Nathan Williams

Reading cut the hyperplanes in a real central arrangement \mathcal{H} into pieces called shards , which reflect order-theoretic properties of the arrangement. We show that shards naturally index certain generators of the fundamental group of the complement of the complexification of \mathcal{H} . Taking only positive expressions in these generators yields a new poset that we call the pure shard monoid . When \mathcal{H} is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator \mathsf{Pop} . In this case, we use \mathsf{Pop} to define an embedding \mathsf{Crackle} of Reading’s shard intersection order into the pure shard monoid. When \mathcal{H} is the reflection arrangement of a finite Coxeter group, we also define a poset embedding \mathsf{Snap} of the shard intersection order into the positive braid monoid; in this case, our three maps are related by \mathsf{Snap}=\mathsf{Crackle}\cdot \mathsf{Pop} .

Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition of charmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions.

We solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.