2009/11/14 by Will Traves · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Differential operator #Differential (mechanical device) #Mathematics #Computer science #Pure mathematics #Physics
paper · doi:10.1007/978-0-8176-4875-6_10
openalex publication_date 2009/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Following Weyl’s account in The Classical Groups we develop an analogue of the (first and second) Fundamental Theorems of Invariant Theory for rings of differential operators: when V is a k-dimensional complex vector space with the standard SL kC action, we give a presentation of the ring of invariant differential operators D(C[V n])SLkC and a description of the ring of differential operators on the G.I.T. quotient, D(C[V n]SLkC), which is the ring of differential operators on the (affine cone over the) Grassmann variety of k-planes in NumberingDepth="0"-dimensional space. We also compute the Hilbert series of the associated graded rings GrD(C[V n])SLk and Gr(D(C[V n]SLkC)). This computation shows that earlier claims that the kernel of themap from D(C[V n])SLkC to D(C[V n]SLkC) is generated by the Casimir operator are incorrect. Something can be gleaned from these earlier incorrect computations though: the kernel meets the universal enveloping algebra of slkC precisely in the central elements of U(slkC).