Moduli of McKay quiver representations I: the coherent component

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.

Download Full-text

Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2017-0010 ◽

2019 ◽

Vol 2019 (754) ◽

pp. 143-178 ◽

Cited By ~ 5

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.

Download Full-text

Combinatorics of Exceptional Sequences in Type A

The Electronic Journal of Combinatorics ◽

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.

Download Full-text