Narrow Results

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an ${\mathrm{\text{SU}}}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at $t=1$. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most $10$. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary ${\mathrm{\text{SL}}}_2$- or ${\mathrm{\text{SU}}}_2$-module as well as its first three Laurent coefficients.

We consider the generalization of Laplace invariants to linear differential systems of arbitrary rank and dimension. We discuss completeness of certain subsets of invariants.