We give a description of the moment graph for Bott-Samelson varieties in arbitrary Lie type. We use this, along with curve neighborhoods and explicit moduli space computations, to compute a presentation for the small quantum cohomology ring of a particular Bott-Samelson variety in Type A. / Ph. D. / Since the early 1990’s, the study of quantum cohomology has been a fascinating, and fruitful field of research with connections to physics, representation theory, and combinatorics. The quantum cohomology of a space X encodes enumerative information about how many curves intersect certain subspaces of X; these counts are called Gromov-Witten invariants. For some spaces X, including the class of spaces we consider here, this count is only ”virtual” and negative Gromov-Witten invariants may arise.
In this dissertation, we study the quantum cohomology of Bott-Samelson varieties. These spaces arise frequently in applications to representation theory and combinatorics, however their quantum cohomology was previously unexplored. The first of our three main theorems describes the moment graph for Bott-Samelson varieties. This is a description of what all the possible curves, stable under certain symmetries, exist in a Bott-Samelson variety. Our second main theorem is a technical result which enables us to compute some GromovWitten invariants directly. Finally, our third main theorem is a description of the quantum cohomology for a certain three-dimensional Bott-Samelson variety.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/83820 |
| Date | 29 June 2018 |
| Creators | Withrow, Camron Michael |
| Contributors | Mathematics, Mihalcea, Constantin Leonardo, Orr, Daniel D., Haskell, Peter E., Shimozono, Mark M. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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