We derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver. 2010 Mathematics Subject Classification: Primary 16G20, secondary 05E05, 14N35

We study the relationship between two stratifications on parameter spaces for coherent sheaves and for quiver representations: a stratification by Harder–Narasimhan types and a stratification arising from the geometric invariant theory construction of the associated moduli spaces of semistable objects. For quiver representations, both stratifications coincide, but this is not quite true for she...

This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of moduli spaces of quiver representations and derived categories arising in toric geometry. The first main goal is to present the noncommutative geometric approach to semiprojective toric varieties via qu...

We compute t–analogs of q–characters of all l–fundamental representations of the quantum affine algebras of type E (1) 6 , E (1) 7 , E (1) 8 by a supercomputer. In particular, we prove the fermionic formula for Kirillov-Reshetikhin modules conjectured by Hatayama et al. [6] for these classes of representations. We also give explicitly the monomial realization of the crystal of the corresponding...

We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of w...

Motivated by a recent conjecture by Hernandez and Leclerc [30], we embed a Fomin-Zelevinsky cluster algebra [20] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties [48]. Graded quiver varieties controlling the image can be identified wi...

We study the proposed duality between the 5-dimensional supergravity/superstring on AdS3×S and the 2-dimensional N = (0, 4) SCFT defined on the boundary of AdSspace. We construct explicitly the N = (0, 4) SCFT by imposing the “quiver projection” developed by Douglas-Moore [14] on the N = (4, 4) SCFT of symmetric orbifold, which is proposed to be the dual of the 6-dimensional supergravity/supers...

Using a result of T. P. Bidigare [Bidigare, 1997], we identity the descent algebra Σk(W ) (over a field k) of a finite Coxeter group W with a subalgebra of kF , an algebra built from the hyperplane arrangement associated to W . Specifically, Σk(W ) is anti-isomorphic to the W -invariant subalgebra (kF) . We use this identification and results about kF to study Σk(W ). We construct a complete sy...