Doctor of Philosophy (PHD)
Mathematics (Arts and Sciences)
Date of Defense
First Committee Member
Second Committee Member
Richard P. Stanley
Third Committee Member
Fourth Committee Member
Victor J. Milenkovic
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.
Alternating Sign Matrices Coxeter Tiling Reflection
McKeown, James, "On the Combinatorics of the Waldspurger Decomposition" (2018). Open Access Dissertations. 2122.