The $\mathrm{GL}(V)$-orbits in the enhanced nilpotent cone $V\times\mathcal{N}(V)$ are (essentially) in bijection with the orbits of a certain parabolic $P\subseteq\mathrm{GL}(V)$ (the mirabolic subgroup) in the nilpotent cone $\mathcal{N}(V)$. We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.
We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Finally we compute the number of points over a finite field of the various s…
We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the numb…
This paper presents analogous results of Hua [7][8] on numbers of representations of quivers over finite fields which respect nilpotent relations under certain assumptions. A closed formula which counts isomorphism classes of absolutely indecomposable representations with given dimension vectors is given and a $q$-deformation of Weyl-Kac denominator identity is established. In principle, if the numbers of representations are known, then the numbers of isomorphism classes of absolutely indecomposable representations are known.
Generalizing Schubert cells in type A and a cell decomposition if Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of an oriented cycle admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation and describe the T-equivariant cohomology using Goresky-Kottwitz-MacPherson-theory. 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 …
For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse-Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of $\mbox{A}$-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of $\mbox{A}_1$. For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are…
We obtain an explicit combinatorial formula for certain parabolic Kostka-Shoji polynomials associated with the cyclic quiver, generalizing results of Shoji and of Liu and Shoji.
We describe the combinatorics of the multisemigroup with multiplicities for the tensor category of subbimodules of the identity bimodule, for an arbitrary non-uniform orientation of a finite cyclic quiver.
We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $\tt A$ flag varieties, their linear degenerations, finite dimensional approximations of the $GL_n$-affine flag variety and affine Grassmannian. We show that these quiver Grassmannians, equipped with our torus action, are GKM-varieties and that their moment graph admits a combinatorial description in terms of coefficients quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able …
If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's resolution quiver. In [IZ92], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example tha…
The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot. The s…
We calculate the additive structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types ADE.
In this paper, we give a recursive formula for the interesting PBW basis $E_{A}$ of composition subalgebras of Ringel-Hall algebras $\fkH_\vartri(n)$ of cyclic quivers after \cite{DengDuXiao2007generic}, and another construction of canonical bases of $\U_v^+(\wih{\fks\fkl}_n)$ from the monomial bases $m^{(A)}$ follow \cite{DuZhaomultiplication}. As an application, we will determined all the canonical basis of $\U_v^+(\wih{\fks\fkl}_2)$ associated with modules of Lowery length $\leqs3$. Finally, we will discuss the relation of canonical bases of Ringel-Hall algebras and those of affine quantum …
We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig's bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland's recent work for type ADE.
This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Maffei's theorem relating Slodowy slices to Nakajima quiver varieties in type A. There is one new observation: the results of Juteau and Mautner, combined with Maffei's theorem, give a geometric proof of a result on decomposition numbers of Schur algebras due to Fang, Henke and Koenig.
Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(Λ_0)$ of ${\mathscr D}(Δ_n)$ of the cyclic quiver $Δ_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $Δ_n$, we obtain a decomposition of the basic representation $L(Λ_…
In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $\overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawa's derived equivalence between crepant resolutions $Y$ and $Y^+$ of $\ove…
A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\mathcal{C}_{Q,m}$, are determined up to isomorphism by this invariant. We use …
In previous work we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups and we investigate this permutation action via GKM techniques. In the case of (type A) flag varieties, or Schubert varieties therein, we recover Tymoczko's results on permutation representations.
We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that a $2d$-dimensional triangulation has no interior $(d + 1)$-simplices if and only if its quiver is a cut quiver of type $A$, in the sense of Iyama and Oppermann. This is a higher-dimensional generalisation of the fact that triangulations of polygons with no interior triangles correspond to orientations of an $A_{n}$ Dynkin diagram. An application of this fir…
Let $Q$ be a finite type quiver i.e. ADE Dynkin quiver. Denote by $Λ$ its preprojective algebra. It is known that there are finitely many indecomposable $Λ$-modules if and only if $Q$ is of type $A_1,A_2,A_3,A_4$. In this paper, extending Lusztig's construction of $U\frak{n}_+$, we study an algebra generated by these indecomposable submodules. It turns out that it forms the universal enveloping algebra of some nilpotent Lie algebra inside the function algebra on Lusztig's nilpotent scheme. The defining relations of the corresponding nilpotent Lie algebra for type $A_1, A_2,A_3,A_4$ are given h…
We generalize Lusztig's nilpotent varieties, and Kashiwara and Saito's geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac-Moody algebras.
