Kathryn Nyman

Ph.D. (2001) Cornell University

First Position

Postdoctoral position at Texas A&M University

Dissertation

Enumeration in Geometric Lattices and the Symmetric Group

Research Area

Combinatorics

Abstract

This work primarily deals with questions relating to the enumeration of chains in geometric lattices. The study of geometric lattices arose in the process of characterizing the lattices of subspaces formed by a finite set of points in projective or affine space, and ordered by inclusion. More generally geometric lattices encode the structure of matroids (or combinatorial geometries), which take an axiomatic approach to the concept of dependence.

The flag Whitney numbers of a geometric lattice count the number of chains of the lattice with elements having specified ranks. We are interested in the linear inequalities satisfied by the flag Whitney numbers.

We present a lower bond on the flag Whitney numbers of a lattice in terms of its rank and number of atoms. We also show that all flag Whitney numbers are simultaneously minimized by the lattice corresponding to the near pencil arrangement of points in R^n. Next we focus on rank 3 geometric lattices and give a collection of inequalities which imply all the linear inequalities satisfied by the flag Whitney numbers of rank 3 geometric lattices. We further describe the smallest closed convex set containing the flag Whitney numbers of rank 3 lattices corresponding to oriented matroids.

The ab-index is a polynomial associated to a lattice which contains all of the information regarding the flag Whitney numbers. We give a recurrence relation for the ab-index of families of lattices satisfying a particular set of conditions. We also give a collection of inequalities for the ab-index of geometric lattices.

Finally, we turn our attention to the group algebra of the symmetric group. The peak set of a permutation $\sigma$ is the set $\{i:\sigma(i–1)<\sigma(i)>\sigma(i+1)\}$. We prove the existence of a subalgebra of Solomon's descent algebra in which elements are sums of permutations that share a common peak set.