Invariant Theory and Differential Operators

Constructive invariant theory was a preoccupation of many nineteenth century mathematicians, but the topic fell out of fashion in the early twentieth century. In the latter twentieth century the topic enjoyed a resurgence, partly due to its connections with the construction of moduli spaces in algebraic geometry and partly due to the development of computational algorithms suitable for implementation in modern symbolic computation packages. In this survey paper we briefly discuss some of the history and applications of invariant theory and apply one particular algorithm that uses Gröbner bases to find invariants of linearly reductive algebraic groups acting on the Weyl algebra. After showing how we can present the ring of invariant differential operators in terms of generators and relations, we turn to the operators on the invariant ring itself. The theory is particularly nice for finite groups acting on polynomial rings, but we also compute an example involving an SL2C-action. In this example, we give a complete description of the generators and relations of D(G(2, 4)), the ring of differential