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Cluster complexes via semi-invariants

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

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

[1]Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Volume 1: techniques of representation theory, London Mathematical Society Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
[2]Buan, A. B., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618.CrossRefGoogle Scholar
[3]Chapoton, F., Fomin, S. and Zelevinsky, A., Polytopal realizations of generalizad associahedra, Canad. Math. Bull. 45 (2002), 537–566.CrossRefGoogle Scholar
[4]Derksen, H. and Weyman, J., Semi-invariants of quivers and saturation for Littlewood–Richardson coefficients, J. Amer. Math. Soc. 13 (2000), 467–479.CrossRefGoogle Scholar
[5]Derksen, H. and Weyman, J., On canonical decomposition for quiver representations, Compositio. Math. 133 (2002), 245–265.CrossRefGoogle Scholar
[6]Happel, D. and Unger, L., Almost complete tilting modules, Proc. Amer. Math. Soc. 107 (1989), 603–610.CrossRefGoogle Scholar
[7]Igusa, K. and Orr, K., Links, pictures and the homology of nilpotent groups, Topology 40 (2001), 1125–1166.CrossRefGoogle Scholar
[8]Kac, V. G., Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), 141–162.CrossRefGoogle Scholar
[9]Schofield, A., Semi-invariants of quivers, J. London Math. Soc. 43 (1991), 383–395.Google Scholar
[10]Schofield, A., General Representations of quivers, Proc. London Math. Soc. (3) 65 (1992), 46–64.CrossRefGoogle Scholar
[11]Steinberg, R., Lectures on Chevalley groups, Yale University Lecture Notes (1964).Google Scholar
[12]Unger, L., The simplicial complex of tilting modules over quiver algebras, Proc. London Math. Soc (3) 73 (1996), 27–46.CrossRefGoogle Scholar