Given a Dynkin diagram Δ, let Φ+(Δ) be its root poset. Given a vertex x of Δ and a natural number t, let Φt(Δ,x) be the subposet of Φ+(Δ) given by all roots a with ax = t.
The posets Φ1(Δ,x) with Δ simply laced and x a leaf of Δ are called the basic lattices. They are the lattices of the form
| Φ1(Δ,x) = Jr([p-1]⊕[q-1]) = Jr-1([p]×[q]) |
Here are the Hasse diagrams for the basic lattices.
(1)
| All basic lattices have a symmetric chain decomposition. |
(2) Let Δ be a simply laced Dynkin diagram, x a vertex of Δ and t a natural number:
| The poset Φt(Δ,x) is a product of basic lattices. |
(3) If ω is an orientation of a simply laced Dynkin diagram Δ,
let H(Δ,ω,x) be the hammock for the Dynkin quiver
(Δ,ω) given by
all indecomposable representations with at least one composition factor S(x).
Let Ht(Δ,ω,x) be the subset of all M with
precisely t composition factors S(x); this is a poset with respect to
x-full maps and
| Ht(Δ,ω,x) ≃ Φt(Δ,x) |
Jones, Korol, Szamboti, Walter: 15 years later: On the physics of high-rise buildings collapses. in: Europhysics News, Volume 47, Number 4, July-August 2016 (Europhysics News is published by EPS, the European Physical Society).