Regular two-graphs and extensions of partial geometries

Geometry;meetkunde

Extended F_4-buildings and the Baby Monster

The Baby Monster group B acts naturally on a geometry E(B) with diagram c.F_4(t) for t=4 and the action of B on E(B) is flag-transitive. It possesses the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2, and (b) three elements of type 1 are pairwise incident to common elements of type 2 iff they are incident to a common element of type 5. It is shown that E(B) is the only (non-necessary flag-transitive) c.F_4(t)-geometry, satisfying t=4, (a) and (b), thus obtaining the first characterization of B in terms of an incidence geometry, similar in vein to one known for classical groups acting on buildings. Further, it is shown that E(B) contains subgeometries E(^2E_6(2)) and E(Fi22) with diagrams c.F_4(2) and c.F_4(1). The stabilizers of these subgeometries induce on them flag-transitive actions of ^2E_6(2):2 and Fi22:2, respectively. Three further examples for t=2 with flag-transitive automorphism groups are constructed. A complete list of possibilities for the isomorphism type of the subgraph induced by the common neighbours of a pair of vertices at distance 2 in an arbitrary c.F_4(t) satisfying (a) and (b) is obtained.Comment: to appear in Inventiones Mathematica

Gamma-polynomials of flag homology spheres

Chapter 1 contains the main definitions used in this thesis. It also includes some basic theory relating to these fundamental concepts, along with examples. Chapter 1 includes an original result, Theorem 1.5.4, answering a question of Postnikov-Reiner-Williams, which characterises the normal fans of nestohedra. Chapter 2 contains the content of the paper [2], of which Theorem 2.0.6 is the main result. As mentioned, [2] shows that the Nevo and Petersen conjecture holds for simplicial complexes in sd(Σd−1). . Chapter 3 includes the content of the paper [1], where we show that the Nevo and Petersen conjecture holds for the dual simplicial complexes to nestohedra in Theorem 3.0.4. Chapter 4 contains the content of the paper [3] in which we prove Conjecture 0.0.4 in Theorem 4.1.2 by showing that tree shifts lower the γ-polynomial of graph-associahedra. Chapter 4 also includes Theorem 4.2.1, which shows that flossing moves also lower the γ-polynomial of graph-associahedra. In Chapter 5 we include smaller results that have been made. This chapter includes a result proving Gal’s conjecture for edge subdivisions of the order complexes of Gorenstein* complexes, and shows that this result can be attributed to the work of Athanasiadis in [4]. Chapter viii INTRODUCTION 5 also includes some work we have done towards answering Question 14.3 of [26] for interval building sets

Noncrossing sets and a Grassmann associahedron

We study a natural generalization of the noncrossing relation between pairs of elements in [n] to k-tuples in [n] that was first considered by Petersen et al. [J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on ([n]k) induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product [k]×[n−k] of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for k=2. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex the noncrossing complex, and the polytope derived from it the dual Grassmann associahedron. We extend results of Petersen et al. [J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphismGk,n≅Gn−k,n. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define a Grassmann–Tamari order on maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersen et al. [J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part