Full Record

Author | Gedeon, Katie |

Title | Kazhdan-Lusztig Polynomials of Matroids and Their Roots |

URL | http://hdl.handle.net/1794/23913 |

Publication Date | 2018 |

Date Accessioned | 2018-10-31 22:34:14 |

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 This dissertation includes both previously published and unpublished co-authored material.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 | 2020-06-15 |

Date Indexed | 2020-06-18 |

Grantor | University of Oregon |

Issued Date | 2018-10-31 00:00:00 |

Sample Search Hits | Sample Images | Cited Works

…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…