THÈSE de DOCTORAT de l'UNIVERSITÉ PARIS 6 Litp /
Litp research reports
74 pages - Novembre/November 1995 - Document en anglais.
Titre / Title: Hauteur sur le treillis des partitions
Abstract : The lattice of partitions is not a rank poset, i.e. the maximal chains between two comparable partitions do not have the same length in general, and thus it is not straightforward to define a height (or rank) function on this lattice. C. Greene has described the lengths of maximal chains. We propose a notion of height, not by using maximal chains, but by comparing partitions to the special partitions in the lattice of partitions. Our basic set of partitions, called basis, will be the set of those partitions which have only one successor. Any partition is determined by all its camparisons with the basis elements, comparisons that we can code as a boolean vector. The height is defined using this vector, but not maximal chains. We first characterize explicitly basis elements, and enumerate them (Chapters 2 and 3). We then compute height for basis elements (results are summarized in chapter 4). Next we give heights for general partitions in chapter 5. Each partition can be characterized by the maximal basis elements which are below it. We characterize them in chapter 6. In chapter 7, we describe some special interesting intervals in the lattice of partitions and we prove that all modular sublattices are distributive. We finish by the study of some distributive sublattices related to a formula (called Pieri's formula) which is widely used in representation theory and in geometry.
Publications internes Litp 1995 / Litp research reports 1995