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1.
2.
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
3.
Auslander-Reiten triangles and quivers are introduced into algebraic
topology.
It is proved that the existence of Auslander-Reiten triangles characterizes Poincaré
duality spaces, and that the Auslander-Reiten quiver is a weak
homotopy invariant.
The theory is applied to spheres whose Auslander-Reiten triangles and quivers are
computed. The Auslander-Reiten quiver over the $d$-dimensional sphere turns out to
consist of $d-1$ copies of ${\mathbb Z} A_{\infty}$. Hence the quiver is a
sufficiently sensitive invariant to tell spheres of different
dimension apart. 相似文献
4.
N. Ressayre 《Mathematische Zeitschrift》2012,270(1-2):263-275
In this work, we improve results of (Ressayre in Geometric invariant theory and generalized eigenvalue problem II, pp 1–25 2008; Ressayre in Ann. Inst. Fourier. 180:389–441 2010) on GIT-cones associated to the action of a reductive group G on a projective variety X. These results are applied to give a short proof of the Derksen–Weyman theorem that parametrizes bijectively the faces of a rational cone associated to any quiver without oriented cycles. An important example of such a cone is the Horn cone. 相似文献
5.
R Parthasarathy 《Proceedings Mathematical Sciences》1988,98(2-3):187-214
Given a finite quiver without oriented cycles, we describe a family of algebras whose module category has the same derived category as that of the quiver algebra. This is done in the more general setting oft-structures in triangulated categories. A completeness result is shown for Dynkin quivers, thus reproving a result of Happel [H]. 相似文献
6.
Peter Jørgensen 《Algebras and Representation Theory》2006,9(4):323-336
In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincaré duality space, each component of the Auslander–Reiten quiver is isomorphic to .
Presented by Yuri Drozd 相似文献
7.
Kazushi Ueda 《Proceedings of the American Mathematical Society》2008,136(8):2745-2747
We prove there is an equivalence of derived categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations, which is obtained from a McKay quiver by removing one vertex and half of the arrows. This result produces examples of distinct quivers with relations which have equivalent derived categories of representations.
8.
Carl Fredrik Berg Adam-Christiaan van Roosmalen 《Journal of Pure and Applied Algebra》2010,214(7):1082-257
Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to for a so-called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form ZQ. 相似文献
9.
D.A. Shmelkin 《Transformation Groups》2007,12(1):153-173
We suggest a geometrical approach to the semi-invariants of quivers based on Luna's slice theorem and the Luna-Richardson
theorem. The locally semi-simple representations are defined in this spirit but turn out
to be connected with stable representations in the sense of GIT, Schofield's perpendicular categories, and Ringel's regular
representations. As an application of this method we obtain an independent short proof of a theorem of Skowronski and Weyman
about semi-invariants of the tame quivers. 相似文献
10.
A notion of a mixed representation of a quiver can be derived from ordinary quiver representation by considering the dual
action of groups on "vertex" vector spaces together with their usual action. A generating system for the algebra of semi-invariants
of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers
of a special form and using a function DP on three matrices, which is a mixture of the determinant and two pfaffians. 相似文献
11.
Raf Bocklandt 《Journal of Pure and Applied Algebra》2008,212(1):14-32
In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist. 相似文献
12.
13.
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
14.
Sarah Scherotzke 《Journal of Pure and Applied Algebra》2011,215(3):232-241
We analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quivers are determined. We also determine components that contain shift periodic complexes. 相似文献
15.
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
16.
We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework
allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching
generalization of Bernstein–Gelfand–Ponomarev reflection functors. The motivations for this work come from several sources:
superpotentials in physics, Calabi–Yau algebras, cluster algebras.
相似文献
17.
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
18.
Yu-Han Liu 《Algebras and Representation Theory》2014,17(2):593-621
We study natural Grothendieck topologies on categories of quivers without and with relations, prove descent theorems for quiver representations, and introduce the notion of torsors over quivers. 相似文献
19.
Cristina Di Trapano 《代数通讯》2013,41(11):4357-4373
We give a new short proof of Skowroński and Weyman's theorem about the structure of the algebras of semi-invariants of Euclidean quivers, in the case of quivers without oriented cycles and in characteristic zero. Our proof is based essentially on Derksen and Weyman's result about the generators of these algebras and properties of Schofield semi-invariants. 相似文献
20.
In this paper we characterize when the path ring associated to a quiver is Gorenstein (in the sense of Iwanaga [9]). Then,
by using the notion of a Gorenstein category (cf. [2]), we extend the classes of quivers whose corresponding category of representations
has finite Gorenstein global dimension. This extension includes non-noetherian quivers.
E. E., S.E., and J.R.G.R., partially supported by the DGI MTM2005-03227.
Estrada’s work was supported by a MEC/Fulbright grant from the Spanish Secretaría de Estado de Universidades e Investigación
del Ministerio de Educación y Ciencia.
Received: 28 February 2006 相似文献