Rationality of Rigid Quiver Grassmannians

Algebras and Representation Theory - We show that any quiver Grassmannian associated with a rigid representation of a quiver is a rational variety using torus localization techniques.     

Quiver Grassmannians associated with string modules

We provide a technique to compute the Euler–Poincaré 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–Poincaré characteristic of quiver Grassmannians associated with indecomposable pre-projective, pre-injective and regular homogeneous representations of an affine quiver of type [(A)\tilde]_p,1\tilde{A}_{p,1}. For p=1, this approach provides another proof of a result due to Caldero and Zelevinsky (in Mosc. Math. J. 6(3):411–429, 2006).     

Homological Algebra of Twisted Quiver Bundles

Several important cases of vector bundles with extra structure(such as Higgs bundles and triples) may be regarded as examplesof twisted representations of a finite quiver in the categoryof sheaves of modules on a variety/manifold/ringed space. Itis shown that the category of such representations is an abeliancategory with enough injectives by the construction of an explicitinjective resolution. Using this explicit resolution, a longexact sequence is found that computes the Ext groups in thisnew category in terms of the Ext groups in the old category.The quiver formulation is directly reflected in the form ofthe long exact sequence. It is also shown that under suitablecircumstances, the Ext groups are isomorphic to certain hypercohomologygroups.     

Desingularization of Quiver Grassmannians via Nakajima Categories

In this paper, we show that generalized Nakajima Categories provide a framework to construct a desingularization of quiver Grassmannians for self-injective algebras of finite representation type. Furthermore, we show that all standard Frobenius models of orbit categories of the bounded derived category considered in Keller, Documenta Math. 10: 551–581, 2005 are equivalent to proj ??, the finitely generated projective modules of the regular Nakajima category ??.     

Desingularization of Quiver Grassmannians for Gentle Algebras

In (Cerulli Irelli et al., Adv. Math. 245(1) 182–207 2013), Cerulli Irelli-Feigin-Reineke construct a desingularization of quiver Grassmannians for Dynkin quivers. Following them, a desingularization of arbitrary quiver Grassmannians for finite dimensional Gorenstein projective modules of 1-Iwanaga-Gorenstein gentle algebras is constructed in terms of quiver Grassmannians for their Cohen-Macaulay Auslander algebras.     

Degenerate Affine Flag Varieties and Quiver Grassmannians

We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.

     

On the Cohomology of Quiver Grassmannians for Acyclic Quivers

Algebras and Representation Theory - For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra $U_{q}^{-}(mathfrak...     

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

     

Euler Characteristics of Quiver Grassmannians and Ringel-Hall Algebras of String Algebras

We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G. Cerulli Irelli (2010). As an application we consider the Ringel-Hall algebra ${\mathcal C}(A)$ of some string algebras A and compute in combinatorial terms the products of arbitrary functions in ${\mathcal C}(A)$ . These results are transferred to covering theory.     

McKay箭图

Using the connection between McKay quiver and Loewy matrix,and the properties of characteristic polynomial of Loewy matrix,we give a generalized way to determine the McKay quiver for a finite subgroup of a generalized linear group.     

A Non-expensive Homological Helly Theorem

We will discuss the following result: for a topological space X with the property that $H_k(U)=0$ for $k⩾d$ and every open subset U of X, a finite family of open sets in X has nonempty intersection if for any subfamily of size j, $1⩽j⩽d+1$, the ($d-j$)-dimensional reduced homology group of its intersection is zero. We also use this theorem to discuss new results concerning transversal affine planes to families of convex sets.     

Cell Decompositions of Quiver Flag Varieties for Nilpotent Representations of the Cyclic Quiver

Generalizing Schubert cells in type A and a cell decomposition of Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of a cyclic quiver admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation with finitely many fixpoints. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules.     

A Homological Version of the Implicit Function Theorem

Our objective in this paper is to prove an Implicit Function Theorem for general topological spaces. As a consequence, we show that, under certain conditions, the set of the invertible elements of a topological monoid X is an open topological group in X and we use the classical topological group theory to conclude that this set is a Lie group.     

Homological characterization of the unknot

Given a knot K in the 3-sphere, let Q_K be its fundamental quandle as introduced by Joyce. Its first homology group is easily seen to be . We prove that H₂(Q_K)=0 if and only if K is trivial, and whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components.More detailed information can be derived from the conjugation quandle: let Q_K^π be the conjugacy class of a meridian in the knot group . We show that , where p is the number of prime summands in a connected sum decomposition of K.     

A normal form for curves in Grassmannians

The purpose of this paper is to give an elementary proof of Griffiths' and Harris' normal form lemma [4, p.385]. The author thanks Gerd Fischer for his encouragement and support This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag.     

Secants of Lagrangian Grassmannians

We study the dimensions of secant varieties of Grassmannian of Lagrangian subspaces in a symplectic vector space. We calculate these dimensions for third and fourth secant varieties. Our result is obtained by providing a normal form for four general points on such a Grassmannian and by explicitly calculating the tangent spaces at these four points.