Nathan Reading

Noncrossing partitions and the shard intersection order
dmtcs:2709 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)

https://doi.org/10.46298/dmtcs.2709
Noncrossing partitions and the shard intersection order
Authors: Nathan Reading ^{1}
0000000307687872
Nathan Reading
1 Department of Mathematics [Raleigh]
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 WEulerian polynomial. Many ordertheoretic 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 dimensionpreserving 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 Wpermutohedron. Restricting the bijection to the order complex of \NC (W) yields a bijection to simplices in a pulling triangulation of the Wassociahedron. 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.
Reading, Nathan, 2010, Noncrossing Partitions And The Shard Intersection Order, Journal Of Algebraic Combinatorics, 33, 4, pp. 483530, 10.1007/s1080101002553.