10/6/2004

Allen Knutson
Berkeley

The algebraic geometry of vertex decompositions and Young tableaux

One can get an inductive handle on simplicial complexes by separating the complex into those faces not using a certain vertex (picture a ball of ice cream) and those that do (picture the solid cone, attached to the lower hemisphere of the ice cream). We'll observe that an analogous decomposition can be performed on algebraic varieties, once they've been degenerated a little bit. This gives a method to prove some complexes to be homeomorphic to balls, or that some algebraic varieties are Cohen-Macaulay, respectively. I'll explore a particular case of these "geometric vertex decompositions", in which the varieties are "vexillary matrix Schubert varieties". These are already familiar in the study of degeneracy loci in singularities of mappings. This forces upon us a new structure on the century-old Young tableaux: they naturally index the facets of a simplicial ball. This work is joint with Ezra Miller and Alex Yong.