On the Structure of Nilpotent Normal Form Modules

The set of vector fields that are in normal form with respect to a given linear part has the structure of a module and is best described by giving the Stanley decomposition of that module. An algorithm is presented that produces a Stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field, given a Stanley decomposition for the ring of invariants. This reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm. Both the inner product normal form (Elphick) and the sl(2) normal form (Cushman–Sanders) are covered within a single theory, and simplified (non-equivariant) versions of each normal form are presented.