Combinatorics Seminar: "Geometric Vertex Decomposability and Toric Ideals of Graphs"

Abstract: A Gröbner degeneration is a useful method to reduce problems involving ideals to the monomial ideal setting. In the square-free case, we can associate a simplicial complex to a monomial ideal and ask whether this complex is vertex decomposable.  Another tool in this direction, called a geometric vertex decomposition, is a generalization of this technique to non-monomial ideals. It was first introduced by Knutson-Miller-Yong to study diagonal degenerations of Schubert varieties. Later results on the topic were mostly formulated in the context of Schubert geometry, until very recent work of Klein-Rajchgot, which established a connection to liaison theory. The interplay between these two theories can be used to analyze degenerations and to construct Gröbner bases. In this talk, I will introduce geometric vertex decomposition and highlight its application to toric ideals of graphs.

Host: Laura Escobar