Quantum cluster characters for valued quivers

These notes contain an extended abstract and references for a minicourse given at the Summer School 'Microlocal and Geometric Methods in Representation Theory' (Schloss Reisensburg, July 2006) of the International Research Training Group Metz-Paderborn. 1. History

Annales de l’institut Fourier, 2011

Introduction 1.1. Main results 1.2. Horn's conjecture and related problems 1.3. The quiver method 1.4. Organization 2. Preliminaries 2.1. Basic notions for quivers 2.2. Semi-invariants for quiver representations 2.3. Representations in general position 2.4. The canonical decomposition 2.5. The combinatorics of dimension vectors 2.6. Perpendicular categories 2.7. Exceptional Sequences 2.8. Stability and GIT-quotients 3. Stability 3.1. Harder-Narasimhan and Jordan-Hölder filtrations 3.2. The σ-stable decomposition 4. Schur sequences 5. The faces of the cone R + Σ(Q, α) 6. More on the σ-stable decompositions 6.1. The set of σ-stable dimension vectors 6.2. σ-stable decomposition for quivers with oriented cycles 7. Littlewood-Richardson coefficients 7.1. The Klyachko cone 7.2. Walls of the Klyachko cone 7.3. Faces of the Klyachko cone of arbitrary codimension 7.4. A multiplicative formula for Littlewood-Richardson coefficients Appendix: Belkale's proof of Fulton's conjecture References

Journal of Algebra, 2000

dedicated to professor shaoxue liu on the occasion of his 70th birthday By counting the numbers of isomorphism classes of representations (indecomposable or absolutely indecomposable) of quivers over finite fields with fixed dimension vectors, we obtain a multi-variable formal identity. If the quiver has no edge-loops, this identity turns out to be a q-analogue of the Kac denominator identity modulus a conjecture of Kac.

Algebra & Number Theory, 2011

We study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we find a geometric realization of the "canonical basis' ' of cluster algebras of type A

Representation Theory – Current Trends and Perspectives

The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero-Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin.

2007

It is well known that the intersection multiplicities of Schubert classes in the Grassmannian are Littlewood-Richardson coefficients. We generalize this statement in the context of quiver representations. Here the intersection multiplicity of Schubert classes is replaced by the number of subrepresentations of a general quiver representation, and the Littlewood-Richardson coefficients are replaced by the dimension of a certain space of semi-invariants.

International Mathematics Research Notices, 2018

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.

Bulletin of the London Mathematical Society, 2013

Let (Q, W) be a quiver with a non degenerate potential. We give a new description of the c-vectors of Q. We use it to show that, if Q is mutation equivalent to a Dynkin quiver, then the set of positive c-vectors of the cluster algebra associated to Q op coincides with the set of dimension vectors of the indecomposable modules over the Jacobian algebra of (Q, W).

Compositio Mathematica, 2009

We situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

Mathematische Zeitschrift, 2019

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.