Preprojective algebras and MV polytopes

@article{Baumann2010PreprojectiveAA,
  title={Preprojective algebras and MV polytopes},
  author={Pierre Baumann and Joel Kamnitzer},
  journal={arXiv: Representation Theory},
  year={2010}
}
The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver. 
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References

SHOWING 1-10 OF 35 REFERENCES
Modules with 1-Dimensional Socle and Components of Lusztig Quiver Varieties in Type A
We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they
Geometric construction of crystal bases
We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As
Crystal bases and quiver varieties
Abstract. We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One can show that, as a crystal, it is isomorphic to the crystal base of
A polytope calculus for semisimple groups
We define a collection of polytopes associated to a semisimple group G. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain
The Preprojective Algebra of a Quiver
The preprojective algebra P k (Q) of a quiver Q plays an important role in mathematics. We are going to present some descriptions of these algebras and their module categories which seem to be
Mirkovic-Vilonen cycles and polytopes
We give an explicit description of the Mirkovie-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We
Total positivity in Schubert varieties
Abstract. We extend the results of [2] on totally positive matrices to totally positive elements in arbitrary semisimple groups.
Representation Theory of Artin Algebras: Contents
1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary
Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras
Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of
On the exceptional fibres of Kleinian singularities
We give a new proof, avoiding case-by-case analysis, of a theorem of Y. Ito and I. Nakamura which provides a module-theoretic interpretation of the bijection between the irreducible components of the
...
...