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Stanley's formula for the number of reduced expressions of a permutation regarded as a Coxeter group element raises the question of how to enumerate the reduced expressions of an arbitrary Coxeter group element. We provide a framework for answering this question by constructing combinatorial objects that represent the inversion set and the reduced expressions for an arbitrary Coxeter group element. The framework also provides a formula for the length of an element formed by deleting a generator from a Coxeter group element. Fan and Hagiwara, et al: showed that for certain Coxeter groups, the short-braid avoiding elements characterize those elements that give reduced expressions when any generator is deleted from a reduced expression. We provide a characterization that holds in all Coxeter groups. Lastly, we give applications to the freely braided elements introduced by Green and Losonczy, generalizing some of their results that hold in simply-laced Coxeter groups to the arbitrary Coxeter group setting.
Advisor: | Green, Richard M., Thiem, Nathaniel |
Commitee: | Finkelstein, Noah, Liebler, Robert, Vojtechovsky, Petr |
School: | University of Colorado at Boulder |
Department: | Mathematics |
School Location: | United States -- Colorado |
Source: | DAI-B 70/07, Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Mathematics |
Keywords: | Coxeter groups, Enumeration, Freely braided, Fully covering, Reduced expressions, Reflection subgroups |
Publication Number: | 3366660 |
ISBN: | 978-1-109-28232-0 |