Noncrossing partitions and the shard intersection order

Reading, Nathan

Journal of Algebraic Combinatorics, Tome 33 (2011) no. 4, pp. 483-530.

Résumé

Summary: We define a new lattice structure $( W,\preceq)$ (W,preceq) 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 $( W,\preceq)$ (W,preceq), like Möbius 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 $( W,\preceq)$ (W,preceq) (i.e. chains in $( W,\preceq)$ (W,preceq)) 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 lattice $( W,\preceq)$ (W,preceq) 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.

Détail

Keywords: keywords noncrossing partition, shard, Coxeter group, weak order