Representation stability and the homology of simplicial complexes of graphs
Final Report Abstract
Consider a property P a graph might have, for example being connected or being disconnected. Many common properties of graphs are preserved either under deleting edges or under adding edges to the graph. Here we only consider properties which are closed under deleting edges, for example the property of being disconnected. For such a property P, let Gn (P) be the set of all graphs with vertex set [n] := { 1, . . . , n } with this property. Since we fix the set of vertices, we might consider a graph to be its set of edges. Moreover, since we required P to be preserved under deleting edges, Gn (P) is closed under taking subsets, and thus a simplicial complex. We call Gn (P) the graph complex of P. In this project we studied the homology of Gn (P) for large n. More precisely, we assume that P is preserved under isomorphism of graphs, and thus in particular under permutations of the vertices. This gives rise to an action of the symmetric group on Gn (P) and also on its homology groups. The objective of this project was to study the occurring representations for large n, with view of stabilization phenomena in the sense of Church and Farb. It turned out that for many properties P such stabilization does indeed occur. However, oftentimes the homology stabilizes to zero for rather trivial reasons. After this finding, we considered a particular property P, namely the property of being a matching (i.e., having at most one edge on each vertex). The corresponding graph complex is called the matching complex and it is a well-studied object. It plays a special role in this theory, because if P is another property which is preserved under adding isolated edges, then Gn (P) is a “module” (in a certain sense) over the matching complex, where the operation is disjoint union of graphs. The homology of the matching complex is closely related to the Koszul homology of the second Veronese subalgebra of the polynomial ring. Using this connection, we were able to use our results about the matching complex to answer a question of Bruns, Conca and Römer about the generators of the Koszul homology as an algebra.
Publications
- The Koszul homology algebra of the second Veronese is generated by the lowest strand
Lukas Katthän, A. Conca und V. Reiner
- Linearity in minimal resolutions of monomial ideals, 2017
Lukas Katthän
- Spanning Lattice Polytopes and the Uniform Position Principle, 2017
Lukas Katthän, J. Hofscheier and B. Nill
