Klara Stokes

The aim of this paper is to use the framework of incidence geometry to develop a theory that permits to model both the inner and outer automorphisms of a group G simultaneously. More precisely, to any group G, we attempt to associate an incidence system whose group of type-preserving automorphisms is Inn(G), the group of inner automorphisms of G, and whose full group of automorphisms is the group Out(G) of outer automorphisms of G, getting what we call an incidence geometric representation theory for groups. Hence, in this setting, the group Inn(G) preserves the types of the associated incidence structure while the group Out(G) is acting non-trivially on the typeset of it, realizing the outer automorphisms of G as correlations. We give examples of incidence geometric representations for the dihedral groups, the symmetric groups, the automorphism groups of the five platonic solids, families of classical groups defined over fields such as the projective linear groups, and finally subgroups of free groups whose outer automorphism group is the largest finite subgroup of their automorphism group.

We introduce the notion of power lattices that unifies and extends the equicardinal geometric lattices, Cartesian products of subspace lattices, and multiset subset lattices, among several others. The notions of shellability for simplicial complexes, q-complexes, and multicomplexes are then unified and extended to that of complexes in power lattices, which we name as P-complexes. A nontrivial class of shellable P-complexes are obtained via P-complexes of the independent sets of a matroid in power lattice, which we introduce to generalize matroids in Boolean lattices, q-matroids in subspace lattices, and sum-matroids in Cartesian products of subspace lattices. We also prove that shellable P-complexes in a power lattice yield shellable order complexes, extending the celebrated result of shellability of order complexes of (equicardinal) geometric lattices by Björner and also, a recent result on shellability of order complexes of lexicographically shellable q-complexes. Finally, we provide a construction of matroids on the lattice of multiset subsets from weighted graphs. We also consider a variation of Stanley-Reisner rings associated with shellable multicomplexes than the one considered by Herzog and Popescu and proved that these rings are sequentially Cohen-Macaulay.

We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.