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 $s_\lambda$, 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.)