Publication Date

2018-06-27

Availability

Open access

Embargo Period

2018-06-27

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2018-04-25

First Committee Member

Drew Armstrong

Second Committee Member

Richard P. Stanley

Third Committee Member

Bruno Benedetti

Fourth Committee Member

Victor J. Milenkovic

Abstract

This thesis is comprised of an introduction, three chapters, and a brief appendix of figures. The introduction develops the theory of root systems and finite groups generated by reflections. For brevity and readability we occasionally argue by fiat, providing references for omitted proofs and further compensating with ample exam- ples. We establish conventions for ordering and coordinatizing root and weight vectors for the classical types and conclude with two central theorems from Waldspurger and Meinrenken. Chapter 1 introduces a combinatorial algorithm for studying the Waldspurger and Meinrenken theorems in the type A setting where the underlying reflection group is the symmetric group, S_n. Our algorithm associates π ∈ S_n with an (n − 1) × (n − 1) matrix denoted WT(π). We characterize the column, row, and diagonal vectors of WT(π) in terms of certain lattice paths. Because componentwise order on WT(S_n) is isomorphic to Bruhat order, we extend the domain of WT to the set of alternating sign matrices to obtain a new combinatorial model of the classical ASM lattice. Chapter 2 uses the map WT and folding techniques to study Waldspurger and Meinrenken’s theorems in types B and C. In particular, we characterize the set of join-irreducible elements of the Dedekind-MacNeille completion of Bruhat order. Chapter 3 uses symmetries from Meinrenken’s theorem to compare three notions of dimension for permutations, the most novel of which relates to SIF permutations. We conclude by considering a dual graph structure on n-cycles, study its degree sequence and presenting a number of conjectures relating to its recursive structure.

Keywords

Alternating Sign Matrices Coxeter Tiling Reflection

Share

COinS