Signed quivers, symmetric quivers, and root systems

Kac 多项式沿根串的稳定性

Stabilization of Kac polynomials along root strings

本文研究固定箭图表示模空间上同调群的稳定化，在温和条件下证明添加足够大的维度向量倍数后维度稳定，并给出显式公式及 Kac 多项式系数的稳定性。

Stratifications of Marsden-Weinstein reductions for representations of quivers and deformations of symplectic quotient singularities

We investigate the Poisson geometry of the Marsden-Weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver. We show that the stratification by representation type equals the stratification by symplectic leaves. The deformed symplectic quotient singularities - spectra of centres of symplectic reflection algebras associated to a wreath product - are shown to be isomorphic to reductions of a certain quiver. This establishes a method to calculate when these deformations are smooth. Furthermore, the isomorphism identifies symplectic leav…

Quiver Grassmannians of extended Dynkin type D - Part 2: Schubert decompositions and F-polynomials

Extending the main result of Part 1, in the first part of this paper we show that every quiver Grassmannian of a representation of a quiver of extended Dynkin type $D$ has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. $F$-polynomials). Al…

Tree normal forms for quiver representations

We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables of smaller dimension vector. This is then specialized to the situation of tree modules, where the existence of a special basis simplifies computations and gives nicer normal forms. Motivated by a conjecture of Kac, we use this to construct cells of indecomposable representations as deformations of tree modules. The second part of the paper develops geometric …

Birational classification of moduli spaces of representations of quivers

Let αbe a Schur root; let h=hcf_v(α(v)) and let p = 1 - < α/h,α/h >. Then a moduli space of representations of dimension vector αis birational to p h by h matrices up to simultaneous conjugacy. Therefore, if h=1,2,3 or 4, then such a moduli space is a rational variety and if h divides 420 it is a stably rational variety.

Non-Schurian indecomposables via intersection theory

For an acyclic quiver with three vertices, we consider the canonical decomposition of a non-Schurian root and associate certain representations of a generalized Kronecker quiver. These representations correspond to points contained in the intersection of two subvarieties of a Grassmannian and give rise to representations of the original quiver, preserving indecomposability. We show that these subvarieties intersect using Schubert calculus. Provided that the intersection contains a Schurian representation, it already contains an open subset of Schurian representations whose dimension is what we…

Counting invariants for the ADE McKay quivers

We consider the moduli space of the McKay quiver representations associated to the binary polyhedral groups G < SU(2) < SU(3). The derived category of such representations is equivalent to the derived category of coherent sheaves on the corresponding ADE resolution Y=G-Hilb(C^3). Following the ideas of Nagao and Nakajima, by making particular choices of parameters in the space of stability conditions on the equivalent derived categories above, we recover Donaldson-Thomas (DT), Pandharipande-Thomas (PT) and Szendroi (NCDT) moduli spaces. We also compute the Gromov-Witten (GW) partition function…

单位根处的箭图簇量子K理论

On the Quantum K-theory of Quiver Varieties at Roots of Unity

本文证明了Nakajima簇的量子差分方程基本解矩阵在单位根处无极点，并给出了量子连接p-曲率谱的显式描述。

根多面体、流多面体与序多面体

Root polytopes, flow polytopes, and order polytopes

本文研究由箭图编码的根多面体，证明其强连通时的自反终端性、平面时与流多面体的极对偶性，以及秩偏序集情形下与标记偏序集多面体的对偶关系，还给出相关环面簇的皮卡群描述与Hibi环面簇的小奇点消解，可应用于镜像对称。

Low dimensional orders of finite representation type

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders [CI05]. These were classified independently by Artin (in terms of ramification data) and Reiten-Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $G \subset \mathrm{GL}_2$, explicitly computing $H^2(G,k^*)$, and then matching…