Abstract for the talk on 27.03.2019 (11:00 h)Seminar on Nonlinear Algebra
Chris McDaniel (Endicott College, Beverly)
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 RˆK˙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 RˆK˙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.