Cataland: Why the Fuß?

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Date
2019-01-21MFO Scientific Program
Research in Pairs 2018Series
Oberwolfach Preprints;2019,01Author
Stump, Christian
Thomas, Hugh
Williams, Nathan
Metadata
Show full item recordOWP-2019-01
Abstract
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.