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Seminar BIREP

50 Years of Mathematics

Discrete Structures in Mathematics - the SFB 343 (1989-2000)

The basic lattices

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])
where Δ = Tpqr is the star with arms of rank p, q, r ≥ 1 and x is the leaf on the arm of rank r. For example, T332 = E6 and T321 = A5.
(Here, [n] is the chain of cardinality n, ⊕ is the disjoint union, × the product of posets, and J(P) is the poset of ideals of a poset P.)

Here are the Hasse diagrams for the basic lattices.

(1)

All basic lattices have a symmetric chain decomposition.
Thus, they are Peck posets (graded posets which are rank symmetric, rank unimodal and strongly Sperner). Of course, the basic lattices are distributive (since they are ideal lattices of posets).

(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.
Here is an example.

(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)

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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).

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Fakultät für Mathematik, Universität Bielefeld
Verantwortlich: C.M.Ringel
E-Mail: ringel@math.uni-bielefeld.de
Last modified: Oktober 13, 2016