On the Shard Intersection Order of a Coxeter Group

Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the case of the symmetric group, we provide combinatorial models for shard intersections of all classical types and use this understanding to prove that the shard intersection order is EL-shellable. Further, inspired by work of Simion and Ullman on the lattice of noncrossing partitions, we show that the shard intersection order on the symmetric group admits a symmetric Boolean decomposition, i.e., a partition into disjoint Boolean algebras whose middle ranks coincide with the middle rank of the poset. Our decomposition also yields a new symmetric Boolean decomposition of the noncrossing partition lattice.