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Transactions of the American Mathematical Society

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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

G. Zwara
Affiliation: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

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