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1.
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. 相似文献
2.
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. 相似文献
3.
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. 相似文献
4.
We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi?, M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
5.
Csaba Sz��nt�� 《Mathematische Zeitschrift》2011,269(3-4):833-846
We deduce using the Ringel?CHall algebra approach explicit formulas for the cardinalities of some Grassmannians over a finite field associated to the Kronecker quiver. We realize in this way a quantification of the formulas obtained by Caldero and Zelevinsky for the Euler characteristics of these Grassmannians. Finally we present a recursive algorithm for computing the cardinality of every Kronecker quiver Grassmannian over a finite field. 相似文献
6.
S. A. Kruglyak L. A. Nazarova A. V. Roiter 《Functional Analysis and Its Applications》2010,44(2):125-138
It is known that finitely representable quivers correspond to Dynkin graphs and tame quivers correspond to extended Dynkin
graphs. In an earlier paper, the authors generalized some of these results to locally scalar (later renamed to orthoscalar)
quiver representations in Hilbert spaces; in particular, an analog of the Gabriel theorem was proved. In this paper, we study
the relationships between indecomposable representations in the category of orthoscalar representations and indecomposable
representations in the category of all quiver representations. For the quivers corresponding to extended Dynkin graphs, the
indecomposable orthoscalar representations are classified up to unitary equivalence. 相似文献
7.
Bin Zhu 《Journal of Algebraic Combinatorics》2008,27(1):35-54
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using
d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations
of valued quivers, we define a d-compatibility degree (−∥−) on any pair of “colored” almost positive real Schur roots which generalizes previous definitions on the noncolored case
and call two such roots compatible, provided that their d-compatibility degree is zero. Associated to the root system Φ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost
positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is
the generalized cluster complex defined by Fomin and Reading.
Supported by the NSF of China (Grants 10471071) and by the Leverhulme Trust through the network ‘Algebras, Representations
and Applications’. 相似文献
8.
In this paper we discuss, in terms of quiver with relations, su?cient and necessary conditions for an algebra to be a quasitilted algebra. We start with an algebra with global dimension at most two and we give a su?cient condition to be a quasitilted algebra. We show that this condition is not necessary. In the case of a strongly simply connected schurian algebra, we discuss necessary conditions, and combining both types of conditions, we are able to analyze if some given algebra is quasitilted. As an application we obtain the quiver with relations of all the tilted and cluster tilted algebras of Dynkin type Ep. 相似文献
9.
Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras. 相似文献
10.
Sarah Scherotzke 《Algebras and Representation Theory》2017,20(1):231-243
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 ??. 相似文献
11.
We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector. 相似文献
13.
Ralf Schiffler 《Advances in Mathematics》2010,223(6):1885-244
We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras.Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type . 相似文献
14.
《Journal of Pure and Applied Algebra》2023,227(1):107156
We introduce a framework of translation quiver varieties which includes Nakajima quiver varieties as well as their graded and cyclic versions. An important feature of translation quiver varieties is that the sets of their fixed points under toric actions can be again realized as translation quiver varieties. This allows one to simplify quiver varieties in several steps. We prove that translation quiver varieties are smooth, pure and have Tate motivic classes. We also describe an algorithm to compute those motivic classes. 相似文献
15.
We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras. 相似文献
16.
We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed
in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams
(also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient
1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is
a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds
by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled”
versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other
half specializes to the ordinary case.
Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups
[EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag
manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double
Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide
a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert
polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05
]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials
in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in
our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del
Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass
from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from
there to our formula of Buch–Fulton type. 相似文献
17.
G. Cerulli Irelli 《Journal of Algebraic Combinatorics》2011,33(2):259-276
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). 相似文献
18.
Xiao Qin DENG Jiang Rong CHEN 《数学学报(英文版)》2005,21(3):613-622
The notion of generic extensions of representations of a Dynkin quiver plays a big role in the study of the structure of the corresponding quantum group. In this paper, we describe the generic extensions of a simple representation by any representation and that of any representation by a simple representation of a Dynkin quiver Q of type D. 相似文献
19.
Magnus Engenhorst 《Algebras and Representation Theory》2017,20(1):163-174
Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite-dimensional algebras with finitely many bricks. 相似文献
20.