Symmetric Ideals, Invariant Hilbert Schemes and Stabilization
- Andreas Kretschmer
Abstract
Symmetric ideals are ideals in a polynomial ring which are stable under all permutations of the variables. They appear at prominent places in algebraic combinatorics and asymptotic commutative algebra. A global study of zero-dimensional symmetric ideals is the study of the invariant Hilbert scheme for the action of the n-th symmetric group on affine n-space. In this talk I will introduce invariant Hilbert schemes in this setting and explain our irreducibility and smoothness results for certain cases. We have also classified all homogeneous symmetric ideals giving rise to a representation of dimension at most 2n and we can decide which of them define singular points of the invariant Hilbert scheme. I will end with several open questions. This is joint work with Sebastian Debus.