Generalizations of Total Positivity

✍ Scribed by Sunita Chepuri

Year: 2020
Tongue: English
Leaves: 171
Series: PhD Thesis
Category: Library

No coin nor oath required. For personal study only.

✦ Table of Contents

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

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