The orbits of the group Bn of upper-triangular matrices acting on 2-nilpotent complex matrices via conjugation are classified via oriented link patterns, generalizing A. Melnikov's classification of the Bn-orbits on upper-triangular such matrices. The orbit closures as well as the "building blocks" of minimal degenerations of orbits are described. The classification uses the theory of representations of finite-dimensional algebras. Furthermore, we initiate the study of the Bn-orbits on arbitrary nilpotent matrices.arXiv:1004.1996v1 [math.RT] 12 Apr 2010 the required representations. This gives a combinatorial classification in terms of oriented link patterns in section 3.3.Since several results on orbit closures for representations of finite-dimensional algebras are available through work of G. Zwara [10,11], we can also characterize the orbit closures of 2-nilpotent matrices in section 4.Finally, we study the conjugation action of upper-triangular matrices on arbitrary nilpotent matrices. We provide a generic normal form for the orbits of this action in section 5.1 and construct a large class of semiinvariants in section 5.2.Acknowledgments: The authors would like to thank K. Bongartz and A. Melnikov for valuable discussions concerning the methods and results of this work.
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL n (C) on the variety of x-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of 2-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. We show that in all non-finite cases, the corresponding quiver algebra is of wild representation type.
We consider a class of finite-dimensional algebras, the so-called "Staircase algebras" parametrized by Young diagrams. We develop a complete classification of representation types of these algebras and look into finite, tame (concealed) and wild cases in more detail. Our results are translated to the setup of graded nilpotent pairs for which we prove certain finiteness conditions.
The GL(V )-orbits in the enhanced nilpotent cone V × N (V ) are (essentially) in bijection with the orbits of a certain parabolic P ⊆ GL(V ) (the mirabolic subgroup) in the nilpotent cone N (V ). We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of the general linear group on the variety of nilpotent matrices in its Lie algebra. Lie-theoretically, it is natural to wonder about the number of orbits of this action. We translate the setup to a representation-theoretic one and obtain a finiteness criterion which classifies all actions with only a finite number of orbits over an arbitrary infinite field. These results are applied to commuting varieties and nested punctual Hilbert schemes.
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