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Conjugation on varieties of nilpotent matrices / vorgelegt von Magdalena Boos. 2012
Inhalt
Introduction
Theoretical background
Methods from Algebraic geometry and Invariant theory
Algebraic group actions
Invariants and algebraic quotients
Semi-invariants and GIT-quotients
Toric varieties
Representation theory of finite-dimensional algebras
Covering theory of quiver algebras
Tame and wild algebras
Degenerations
The concrete setup
(Oriented) Link patterns
Known results
Results of M. Jordan and M. Gerstenhaber
Results of A. Melnikov
Results of L. Hille and G. Röhrle
Representation-theoretic approach
Nilpotency degree 2
Classification of the orbits
Parabolic orbits
Unipotent orbits
The examples n=3 and n=4
Homomorphisms, extensions and their combinatorial interpretation
Closures of Borel orbits
Minimal, disjoint pieces of degenerations
Minimal degenerations in general
Dimensions and the open orbit
Minimal singularities
Closures of parabolic orbits
Minimal, disjoint degenerations
Dimensions of orbits
Finite classifications in higher nilpotency degrees
Maximal parabolic action for x=3
Classification of the orbits
Orbit closures
Dimensions and the open orbit
The parabolic subgroup of block sizes (2,2)
A finiteness criterion
A wildness criterion
The representation type of the corresponding algebras
Concrete 2-parameter families via tree modules
Generic classification in the nilpotent cone
Generic normal forms
(Semi-) Invariants
Generation of semi-invariant rings
Towards an algebraic U-quotient of the nilpotent cone
A quotient criterion for unipotent actions
The examples n=2 and n=3
Toric invariants
Reductions
General description of toric invariants
Generic separation of the U-orbits
The toric variety X
Toric operation(s) on the U-invariant ring
Explicit description of toric invariants
Interrelation between N//U and X
The case n=4
Towards a GIT-quotient for the Borel action
The examples n=2 and n=3
Generic separation of the same weight
Translation to the language of quiver moduli
Appendix
Singular computations
The parabolic subgroup of block sizes (4,3)
Bibliography