Invariants of group representations, dimension/degree duality and normal forms of vector fields

We develop an analytic approach to the problem of polynomial first integrals for linear vector fields. As an application we obtain a new proof of the theorem of Maurer and Wietzenböck about finiteness of the number of generators of the ring of constants of a linear derivation in the polynomial ring.

In the case of linear nilpotent vector field X with one Jordan cell we deal with an irreducible representation Sym${}^{n}V$, $V=C^2$, of the Lie algebra $\mathrm{sl}\left(2\right)$. The homogeneous polynomial first integrals of X of degree d correspond to highest weight vectors in the representation Sym${}^d\left({\mathrm{Sym}}^{n}V\right)$. We present a generating function for the multiplicities of the splitting of the latter representation into irreducible ones.

The dim/deg duality is an isomorphism Sym${}^d\left({\mathrm{Sym}}^{n}V\right)\simeq$ Sym${}^n\left({\mathrm{Sym}}^{d}V\right)$. We give a functional analytic construction of the duality map.

Using a transvectant formula we obtain a new relation for the elementary symmetric polynomials.

Finally, we propose an alternative approach to the analyticity property of the normal form reduction of a germ of vector field with nilpotent linear part in a case considered by Stolovich and Verstringe.