Quiver Representations and Quiver Varieties

2016 ◽  
Author(s):  
Alexander Kirillov
Keyword(s):  
Quiver Representations ◽  
Quiver Varieties

scholarly journals Group actions on quiver varieties and applications

2019 ◽  
Vol 30 (02) ◽  
pp. 1950007 ◽  
Author(s):  
Victoria Hoskins ◽  
Florent Schaffhauser
Keyword(s):  
Finite Groups ◽  
Moduli Spaces ◽  
Group Actions ◽  
Group Cohomology ◽  
Quiver Representations ◽  
Quiver Varieties ◽  
Algebraic Actions

We study algebraic actions of finite groups of quiver automorphisms on moduli spaces of quiver representations. We decompose the fixed loci using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperkähler quiver varieties, as fixed loci of such actions.

scholarly journals On the equations and classification of toric quiver varieties

2016 ◽  
Vol 146 (2) ◽  
pp. 265-295 ◽  
Author(s):  
Mátyás Domokos ◽  
Dániel Joó
Keyword(s):  
Projective Space ◽  
Moduli Spaces ◽  
Toric Variety ◽  
Projective Variety ◽  
Homogeneous Ideal ◽  
Quiver Representations ◽  
Canonical Embedding ◽  
Quiver Varieties ◽  
Fixed Dimension

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.

scholarly journals On Generalized Minors and Quiver Representations

2018 ◽  
Vol 2020 (3) ◽  
pp. 914-956 ◽  
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Finite Type ◽  
Dynkin Diagram ◽  
Cluster Algebra ◽  
Coordinate Ring ◽  
Coxeter Element ◽  
Quiver Representations ◽  
Group Representations ◽  
Weight Group ◽  
Highest Weight ◽  
Bruhat Cell

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

scholarly journals Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

2019 ◽  
Vol 2019 (754) ◽  
pp. 143-178 ◽  
Author(s):  
Sven Meinhardt ◽  
Markus Reineke
Keyword(s):  
Moduli Space ◽  
Stability Condition ◽  
Intersection Cohomology ◽  
Quiver Representations ◽  
Compactly Supported ◽  
Generic Stability ◽  
Relative Version ◽  
Zero Potential ◽  
Stable Locus

Abstract The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

scholarly journals Quantum groups via cyclic quiver varieties I

2015 ◽  
Vol 152 (2) ◽  
pp. 299-326 ◽  
Author(s):  
Fan Qin
Keyword(s):  
Quantum Groups ◽  
Bilinear Form ◽  
Canonical Basis ◽  
Natural Basis ◽  
Grothendieck Ring ◽  
Quiver Varieties ◽  
Enveloping Algebra ◽  
Dual Canonical Basis ◽  
Quantized Enveloping Algebra ◽  
Cyclic Quiver

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ 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 $\mathbb{A}\mathbb{D}\mathbb{E}$.

scholarly journals Combinatorics of Exceptional Sequences in Type A

10.37236/6251 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Alexander Garver ◽  
Kiyoshi Igusa ◽  
Jacob P. Matherne ◽  
Jonah Ostroff
Keyword(s):  
Dynkin Diagram ◽  
Type A ◽  
Linear Extensions ◽  
Quiver Representations ◽  
Noncrossing Partitions ◽  
Underlying Graph ◽  
Exceptional Sequences

Exceptional sequences are certain sequences of quiver representations.  We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions. 

scholarly journals On the number of points of nilpotent quiver varieties over finite fields

2020 ◽  
Vol 53 (6) ◽  
pp. 1501-1544
Author(s):  
Tristan BOZEC ◽  
Olivier SCHIFFMANN ◽  
Eric VASSEROT
Keyword(s):  
Finite Fields ◽  
Quiver Varieties
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