Invariant Coinvariant Rings, Strong Lefschetz Properties, and a Conjecture of G. Almkvist

For a finite group W acting linearly on a polynomial ring R, and any subgroup K < W, we define the invariant coinvariant ring $R^K_W$ to be the quotient of the ring of K-invariant polynomials by the ideal generated by the W-invariant polynomials. In case W is a Weyl group and K is a parabolic subgroup the invariant coinvariant ring can be identified with the cohomology ring of a smooth complex projective manifold called a Grassmannian. These cohomology rings have nice algebraic properties, e.g. Poincaré duality, strong Lefschetz, Schubert calculus, and it seems natural to ask which other group pairs K < W have invariant coinvariant rings with these properties. It turns out that if K and W are both complex reflection groups, then $R^K_W$ always satisfies Poincaré duality, whereas strong Lefschetz can fail, even if K is "parabolic" (we conjecture this does not happen in the real case). Moreover in the complex case, a working Schubert calculus seems to be lacking, even in the simplest cases. I will attempt to fill in the details of this story, with plenty of examples, and then describe a connection to the combinatorics of partitions, and a remarkable conjecture of G. Almkvist.