Abstract
We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.
Highlights
The case k = 1 clearly yields Sperner’s original result. We can rephrase these results in a poset-theoretical way, so that they can be read as “the maximal size of a union of k antichains in the Boolean lattice does not exceed the sum of its k largest rank numbers”
Examples of strongly Sperner posets are the Boolean lattices, the lattices of divisors of some integer [13], or the Bruhat posets associated with finite Coxeter groups [30]
Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices 925 can be viewed as certain permutations of n integers in d colors, and can be grouped according to the image of the first integer in the first color
Summary
Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices 925 can be viewed as certain permutations of n integers in d colors, and can be grouped according to the image of the first integer in the first color. This is clearly a decomposition of the group (and of the poset), and the subposets induced by the parts correspond to direct products of noncrossing partition lattices of type G(d, d, n ) or G(1, 1, n ) for n < n.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
