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Register a new userComparison of quiver varieties, loop Grassmannians and nilpotent cones in type A
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In type A we find equivalences of geometries arising in three settings: Nakajimas (``framed) quiver varieties, conjugacy classes of matrices and loop Grassmannians. These are now all given by explicit formulas. As an application we provide a geometric version of symmetric and skew $(GL(m), GL(n))$ dualities.
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Following [20], a desingularization of arbitrary quiver Grassmannians for finite dimensional Gorenstein projective modules of 1-Gorenstein gentle algebras is constructed in terms of quiver Grassmannians for their Cohen-Macaulay Auslander algebras.
We determine which nilpotent orbits in $E_6$ have normal closure and which do not. We also verify a conjecture about small representations in rings of functions on nilpotent orbit covers for type $E_6$.
An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type $A$ the former is obtained by Frenkel-Kac construction from the latter.
We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.
Let $mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $mathfrak{g}$ over $mathbb C$, induced from a Levi subalgebra whose simple roots are orthogonal short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $overline{ mathcal{O}}$ in $S mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broers result for the subregular nilpotent orbit. Along the way we give another proof of Broers result that $overline{ mathcal{O}}$ is normal. We also prove a result connecting a property of invariants related to flat bases to the question of when one copy of the adjoint representation is in the ideal in $S mathfrak{g}^*$ generated by another copy of the adjoint representation and the fundamental invariants.
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