The inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of type An−1 is given as a sum over a set of quantum walks in the quantum Bruhat graph, QBG(An−1). We establish bounds on the number of quantum steps and simple stationary steps in these quantum walks. By a result of Kato, we map this formula to the equivariant quantum K-theory of partial flag manifolds G/P to give an alternate proof of [KLNS24, Theorem 8]. / Doctor of Philosophy / The quantum Bruhat graph, is a directed graph with vertex set W . Beginning with an arbitrary element of W , at each position, we may either move to a new element of W along a directed edge (a non-stationary step), or stay at the current element (a stationary step).
A quantum walk is the sequence that records the element W at each position. We establish bounds on the number of occurrences of particular kinds of stationary and non-stationary steps called simple stationary steps and quantum steps respectively. These bounds are relevant to calculations of Chevalley formulas in K-Theory.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/118915 |
| Date | 07 May 2024 |
| Creators | Shaplin III, Richard Martin |
| Contributors | Mathematics, Orr, Daniel D., Shimozono, Mark M., Mihalcea, Constantin Leonardo, Loehr, Nicholas A. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | Creative Commons Attribution 4.0 International, http://creativecommons.org/licenses/by/4.0/ |
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