Polynomial representations of the Witt Lie algebra

Abstract

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.

25 pages

References in corpus (10)

• Gröbner methods for representations of combinatorial categories
• Representation stability and finite linear groups
• Syzygies of Segre embeddings and Delta-modules
• Bounded-rank tensors are defined in bounded degree
• Algebraic structures on cohomology of configuration spaces of manifolds with flows
• Uniformly Presented Vector Spaces
• Maps from the enveloping algebra of the positive Witt algebra to regular algebras
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• Representation stability for the Kontsevich space of stable maps
• On finite dimensionality of homology of subalgebras of vector fields