Abstract: Given an ideal in a polynomial ring over a field, the Betti numbers count the ranks of the free modules in a minimal free resolution of the ideal. These numbers form an important collection of invariants from which other algebraic and geometric invariants can be readily obtained. In addition to the Betti numbers, the structure of a minimal free resolution is also of interest, including any inherent symmetry due to the presence of a group action. The motivating example for this talk are the ideals of so-called 'star configurations' and their symbolic powers. By work of Geramita, Harbourne, Migliore, and Nagel, the study of these ideals reduces to certain monomial ideals. In recent joint work with Biermann, De Alba, Murai, Nagel, O'Keefe, Römer, and Seceleanu, we introduce a larger class of monomial ideals, that we call 'symmetric shifted ideals', and provide a description of the Betti numbers as well as the free modules in a minimal free resolution as representations of the symmetric group.
427 Thackeray Hall