Abstract
The main objects of noncrossing Catalan combinatorics associated to a finite Coxeter system are noncross- ing partitions, sortable elements, and cluster complexes. The first and the third of these have known Fuss–Catalan generalizations. We provide new viewpoints for these, introduce a corresponding generalization of sortable elements as elements in the positive Artin monoid, and show how this perspective ties together all three generalizations.
Highlights
Fix a finite Coxeter system (W, S) and a Coxeter element c ∈ W —that is, a product of the simple generators S in any order
There are several components missing in this story. The aim of this extended abstract is to complete the m-eralization of noncrossing Catalan objects using the spherical Artin group corresponding to the finite Coxeter system (W, S)
We prove the existence and uniqueness of this relation using an m-eralization of the subword complex approach to c-cluster complexes given in [7, 13]
Summary
Fix a finite Coxeter system (W, S) and a (standard) Coxeter element c ∈ W —that is, a product of the simple generators S in any order. There are several components missing in this story The aim of this extended abstract is to complete the m-eralization of noncrossing Catalan objects using the spherical Artin group corresponding to the finite Coxeter system (W, S). The second missing component is that there is no simple combinatorial definition of the m-eralized c-cluster complex when the Coxeter element c is not bipartite. With the m-eralizations of noncrossing partitions, sortable elements, and subword complexes in hand, we prove the following theorem. Reading’s m-eralized c-cluster complex can be used to define a Cambrian graph for bipartite Coxeter elements, no corresponding poset has been considered in the literature for m > 1. One intuition comes from the isomorphism between shard order restricted to c-sortable elements and the noncrossing partition lattice: just as D. We place the program of m-eralizing noncrossing Coxeter–Catalan combinatorics in the context of the corresponding positive Artin monoid
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