Decomposition of polynomial ring as $S_n$-module I want to whether there is a containment relation between the $S_n$-modules $\mathbb{C}S_n$ and $\mathbb{C}[x_1,\ldots ,x_n]$. Is it true that $\mathbb{C}[x_1,\ldots ,x_n]$ contains an isomorphic copy of $\mathbb{C}S_n$? In other words, suppose $\mathbb{C}[x_1,\ldots ,x_n]$ decomposes into irreducible modules as $\bigoplus_{\lambda\vdash n }r_\lambda V_\lambda$. Then what can we say about $r_\lambda$? Is $r_\lambda\geq \dim V_\lambda$?
EDIT: As  Jeremy Rickard has pointed out that the above claim is true and $r_\lambda \geq \dim V_\lambda $.
My next question: What can be said about $r_\lambda$? How big are they compared to $\dim V_\lambda$?
We know that $r_\lambda =\dim \text{Hom}(V_\lambda, \mathbb{C}[x_1,\ldots ,x_n]) $ and $V_\lambda$ is the space spanned by the polynomials $F_T$, where $F_T=\prod_{i<j} (x_i-x_j)$ where product is over $i,j$ in the same coloumn of the tableau $T$. Can a basis be constructed from this description?
 A: This is sniffing around a very big area, so it's hard to know exactly what to say.  The $r_\lambda$'s are sort of the wrong question: they are always $\infty$.  It's much better to look at the multiplicity space as a module over the symmetric polynomials.  The remarkable fact is that it's a free module and its rank is the same as $\dim V_\lambda$. This is a special case about a much more general fact about Coxeter groups, acting in their reflection representation: you always get that $\mathbb{C}[V]\cong \mathbb{C}[G]\otimes \mathbb{C}[V]^{G}$.  
You'll easily find the statement that the invariants are a polynomial ring (this is a special case of the Chevalley-Shepard-Todd theorem), but this is equivalent to the freeness of $\mathbb{C}[V]$, and once you know that's free, you know that the multiplicities of $G$-reps in the quotient by any maximal ideal in the invariants gives the ranks of the multiplicity spaces. A generic maximal ideal in the invariants is the vanishing ideal of a free orbit, so the quotient by it is a regular representation.
