Informal Description of
the Classification of d-Complete Posets

This page assumes that What do d-complete posets look like? has been read.

Removal of the 'global tree-edges' from a d-complete poset will leave a disjoint union of 'slant irreducible components'. Roughly speaking, each slant irreducible component consists of a local region of four element diamonds, together with a few more elements which are required by a certain condition.

The top tree of any connected poset P with a unique maximal element is defined to consist of all elements y such that the set {x: x > = y} is a totally ordered subset of P. It can be shown that the top tree of any slant irreducible component is a "Y-shaped" graph. Known connections with Lie theory indicate that one should think of each top tree as being a Dynkin diagram for a simply laced Kac-Moody algebra. Let's say that a Dynkin diagram is of general type E if it is Y-shaped, has exactly one branch of length 1, and is not of type A.

There are 15 classes of possible slant irreducible components. In all but Classes 1-3, the top tree of each slant irreducible component is of general type E. Each of the 15 classes is further divided into subclasses, and each subclass possesses a unique maximal member. In 8 of the 15 classes, the subclasses are indexed by a single positive integer parameter. Two such parameters are required in 5 classes, and 3 parameters are needed in Class 3. Class 15 consists of one poset, which is distantly related to the famous 27 lines on a cubic curve situation (or a an irrep of E7).

Class 1 consists of shapes and Class 2 consists of shifted shapes. Class 3 posets are relatively simple. Each poset in one of the other classes corresponds to a certain especially nice kind of Weyl group element in a simply laced Kac-Moody Weyl group whose Dynkin diagram is of general type E.

The order diagrams shown on the following pages are the order diagrams for the unique maximal poset within each class. There are 7 kinds of symbols used in these diagrams to denote 7 kinds of elements, as is explained in Section 7 of the paper 'Classification.'

Move on and look at the:

First Half of Slant Irreducibles List.

Second Half of Slant Irreducibles List.

or return to Recent Research Table of Contents.