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 This dissertation includes both previously published and unpublished co-authored material.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

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…mathematically or otherwise. Thank you for your support and encouragement.
Finally, I owe a huge debt of gratitude to my friends and all the people who
have supported me during my time at the *University* *of* *Oregon*. It would be
impossible to list the name of every…