Betti diagrams of graphsAlgebra & Discrete Mathematics
|Speaker:||Matthew Stamps, Alto University|
|Start time:||Mon, Jun 18 2012, 2:10PM|
The emergence of Boij-Söderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently gave a surprising proof that every Betti diagram of an ideal with a linear minimal resolution arises from the Stanley-Reisner ideal of a simplicial complex. In this talk, we will show that the correspondence is bijective in the special case of ideals with 2-linear minimal resolutions. We will also prove a more general statement that every Betti diagram of a module with a 2-linear minimal resolution arises from a direct sum of edge ideals of chordal graphs. The main observation is that these Betti diagrams are the lattice points of a normal, reflexive polytope which can be constructed recursively from threshold graphs. This is joint work with Alexander Engström.