Kazhdan-Lusztig Polynomials of Matroids and Their Roots
| dc.contributor.advisor | Proudfoot, Nicholas | |
| dc.contributor.author | Gedeon, Katie | |
| dc.date.accessioned | 2018-10-31T22:34:14Z | |
| dc.date.available | 2018-10-31T22:34:14Z | |
| dc.date.issued | 2018-10-31 | |
| dc.description.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 This dissertation includes both previously published and unpublished co-authored material. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/23913 | |
| dc.language.iso | en_US | |
| dc.publisher | University of Oregon | |
| dc.rights | All Rights Reserved. | |
| dc.subject | Kazhdan-Lusztig polynomials | en_US |
| dc.subject | Matroid theory | en_US |
| dc.subject | real-rootedness | en_US |
| dc.title | Kazhdan-Lusztig Polynomials of Matroids and Their Roots | |
| dc.type | Electronic Thesis or Dissertation | |
| thesis.degree.discipline | Department of Mathematics | |
| thesis.degree.grantor | University of Oregon | |
| thesis.degree.level | doctoral | |
| thesis.degree.name | Ph.D. |
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