $U$ and $V$ we obtain $$(\chi_U,\chi_V)=\mathrm{dim}(\mathrm{Hom}_{\mathbf{C} G}(U,V))=\begin{cases} 1 \quad \hbox{if $U \cong V$, and} \\ 0 \quad \text{else.} \end{cases}$$ Thus the set of characters of a complete set of non-isomorphic irreducible $\mathbf{C} G$-modules is an orthonormal basis of the space of class functions on $G$ (this fact is sometimes called the orthogonality relations for characters).

5. Let $\mathrm{Rep}(G)$ be the subring of the $\mathbf{C}$-algebra of class functions on $G$ generated by the characters $\chi_V$ of the irreducible $\mathbf{C} G$-modules (that this really is the representation ring of $G$ follows from semi-simplicity and the calculations in point 2. above, but we do not need this fact nor the proper definition of the representation ring). This is a free $\mathbf{Z}$-module with orthonormal basis given by the characters of the irreducible $\mathbf{C} G$-modules. In particular, writing simply $\mathbf{C} \mathrm{Rep}(G)$ for its tensor product with $\mathbf{C}$, we have $\mathbf{C} \mathrm{Rep}(G)$ equal to the $\mathbf{C}$-algebra of $\mathbf{C}$-valued class functions on $G$. Now suppose we have any orthonormal subset $S \subseteq \mathrm{Rep}(G)$ with the three properties: a) The cardinality of $S$ is the number of conjugacy classes in $G$, b) $S$ is an orthonormal set, and c) for all $s \in S$, $s(1)>0$. From a) and b) it then follows that $S$ is an orthonormal basis of $\mathrm{Rep}(G)$ and then c) implies that $S$ is equal to the set of characters of the irreducible $\mathbf{C}G$-modules. This observation is the key to the link between the ring of symmetric functions (and various deformations and generalizations involving orthogonal sets of special functions) and the representation theory of the symmetric groups (and various deformations and generalizations).

6. We now study the case $G=S_n$ in more detail. The conjugacy classes in $S_n$ are determined by cycle structure, and are thus labeled by integer partitions $\lambda$ of $n$ in the usual (obvious!) fashion, and we will write $\delta_\lambda$ for the function on $G$ defined by $$\delta_\lambda(w)=\begin{cases} 1 \quad \hbox{if $w$ has cycle structure given by $\lambda$, and} \\ 0 \quad \text{else.} \end{cases}$$ The $\delta_\lambda$'s are an orthogonal basis of the $\mathbf{C}$-algebra of class functions on $S_n$: a direct calculation gives $$(\delta_\lambda,\delta_\mu)=\begin{cases} z_\lambda^{-1} \quad \hbox{if $\lambda=\mu$, and} \\ 0 \quad \text{else,} \end{cases}$$ where $$z_\lambda=\prod_{i=1}^\infty i^{m_i} m_i!$$ is the order of the centralizer of an element $w \in S_n$ with cycle type $\lambda$, with $m_i$ equal to the number of parts of $\lambda$ equal to $i$. The representation ring $\mathrm{Rep}(S_n)$ is a certain subring of the $\mathbf{Z}$-span of these $\delta_\lambda$'s, but it is not yet obvious how to identify it nor the subset of characters of irreducible representations. To do this requires a new idea: we should put together all the rings $\mathrm{Rep}(S_n)$ and study them simultaneously. To do this we need induction of class functions (and Frobenius reciprocity).

7. Let $H \subseteq G$ be a subgroup of a finite group $G$, and let $c:H \to \mathbf{C}$ be a class function on $H$. Choose left coset reps $g_1,\dots,g_m$ for $H$ in $G$, with $G=\coprod_{i=1}^m g_i H.$ We put $$\mathrm{Ind}_H^G(c)(g)=\sum_{i, \ g_i^{-1} g g_i \in H} c(g_i^{-1} g g_i).$$ This is a class function on $G$ which does not depend on our choice of coset reps $g_i$. The fundamental fact is now Frobenius reciprocity $$(\mathrm{Ind}_H^G (c),d)=(c,\mathrm{Res}^G_H(d)),$$ for all class functions $d$ on $G$, which you can prove by direct calculation (we are using self-explanatory notation for the restriction of functions on $G$ to functions on $H$).

8. We'll now abbreviate $R_n=\mathrm{Rep}(S_n)$ and put $R=\bigoplus_{n=0}^\infty R_n$. In fact, the ring structure on $R_n$, while very interesting (Kronecker products!), is irrelevant for the remainder of this story. We'll put another ring structure on $R$ that is more useful for our purposes, and in particular which allows us to construct a $\mathbf{Z}$-basis of $R$ inductively. Namely, let $\chi \in R_m$ and $\psi \in R_n$. Then the formula $$(\chi \times \psi)(v,w)=\chi(v) \psi(w) \quad \hbox{for $v \in S_m$ and $w \in S_n$}$$ defines a class function on $S_m \times S_n$ (if $\chi=\chi_U$ and $\psi=\chi_V$ then $\chi \times \psi=\chi_{U \otimes V}$) which we may induce to $S_{m+n}$ to obtain a class function $\chi \cdot \psi \in R_{m+n}$. This gives a commutative and associative algebra structure on $R$. We define $\eta_n \in R_n$ by $$\eta_n(w)=1 \quad \hbox{for all $w \in S_n$}$$ (this is the character of the trivial representation of $S_n$) and for any partition $\lambda$ of $n$ we put $\eta_\lambda=\prod \eta_i^{m_i}$ (where as usual $m_i$ is the number of occurrences of $i$ in $\lambda$). We have constructed particular elements of $R$ that will turn out (see below) to give a $\mathbf{Z}$-basis corresponding to the homogeneous symmetric functions.

9. The ring of symmetric functions. Since you are reading Stanley I'm not going to say anything about this ring or its properties, except the facts we need: it has a positive-definite inner product $(\cdot,\cdot)$ with respect to which the Schur functions $s_\lambda$ (where $\lambda$ ranges over all integer partitions) are an orthonormal $\mathbf{Z}$-basis. If we tensor the ring of symmetric functions by $\mathbf{C}$ (though any field containing $\mathbf{Q}$ would work) then the power sum symmetric functions $p_\lambda$ are an orthogonal basis, with square norm given by $$(p_\lambda,p_\lambda)=z_\lambda.$$ Thus the map sending $\delta_\lambda$ to $z_\lambda^{-1} p_\lambda$ is an isometry from $\mathbf{C} R$ to $\mathbf{C} \Lambda$. Using Frobenius reciprocity one checks that it is a $\mathbf{C}$-algebra isomorphism sending $\eta_\lambda$ to $h_\lambda$. It follows from the determinant formula $s_\lambda=\mathrm{det}(h_{\lambda_i-i+j})$ that the Schur functions are in the image of $R$ under this isomorphism, and hence they correspond to characters $\chi_\lambda \in R$. Then one verifies that $\chi_\lambda(1)>0$ using a bit of combinatorics, and you're done.

Answer 2

After digging out my copy of Stanley, it seems that the necessary background really is just the basics. So some recommendations would be:
-James and Liebeck: Representations and Characters of Groups (the most comprehensive and general one specifically for finite groups as needed here)
-Sagan: The Symmetric Group - Representations, Combinatorial Algorithms, and Symmetric Functions (this does include the basics, but proceeds faster. It will also have a lot of overlap with the contents of Stanley, as indicated by the title)
-Fulton and Harris: Representation Theory - A First Course (somewhat more general the James & Liebeck)

Finally, for a very quick introduction to the main concepts, there is a set of course notes I wrote some years ago https://pure.au.dk/ws/files/120581284/intro_to_character_theory.pdf (unfortunately, that is the first version without a bunch of improvements I made later, and I don't have a good way to put the newer version online. It is missing some stuff compared to what Stanley assumes, most notably induced representations, but it gets through a lot of the basics fairly quickly, and does not assume a ton of algebra background. If you or anyone else is interested in the revised version, just shoot me an email and I can send a copy).