Articles in progress

Geometric characterizations of the representation type of hereditary algebras and canonical algebras.
The theta-stable decomposition for irreducible components of representation varieties.

Submitted articles

GIT-fans for quivers (pdf). arXiv:0805.1440 [math.RT]

ABSTRACT: In this paper we go over the construction of the GIT-fans for quivers without oriented cycles. We follow closely the steps outlined by Ressayre in "The GIT-Equivalence for G-Line Bundles" but we avoid Dolgachev-Hu's finitness theorem. Our arguments are based on King semi-stability criterion for quiver representations and Schofield's theory of general representations.

On orbit closures for infinite type quivers (pdf). arXiv:0709.3613 [math.RT]

ABSTRACT: In "Unibranch orbit closures in module varieties", Zwara proved that for quivers of types A, D, and E the orbit closures of quiver representations are unibranch varieties. We show that the converse is also true. Zwara has found a representation of the Kronecker quiver whose orbit closure is not unibranch. In this short note, we explain how to extend Zwara's example to all infinite type quivers without oriented cycles.

Refereed articles (accepted of published)

Cluster fans, stability conditions, and domains of semi-invariants, (2009), 19 pp, (pdf ). arXiv:0811.1290 [math.RT]
Transactions of the American Mathematical Society (to appear).

ABSTRACT: Given a quiver without oriented cycles, one can construct its cluster algebra, and hence, its cluster fan. In fact, the underlying combinatorics of a cluster algebra is governed by its (possibly infinite) cluster fan. The cluster fan of a Dynkin quiver turns out to be the normal fan of a polytope which can be regarded as a generalized version of the Stasheff polytope (it is the Stasheff polytope in type A) as shown by Chapoton, Fomin and Zelevinsky. However, not much is known about the cluster fan of an arbitrary quiver. In this paper we give an interpretation of the cluster fan of a quiver Q in terms of stability conditions and domains of semi-invariants of Q. Along the way, we give new proofs of Schofield's results on perpendicular categories and semi-invariants of quivers. We also explain how our results can be used to recover Igusa-Orr-Todorov-Weyman theorem on cluster fans and doamins of semi-invariants for Dynkin quivers.

Orbit semigroups and the representation type of quivers (pdf). arXiv:0708.3413 [math.RT]
Journal of Pure and Applied Algebra, 213 (2009), pp. 1418-1429, doi:10.1016/j.jpaa.2008.12.003.

ABSTRACT: It is an important and interesting task to find geometric characterizations of the representation type of a quiver (or more generally, of a finite-dimensional algebra). It this paper we show that a finite connected quiver is Dynkin or Euclidean if and only if the orbit semigroups of all of its representations are saturated. We also show that orbit semigroups are saturated in the thin sincere case.

Quivers, long exact sequences and Horn type inequalities II (pdf). arXiv:0805.1439 [math.RT]
Glasgow Mathematical Journal, 51 (2009), 1-17, doi:10.1017/S0017089508004631.

ABSTRACT: In this paper, we find necessary and sufficient Horn type inequalities for the existence of long exact sequences of finite abelian p-groups without zeros at the ends. As a particular case of our results, we recover Fulton's result on the eigenvalues of majorized Hermitian matrices.

Quivers, long exact sequences and Horn type inequalities (pdf). arXiv:math/0410423 [math.RT]
Journal of Algebra, 320 (2008), no. 1, 128-157.

ABSTRACT: In 1912, Weyl asked for a description of the eigenvalues of a sum of two Hermitian matrices in terms of the eigenvalues of the summands. In 1962, Horn recursively constructed a list of inequalities for the eigenvalues of two Hermitian matrices and their sum, which he conjectured to be necessary and sufficient. In 2000, Horn's conjecture was finally proved. Several other problems turn out to be related and have the exact same answer as Weyl's eigenvalue problem, including the non-vanishing of the Littlewood-Richardson coefficients and the existence of short exact sequences of finite abelian p-groups. We generalize these three problems. We obtain a list of necessary and sufficient inequalities for the existence of long exact sequences of m finite abelian p-groups, using methods from quiver invariant theory. We explain how this result is related to some generalized Littlewood-Richardson coefficients and to eigenvalues of Hermitian matrices satisfying certain (in)equalities.

Counterexamples to Okounkov's log-concavity conjecture (with Harm Derksen and Jerzy Weyman). math.RT/0610819
Compositio Mathematica, 143 (2007), 1545-1557.

ABSTRACT: Motivated by physical considerations, Okounkov conjectured that the Littlewood-Richardson coefficients are log-concave as a function of their highest weights. This conjecture, if true, would immediately imply the Knutson-Tao saturation theorem, a conjecture of Fulton proved by Belkale, and the log-concavity theorem for skew-Schur functions proved by Lam-Postnikov-Pylyavskyy. As it turns out, Okounkov's conjecture can be reformulated in terms of the more general language of quiver theory. In fact, it is the rich combinatorics and geometry of quiver representations that helps to see why Okounkov's conjecture is bound to fail and find explicit counterexamples.

Eigenvalues of Hermitian matrices and cones arising from quivers (pdf).
International Mathematics Research Notices 2006, Art. ID 59457, 27 pp.

ABSTRACT: Buch, answering a question raised by Barvinok, has showed that the set of the possible eigenvalues of Hermitan matrices with positive semi-definite sum of bounded rank is a rational convex polyhedral cone and found its facets. In this paper, we bring this problem into the general framework of quiver theory and give a new proof of Buch's result. Moreover, we compute the dimension of the cone in question and find its lattice points. Our description of the lattice points generalizes the Knutson-Tao saturation theorem for Littlewood-Richardson coefficients.