Time: 9h00, Thursday, October 17, 2019,
Location: Room 301, Building A5, Institute of Mathematics
Abstract: Chains of ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups arise naturally in various fields of mathematics, including algebraic statistics and representation theory. In this talk, I will discuss the asymptotic behavior of some invariants along such chains, namely, the Krull dimension, the projective dimension, and the Castelnuovo-Mumford regularity. The Krull dimension is eventually a linear function whose slope can be described explicitly. We conjecture that the projective dimension and the Castelnuovo-Mumford regularity also grow eventually linearly, and provide linear bounds for these invariants. If time permits, I will also discuss a combinatorial minimization problem that is inspired by the classical Kruskal-Katona theorem and the study of chains of symmetric ideals. This is joint work with U we Nagel, Hop D. Nguyen, and Tim Römer