Abstract
Multilocal higher‐order invariants, which are higher‐order invariants defined at distinct points of representation space, for the classical groups are derived in a systematic way. The basic invariants for the classical groups are the well‐known polynomial or rational invariants as derived from the Capelli identities. Higher‐order invariants are then constructed from the former ones by means of total derivatives. At each order, it appears that the invariants obtained in this way do not generate all invariants. The necessary additional invariants are constructed from the invariant polynomials on the Lie algebra of the Lie transformation groups.
Highlights
The study of invariants stands in contrast with the more structural approaches in differential geometry as, for example, formulated in the books of Kobayashi and Nomizu [13]
Because (x12 + x22 +x32) is invariant, we find the extra invariant ᏽ(x, x). We show that this happens at each level for all the classical groups
We review the basic results on invariants for the classical linear groups such as given in the classical book of Weyl on the subject [25] or more recently by Fulton and Harris [9]
Summary
The study of invariants stands in contrast with the more structural approaches in differential geometry as, for example, formulated in the books of Kobayashi and Nomizu [13]. Let γ(s) be a curve, φt(γ(s)) is the composition with the local diffeomorphism for a fixed t and the action on Jk(V ) is given by φ(tk)∗jkγ = jk(φ(γ)) This determines what is called the complete lift X(k), generator of φ(k), on Jk(V ). The complete lift of a vector field on Ᏹr on a jet bundle over a specific layer Jkα (Vα) is defined as in Definition 4.1 where the total derivative is Tα. It follows that, on a regular subset ᐃ, the rank of the sheaf of germs of C∞-invariant functions equals the codimension of the level surface of any set of generators. Using the natural prolongation of one-parameter local diffeomorphisms on the target space to the jet bundles, we find the prolongation of fundamental vector fields of a group action.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
