acase@tulane.edu / Let G be a reductive group, B be a Borel subgroup, and let K be a symmetric
subgroup of G. The study of B orbits in a symmetric variety G/K or, equivalently, the
study of K orbits in a flag variety G/B has importance in the study of Harish-Chandra
modules; it comes with many interesting Schubert calculus problems. Although this
subject is very well studied, it still has many open problems from combinatorial point
of view. The most basic question that we want to be able to answer is that how
many B orbits there are in G/K. In this thesis, we study the enumeration problem
of Borel orbits in the case of classical symmetric varieties. We give explicit formulas
for the numbers of Borel orbits on symmetric varieties for each case and determine
the generating functions of these numbers. We also explore relations to lattice path
enumeration for some cases. In type A, we realize that Borel orbits are parameterized by the lattice paths in a pxq grid moving by only horizontal, vertical and diagonal steps weighted by an appropriate statistic. We provide extended results for type C
as well. We also present various t-analogues of the rank generating function for the
inclusion poset of Borel orbit closures in type A. / 1 / Ozlem Ugurlu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_79051 |
| Date | January 2018 |
| Contributors | UGURLU, Ozlem (author), Can, Mahir (Thesis advisor), School of Science & Engineering Mathematics (Degree granting institution) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | electronic, 138 |
| Rights | No embargo, Copyright is in accordance with U.S. Copyright law. |
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