Larry Smith (Georg-August-Universität Göttingen)
larry@sunrise.uni-math.gwdg.de
Joint work with Dagmar M. Meyer.
In 1964 Robert Steinberg proved the following result: Let ρ : G → GL(n, C) be a complex representation of a finite group G. Then ρ(G) is generated by pseudoreflections if and only if the ring of coinvariants is a Poincaré duality algebra.
Well almost: Steinberg does not use the term Poincaré duality algebra nor coinvariants, but in modern language that is what he proved. His proof makes use of an existence theorem for meromorphic solutions of partial differential equations with holomorphic coefficients in a central way, so sheds no light on what happens in characteristic other than zero.
It turns out that there is an ideal reformulation of Steinberg's result, namely the ideal in C[V] generated by all the G-invariant forms of strictly positive degree is irreducible if and only if ρ(G) is generated by pseudoreflections. This provides us with the start of our journey through Macaulay's theory of irreducible ideals up to the Hit Problem for rings of invariants.