The zero set of semi-invariants for extended Dynkin quivers

Riedtmann, Ch.; Zwara, G.

doi:10.1090/S0002-9947-08-04613-8

The zero set of semi-invariants for extended Dynkin quivers
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by Ch. Riedtmann and G. Zwara PDF

Trans. Amer. Math. Soc. 360 (2008), 6251-6267 Request permission

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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