# Johann Selewa: Stanley-Reisner rings of abstract simplicial complexes

**Time: **
Fri 2019-10-04 13.00 - 14.00

**Location: **
Sal 306, hus 6, Kräftriket

**Participating: **
Johann Selewa

### Abstract

Let G be a graph on a finite vertex set, and let R be the polynomial ring (over the complex numbers) whose variables are indexed by the vertices of G. To G we associate its Stanley-Reisner I ideal generated by the monomials corresponding to edges *not* in G. The quotient M = R/I is called the Stanley-Reisner ring of G, and is a module over R. Its homological properties give information about G. It is for example known that the (multi-graded) Hilbert-Poincaré series HS(M) of M is a rational function whose denominator encodes the minimal free resolution and Betti numbers of M. These in turn shed homological light on the a priori completely combinatorial description of G.

In this talk we will study HS(M) by defining an action of the automorphism group aut(G) of G. This action lets us study the symmetries of G in terms of homology, which is susceptible to computations. By replacing G with any finite abstract simplicial complex we deepen the study of graph symmetries. Abstract simplicial complexes are generalisations of graphs, and there are natural such complexes associated to graphs which unveil more complicated symmetries.