Noncrossing partitions and the shard intersection order

TL;DR

The paper introduces the shard intersection order, a new lattice structure on finite Coxeter groups, connecting noncrossing partitions with polyhedral geometry and providing new proofs and properties.

Contribution

It defines the shard intersection order, proves it forms a lattice, and establishes bijections with triangulations of W-permutohedron and W-associahedron.

Findings

Shard intersection order is a graded lattice with W-Eulerian polynomial rank generating function.

Many properties of the shard intersection order mirror those of noncrossing partitions.

Bijections link chains in the order to simplices in polyhedral triangulations.

Abstract

We define a new lattice structure on the elements of a finite Coxeter group W. This lattice, called the shard intersection order, is weaker than the weak order and has the noncrossing partition lattice NC(W) as a sublattice. The new construction of NC(W) yields a new proof that NC(W) is a lattice. The shard intersection order is graded and its rank generating function is the W-Eulerian polynomial. Many order-theoretic properties of the shard intersection order, like Mobius number, number of maximal chains, etc., are exactly analogous to the corresponding properties of NC(W). There is a natural dimension-preserving bijection between simplices in the order complex of the shard intersection order (i.e. chains in the shard intersection order) and simplices in a certain pulling triangulation of the W-permutohedron. Restricting the bijection to the order complex of NC(W) yields a bijection to simplices in a pulling triangulation of the W-associahedron. The shard intersection order is defined indirectly via the polyhedral geometry of the reflecting hyperplanes of W. Indeed, most of the results of the paper are proven in the more general setting of simplicial hyperplane arrangements.