42 research outputs found
Quiver Grassmannians associated with string modules
We provide a technique to compute the Euler characteristic of a class of
projective varieties called quiver Grassmannians. This technique applies to
quiver Grassmannians associated with "orientable string modules". As an
application we explicitly compute the Euler characteristic of quiver
Grassmannians associated with indecomposable preprojective, preinjective and
regular homogeneous representations of an affine quiver of type
. For , this approach provides another proof of a result
due to P. Caldero and A. Zelevinsky in \cite{CZ}.Comment: Minor changes. Accepted at the Journal Of Algebraic Combinatoric
Geometry of quiver Grassmannians of Kronecker type and canonical basis of cluster algebras
We study quiver Grassmannians associated with indecomposable representations
of the Kronecker quiver. We find a cellular decomposition of them and we
compute their Betti numbers. As an application, we give a geometric realization
of the "canonical basis" of cluster algebras of Kronecker type (found by
Sherman and Zelevinsky) and of type .Comment: 21 page
Degenerate flag varieties of type A and C are Schubert varieties
We show that in type A or C any degenerate flag variety is in fact isomorphic
to a Schubert variety in an appropriate partial flag manifold.Comment: The new version includes an appendix where we discuss
desingularizations. 14 page
Desingularization of quiver Grassmannians for Dynkin quivers
A desingularization of arbitrary quiver Grassmannians for representations of
Dynkin quivers is constructed in terms of quiver Grassmannians for an algebra
derived equivalent to the Auslander algebra of the quiver.Comment: 22 pages; typos corrected; section 7 restructured, improved and
corrected to take care of reducible quiver Grassmannian
Homological approach to the Hernandez-Leclerc construction and quiver varieties
In a previous paper the authors have attached to each Dynkin quiver an
associative algebra. The definition is categorical and the algebra is used to
construct desingularizations of arbitrary quiver Grassmannians. In the present
paper we prove that this algebra is isomorphic to an algebra constructed by
Hernandez-Leclerc defined combinatorially and used to describe certain graded
Nakajima quiver varieties. This approach is used to get an explicit realization
of the orbit closures of representations of Dynkin quivers as affine quotients.Comment: 12 page
Schubert Quiver Grassmannians
Quiver Grassmannians are projective varieties parametrizing subrepresentations
of given dimension in a quiver representation. We define a class of quiver Grassmannians
generalizing those which realize degenerate flag varieties. We show that each irreducible
component of the quiver Grassmannians in question is isomorphic to a Schubert variety.We
give an explicit description of the set of irreducible components, identify all the Schubert
varieties arising, and compute the Poincar´e polynomials of these quiver Grassmannians
Degenerate flag varieties and Schubert varieties: a characteristic free approach
We consider the PBW filtrations over the integers of the irreducible highest
weight modules in type A and C. We show that the associated graded modules can
be realized as Demazure modules for group schemes of the same type and doubled
rank. We deduce that the corresponding degenerate flag varieties are isomorphic
to Schubert varieties in any characteristic.Comment: 23 pages; A few typos corrected; Authors affiliation adde
Parabolic orbits of -nilpotent elements for classical groups
We consider the conjugation-action of the Borel subgroup of the symplectic or
the orthogonal group on the variety of nilpotent complex elements of nilpotency
degree in its Lie algebra. We translate the setup to a
representation-theoretic context in the language of a symmetric quiver algebra.
This makes it possible to provide a parametrization of the orbits via a
combinatorial tool that we call symplectic/orthogonal oriented link patterns.
We deduce information about numerology. We then generalize these
classifications to standard parabolic subgroups for all classical groups.
Finally, our results are restricted to the nilradical.Comment: comments welcom