On the complement of the dense orbit for a quiver of type \(\mathbb A\). (English) Zbl 1293.16009
Summary: Let \(\mathbb A_t\) be the directed quiver of type \(\mathbb A\) with \(t\) vertices. For each dimension vector \(d\), there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by H. Knight and A. Zelevinsky [Adv. Math. 117, No. 2, 273-293 (1996; Zbl 0915.16009)], and by C. M. Ringel [AMA, Algebra Montp. Announc. 1999, No. 1, Paper No. 2 (1999; Zbl 1010.16014)]. Moreover, we compare with the fan associated to the quiver \(\mathbb A_t\) and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit, we determine the irreducible components and their codimension. Finally, we consider several particular examples.
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 14L35 | Classical groups (algebro-geometric aspects) |
| 20G05 | Representation theory for linear algebraic groups |
Keywords:
representation spaces; dense orbits; numbers of orbits; irreducible components; multisegment duality; representations of quiversReferences:
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