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Cluster complexes via semi-invariants

Part of: Algebraic groups Representation theory of rings and algebras General commutative ring theory

Published online by Cambridge University Press:  01 July 2009

Kiyoshi Igusa ,
Kent Orr ,
Gordana Todorov  and
Jerzy Weyman
Kiyoshi Igusa
Affiliation:
Department of Mathematics, Brandeis University, 415 South Street, Waltham, MA 02454-9110, USA (email: igusa@brandeis.edu)
Kent Orr
Affiliation:
Department of Mathematics, Indiana University, 831 E. 3rd Street, Bloomington, IN 47405-7106, USA (email: korr@indiana.edu)
Gordana Todorov
Affiliation:
Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA (email: todorov@neu.edu)
Jerzy Weyman
Affiliation:
Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA (email: j.weyman@neu.edu)
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Abstract

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We define and study virtual representation spaces for vectors 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.

Keywords

cluster complexsemi-invariants of quiverstilting objectprojective presentationpresentation spaces

MSC classification

Secondary: 14L24: Geometric invariant theory 16G20: Representations of quivers and partially ordered sets 13A50: Actions of groups on commutative rings; invariant theory 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009 , pp. 1001 - 1034
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

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