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Decompositions of Modules Associated to Finite Partially Ordered Sets
AbstractFor a finite partially ordered set L and a field F, let F L be the associated vector space with L as basis. In the case of ranked posets this space decomposes into eigenspaces under the maps afforded by the order relation. In this note we show how to construct generating sets for such decompositions and discuss their combinatorial significance
Cluster Complexes via Semi-Invariants
We define and study virtual representation spaces having both positive and
negative dimensions at the vertices of a quiver without oriented cycles. We
consider the natural semi-invariants on these spaces which we call virtual
semi-invariants and prove that they satisfy the three basic theorems: the First
Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition
Theorem. In the special case of Dynkin quivers with n vertices this gives the
fundamental interrelationship between supports of the semi-invariants and the
Tilting Triangulation of the (n-1)-sphere.Comment: 34 page
Subdivisional spaces and graph braid groups
We study the problem of computing the homology of the configuration spaces of
a finite cell complex . We proceed by viewing , together with its
subdivisions, as a subdivisional space--a kind of diagram object in a category
of cell complexes. After developing a version of Morse theory for subdivisional
spaces, we decompose and show that the homology of the configuration spaces
of is computed by the derived tensor product of the Morse complexes of the
pieces of the decomposition, an analogue of the monoidal excision property of
factorization homology.
Applying this theory to the configuration spaces of a graph, we recover a
cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this
model enhances it with various convenient functorialities, exact sequences, and
module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version
published in Documenta Mathematic
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