We define a new lattice structure (W,\preceq ) on the elements of a finite Coxeter group W. This lattice, called the \emphshard 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 ), 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 ) (i.e. chains in <mbox>(W,\preceq )</mbox>) 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 ) 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.
Discrete Mathematics & Theoretical Computer Science