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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The zero set of semi-invariants for extended Dynkin quivers


Authors: Ch. Riedtmann and G. Zwara
Journal: Trans. Amer. Math. Soc. 360 (2008), 6251-6267
MSC (2000): Primary 14L24; Secondary 16G20
Published electronically: July 21, 2008
MathSciNet review: 2434286
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Abstract: We show that the set of common zeros $ \mathcal{Z}_{Q,\mathbf{d}}$ of all semi-invariants vanishing at 0 on the variety $ \operatorname{rep}(Q,\mathbf{d})$ of all representations with dimension vector $ \mathbf{d}$ of an extended Dynkin quiver $ Q$ under the group $ \operatorname{GL}(\mathbf{d})$ is a complete intersection if $ \mathbf{d}$ is ``big enough''. In case $ \operatorname{rep}(Q,\mathbf{d})$ does not contain an open $ \operatorname{GL}(\mathbf{d})$-orbit, which is the case not considered so far, the number of irreducible components of $ \mathcal{Z}_{Q,\mathbf{d}}$ grows with $ \mathbf{d}$, except if $ Q$ is an oriented cycle.


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Additional Information

Ch. Riedtmann
Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email: christine.riedtmann@math-stat.unibe.ch

G. Zwara
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Email: gzwara@mat.uni.torun.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04613-8
PII: S 0002-9947(08)04613-8
Keywords: Semi-invariants, quivers, representations
Received by editor(s): October 12, 2006
Published electronically: July 21, 2008
Additional Notes: The second author gratefully acknowledges support from the Polish Scientific Grant KBN No. 1 P03A 018 27 and the Swiss Science Foundation.
Article copyright: © Copyright 2008 American Mathematical Society