Polynomial Representations of the Witt Lie Algebra

The Witt algebra ${\mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$ (or of algebraic vector fields on $\textbf{A}^{n}$). A representation of ${\mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $\textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${\mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $\textbf{Fin}^{\textrm{op}}$, where $\textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${\mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $\textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.