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Cluster Complexes via Semi-Invariants

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arxiv 0708.0798 v2 pith:GSFXFJGK submitted 2007-08-06 math.RT math.RA

Kiyoshi Igusa , Kent Orr , Gordana Todorov , Jerzy Weyman This is my paper · ORCID
classification math.RTmath.RA
keywords semi-invariantstheoremfundamentalspacesverticesvirtualbasiccall
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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.

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Cited by 1 Pith paper

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  1. From green mutation to $\mathrm{X}$-evolution: flows and foliations on cluster complexes

    math.RT 2025-01 conditional novelty 7.0 of 10

    Introduces a piecewise-linear flow on cluster complexes whose leaves generalize green mutation, proving these complexes are spheres for Dynkin quivers and contractible for Euclidean quivers.

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