Reflection arrangements, syzygies, and the containment problem

Abstract: Invariant theory, that is the art of finding polynomials invariant under the action of a given group, has played a major role in the historical development of commutative algebra. In this theory reflection groups are singled out for having rings of invariants that are isomorphic to polynomial rings. From a geometric perspective, reflection groups give rise to beautiful and very symmetric arrangements of hyperplanes termed reflection arrangements.

This talk will take a close look at the ideals defining the singular loci of reflection arrangements, which are in turn symmetric subspace arrangements. We describe their syzygies in terms of invariant polynomials for the relevant reflection groups. We leverage this information to settle many aspects of the containment problem asking for containments between the ordinary and the symbolic powers of the ideals in this family. This talk is based on joint work with Ben Drabkin.

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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