  The Witt algebra W_n is the Lie algebra of all derivations of the n-variable
polynomial ring V_n=C[x_1, ..., x_n] (or of algebraic vector fields on A^n). A
representation of W_n is polynomial if it arises as a subquotient of a sum of
tensor powers of V_n. Our main theorems assert that finitely generated
polynomial representations of 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 Fin^op, where Fin is
the category of finite sets. We also show that polynomial representations of
W_n are equivalent to polynomial representations of the endomorphism monoid of
A^n. These equivalences are a special case of an operadic version of
Schur--Weyl duality, which we establish.
