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Author
Title Kazhdan-Lusztig Polynomials of Matroids and Their Roots
URL
Publication Date
Date Accessioned
Degree PhD
Discipline/Department Department of Mathematics
Degree Level doctoral
University/Publisher University of Oregon
Abstract The Kazhdan-Lusztig polynomial of a matroid M, denoted P_M( t ), was recently defined by Elias, Proudfoot, and Wakefield. These polynomials are analogous to the classical Kazhdan-Lusztig polynomials associated with Coxeter groups. For example, in both cases there is a purely combinatorial recursive definition. Furthermore, in the classical setting, if the Coxeter group is a Weyl group then the Kazhdan-Lusztig polynomial is a Poincare polynomial for the intersection cohomology of a particular variety; in the matroid setting, if M is a realizable matroid then the Kazhdan-Lusztig polynomial is also the intersection cohomology Poincare polynomial of a variety corresponding to M. (Though there are several analogies between the two types of polynomials, the theory is quite different.) Here we compute the Kazhdan-Lusztig polynomials of several graphical matroids, including thagomizer graphs, the complete bipartite graph K_{2,n}, and (conjecturally) fan graphs. Additionally, we investigate a conjecture by the author, Proudfoot, and Young on the real-rootedness for Kazhdan-Lusztig polynomials of these matroids as well as a conjecture on the interlacing behavior of these roots. We also show that the Kazhdan-Lusztig polynomials of uniform matroids of rank n − 1 on n elements are real-rooted. This dissertation includes both previously published and unpublished co-authored material.
Subjects/Keywords Kazhdan-Lusztig polynomials; Matroid theory; real-rootedness
Contributors Proudfoot, Nicholas (advisor)
Language en
Rights All Rights Reserved.
Country of Publication us
Record ID handle:1794/23913
Repository oregon
Date Retrieved
Date Indexed 2020-06-18
Grantor University of Oregon
Issued Date 2018-10-31 00:00:00

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…PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 III. IV. 2.1. Matroids and Their Kazhdan-Lusztig Polynomials . . . . . . . 5 2.2. Equivariant Matroids and The Equivariant Kazhdan-Lusztig Polynomial…

…39 xii LIST OF TABLES Table Page 4.1. Kazhdan-Lusztig polynomials of some uniform matroids and their roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2. Roots of PM (t) for some uniform matroids…

…37 A.1 Kazhdan-Lusztig polynomials for the thagomizer matroid τn . . . . . 46 A.2 Kazhdan-Lusztig polynomials for the fan matroid ∆n . . . . . . . . . . 47 xiii CHAPTER I INTRODUCTION Kazhdan-Lusztig polynomials of matroids were first studied in…

…x5B;EPW] where the authors laid out the analogy between this new theory and the classical theory of Kazhdan-Lusztig polynomials of Coxeter groups. The most compelling aspect of these Kazhdan-Lusztig polynomials of matroids is that although they…

…we have two main goals: 1. To give a closed form of the coefficients of Kazhdan-Lusztig polynomials for some families of matroids. 2. To study the behavior of the roots of these polynomials. The closed form of the coefficients of Kazhdan-Lusztig

polynomials of matroids has been a subject of interest since polynomials of this type first appeared in [EPW]. In the appendix of that paper, the authors (along with Young) explicitly computed the coefficients of the Kazhdan-Lusztig

Kazhdan-Lusztig polynomials. Chapter III is dedicated to this first goal. Here, we give a closed form of the coefficients for the Kazhdan-Lusztig polynomials of the matroids associated to thagomizer graphs in Theorem 3.1, the complete bipartite 1 graph…

…and Young. We next turn our attention to the roots of these polynomials. A priori, there is no reason to think that studying the roots of Kazhdan-Lusztig polynomials of matroids would bear fruit. The roots themselves have no known interpretation…

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