Consider the ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a module over the ring of symmetric polynomials $\Lambda_{\mathbb{k}}$. Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ a free $\Lambda_{\mathbb{k}}$-module?
Can you write down "good" generators explicitly? (I think that it has to be something very classical in representation theory).
Comment:
My initial question was whether this module flat. But since all flat Noetherian modules over polynomial ring are free (correct me if it is wrong), it is the same question.
There is much more general question, which seems unlikely to have good answer. Let $G$ be a finite group and $V$ finite dimensional representation of G. Consider projection $p: V \rightarrow V/G$. When is $p$ flat?
Best Answer
Yes, that's indeed a classical result of representation theory: ${\mathbb Z}[x_1,...,x_n]$ is graded free over ${\Lambda}_n$ of rank $n!$ (the graded rank is the quantum factorial $[n]_q!$), and a basis is given by Schubert polynomials defined in terms of divided difference operators.
See for example the original article of Demazure, in particular Theorem 6.2.
Passing to the quotient, one obtains the graded ring ${\mathbb Z}[x_1,...,x_n]/\langle\Lambda_n^+\rangle$ which is isomorphic to the integral cohomology ring of the flag variety of ${\mathbb C}^n$, and the ${\mathbb Z}$-basis of Schubert polynomials coincides with the basis of fundamental classes of Bruhat cells in the flag variety. This is explained in Fulton's book 'Young Tableaux', Section 10.4.