The ring $S=\mathbb{C}[x\_1,x\_2,\dots,x\_n]^{S\_n}$ of symmetric polynomials has a number of commonly used bases, but the undisputed world champion of these is the basis consisting of [Schur polynomials](http://en.wikipedia.org/wiki/Schur_polynomial) $s\_\lambda$, where $\lambda$ ranges over non-increasing sequences $\lambda\_1 \geq \lambda\_2 \geq \cdots \geq \lambda\_n \geq 0$ of non-negative integers. For a partition $\mu$ of $n$, let $V\_\mu$ be the corresponding irreducible $S\_n$-module, and let $M(\mu)=(\mathbb{C}[x\_1,x\_2,\dots,x\_n] \otimes V\_\mu)^{S\_n}$ be the ($S$-module of) $S\_n$-invariant polynomial functions on $\mathbb{C}^n$ with values in $V\_\mu$. Is there a $\mathbb{C}$-basis of $M(\mu)$ that deserves top billing?


(A bit of background: the dimensions of the homogeneous components of $M(\mu)$ can be computed from the *exponents* of $V\_\mu$, that is, the degrees in which it appears in the coinvariant algebra. There are combinatorial expressions known for these numbers---see e.g. Stembridge's paper "On the eigenvalues of representations of reflection groups and wreath
products", Pacific J. Math. 140 (1989), 353--396 and the references therein, but they are not obtained by writing down a particularly nice basis.)



Canonical bases for modules over the ring of symmetric polynomials
In my paper Cyclage, catabolism, and the affine Hecke algebra 
<http://arxiv.org/abs/1001.1569>
I exhibit a canonical basis for $\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ and more generally a canonical basis for $\mathbb{C}[x\_1,x\_2,\dots,x\_n] \otimes V\_\mu$ coming from the extended affine Hecke algebra of type A. The subset of this canonical basis corresponding to cells of shape $(n)$ is a basis for the $S\_n$-invariants in this module. For the special case $M(n)$, these canonical basis elements do correspond to Schur functions -- they are Schur functions in the Bernstein generators times $e^+$, where $e^+$ spans the trivial representation for the finite Hecke algebra (see Theorem 6.1).


I have not thought too much about the combinatorics of the canonical basis for $M(\mu)$, but it may be possible to work this out explicitly, including an explicit description of the $S\_n$ invariants (see Example 9.21 for a little bit about this). This paper is long and may take some time to get through. Feel free to contact me at the email address at the very bottom of the paper if you have any questions or want to discuss this in detail.


