Generalizations of Total Positivity

Acknowledgements
Dedication
Abstract
List of Figures
Introduction
	Totally Positive Matrices
	Totally Nonnegative Varieties
	Overview
	Main Results
k-Positivity of Kazhdan-Lusztig Immanants
	k-Positive Matrices
	Kazhdan-Lusztig Immanants
	Question and Motivation
	Main Theorem
	Formula for 1324- and 2143-avoiding Permutations
	Dodgson Condensation
		Young diagrams
		General case
	Proof of Proposition 2.6.12
	Proof of Proposition 2.6.14
	Pattern Avoidance Conditions
	Future Directions
Cluster Algebras and k-Positivity
	Cluster Algebra Background
	Double Rim Hook Cluster Algebras
	Applications to Kazhdan-Lusztig Immanants
	Cluster Algebras and k-Positivity Tests
	Double Rim Hook Cluster Algebras and 2-Positivity
Total Positivity and Networks
	Totally Nonnegative Matrices and Planar Directed Networks
	Totally Nonnegative Grassmannians and Planar Directed Networks
	Planar Directed Networks to Plabic Networks
	Moves and Reductions
	Postnikov Diagrams
Plabic Networks on the Cylinder
	Planar Directed Networks on a Cylinder
	Face Weights on the Cylinder
	Changing Orientation on the Cylinder
Plabic R-matrix
	Cylindric k-loop Plabic Graphs
	Edge-weighted Plabic R-matrix
	Face-weighted Plabic R-matrix
	Postnikov Diagram Proofs
		Proof of Theorem 6.1.2
		Proof of Theorem 6.1.5
	Proof of Theorems 6.2.5 and 6.3.3
Spider Web Quivers
	Spider Web Quivers and a Mutation Sequence
	x-dynamics
	y-dynamics
Plabic R-matrix, Revisited
References