Author ORCID Identifier
Date Available
8-8-2023
Year of Publication
2023
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Arts and Sciences
Department/School/Program
Mathematics
First Advisor
Dr. Margaret Readdy
Abstract
In this dissertation we study the geometry of pipe dream complexes with the goal of gaining a deeper understanding of Schubert polynomials. Given a pipe dream complex PD(w) for w a permutation in the symmetric group, we show its boundary is Whitney stratified by the set of all pipe dream complexes PD(v) where v > w in the strong Bruhat order. For permutations w in the symmetric group on n elements, we introduce the pipe dream complex poset P(n). The dual of this graded poset naturally corresponds to the poset of strata associated to the Whitney stratification of the boundary of the pipe dream complex of the identity element. We examine pipe dream complexes in the case a permutation is a product of commuting adjacent transpositions. Finally, we consider pattern avoidance results. For 132-avoiding permutations, the Rothe diagram forms a Young diagram. In the case a permutation w has exactly one 132-pattern, the associated pipe dream complex is an m-dimensional simplex, where m = n choose 2 − l(w) − 1 and l(w) is the length of w. In the case of exactly two 132 patterns, there are three possible configurations. We include generalizations of these cases.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2023.367
Recommended Citation
Reese, Benjamin, "Geometry of Pipe Dream Complexes" (2023). Theses and Dissertations--Mathematics. 105.
https://uknowledge.uky.edu/math_etds/105
