Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
Abstrak
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.
Topik & Kata Kunci
Penulis (1)
Henri Mühle
Akses Cepat
- Tahun Terbit
- 2020
- Sumber Database
- DOAJ
- DOI
- 10.46298/dmtcs.6363
- Akses
- Open Access ✓