Current support: Simons Foundation Grant, Award Number 711639 [2020-2025]. Past support: NSA grant H98230-15-1-0022 [2015-2017] and NSF grant DMS-1101383 [2011-2015].
My research interests lie at the interface of representations theory of finite-dimensional algebras and geometric invariant theory (``GIT''). The papers below focus on theoretical aspects of quiver invariant theory, as well as applications to Brascamp-Lieb theory in harmonic analysis, Edmonds problem in algebraic complexity, and simultaneous robust subspace recovery in machine learning.
Background: The motivation in this project goes back to the celebrated Brascamp-Lieb (BL) inequalities in harmonic analysis. The optimal constants
in the BL inequalities, also known as BL constants, permeate many areas of mathematics and computer science. The fundamental work of J. Bennett, A. Carbery, M. Christ,
and T. Tao hints at a relationship between BL constants and semi-stability of quiver representations.
In joint work with H. Derksen, we have developed a quiver invariant theoretic approach to BL theory by introducing the capacity of a quiver datum (V,σ), denoted by cap(V, σ), where V is a representation and σ is a weight (i.e. an assignment of integers to the vertices) of a quiver Q. When Q is the m-subspace quiver, the capacity of quiver data is intimately related to the BL constants that occur in the m-multilinear BL inequalities.
Our first result states that cap(V, σ) > 0 if and only if V is σ-semi-stable. This has important consequences for quiver theory since it yields a deterministic polynomial time algorithm for checking semi-stability of quiver representations. The importance of the existence of such an algorithm stems from the fact that, in general, quiver semi-stability requires one check a number of linear homogeneous inequalities that can grow exponentially.
We have obtained a string of other structural results, including a character formula for the capacity/BL constants of quiver data. Our main tool is a quiver version of a celebrated result of Kempf-Ness on closed orbits in invariant theory. This result leads us to consider certain real algebraic varieties, denoted by G(σ,V), that carry information relevant to our main objects of study. It allows us to express cap(V, σ) in terms of the character induced by σ, denoted by χ, and any point A of G(σ,V) as follows:
A central problem in algebraic complexity, posed by J. Edmonds, asks to decide if the span of a given l-tuple of NxN complex matrices
contains a non-singular matrix.
In this paper, we provide a quiver invariant theoretic approach this problem. Viewing the l-tuple of matrices as a representation W of the l-Kronecker quiver, Edmonds problem can be rephrased as asking to decide if there exists a semi-invariant of weight (1,-1) that does not vanish at W. In other words, Edmonds problem is asking to decide if the weight (1,-1) belongs to the orbit semigroup of W.
Let Q be an arbitrary acyclic quiver and W a representation of Q. We study the membership problem for the orbit semi-group of W by focusing on the so-called W-saturated weights. We first show that for any given W-saturated weight σ, checking if σ belongs to the orbit semigroup of W can be done in deterministic polynomial time.
Next, let (Q, R) be an acyclic bound quiver with bound quiver algebra A and assume that W satisfies the relations in R. We show that if A/ Ann_A(W) is a tame algebra then any weight σ in the weight semigroup of W is W-saturated.
Our results provide a systematic way of producing families of tuples of matrices for which Edmonds problem can be solved effectively.
We consider the problem of simultaneously finding lower-dimensional subspace structures in a given tuple of possibly corrupted, high-dimensional data sets all of them of the same size. We refer to this problem as simultaneous robust subspace recovery (``SRSR'') and provide a quiver invariant theoretic approach to it. We show that SRSR is a particular case of the more general problem of effectively deciding whether a quiver representation is semi-stable (in the sense of Geometric Invariant Theory) and, in case it is not, finding a subrepresentation certifying in an optimal way that the representation is not semi-stable. In this paper, we show that SRSR and the more general quiver semi-stability problem can be solved effectively.Last updated: 9/22/2020, 22 pages.
Let Q be a bipartite quiver, V a real representation of Q, and σ an integral
weight of Q orthogonal to the dimension vector of V. Guided by quiver invariant theoretic considerations,
we introduce the Brascamp-Lieb operator associated to (V,σ) and study its capacity, denoted by D(V,σ).
When Q is the m-subspace quiver, the capacity of quiver data is intimately related to the Brascamp-Lieb constants
that occur in the m-multilinear Brascamp-Lieb inequality in analysis.
We show that the positivity of D(V,σ) is equivalent to the σ-semi-stability of V. We also find a character formula for D(V,σ) whenever it is positive. Our main tool is a quiver version of a celebrated result of Kempf-Ness on closed orbits in invariant theory. This result leads us to consider certain real algebraic varieties that carry information relevant to our main objects of study. It allows us to express the capacity of quiver data in terms of the character induced by σ and sample points of the varieties involved. Furthermore, we use this character formula to prove a factorization of the capacity of quiver data. We also find necessary and sufficient conditions for the existence and uniqueness of gaussian extremals for (V,σ). Finally, we explain how to find the gaussian extremals of a gaussian-extremisable datum (V,σ) using the algebraic variety associated to (V,σ).
Background: One of the fundamental problems in the representation theory of algebras is to classify their indecomposable representations.
Based on the complexity of these representations, algebras come in three flavors: representation-finite, tame, and wild.
The classification of all indecomposable representations for wild algebras is regarded as a hopeless problem. Therefore, in order to make progress,
especially in the presence of wild algebras which occur very frequently in representation theory and other areas, one is naturally
led to consider special but large classes of representations. A common theme throughout my research has been to use objects and methods from
invariant theory to provide a geometric framework for parametrizing large classes of representations. The invariant-theoretic objects that I
have been studying in the papers listed below are:
-on the algebraic side, they parametrize finite direct sums of (rather special) Schur representations -on the geometric side, they are projective varieties which, as shown by Hille and Huisgen-Zimmermann, can be arbitrarily complicated in the sense
that any projective variety can be realized as such a moduli space; in particular, no special features of moduli spaces of representations
can be expected in general unless one imposes conditions on either the algebra in question or the classes of representations to be parametrized
The overarching goal of this project is to find characterizations of the GIT-tameness of an algebra in terms of its moduli spaces of representations. This research will lead to an analog of Drozd’s theorem at the level of moduli spaces of representations of algebras. It will also provide solutions (of a geometric nature) to classification problems for wild GIT-tame algebras. To achieve this goal, several general reduction techniques for dealing with these objects have been developed. These reduction results enable us to:
-on the algebraic side, they parametrize finite direct sums of (rather special) Schur representations
-on the geometric side, they are projective varieties which, as shown by Hille and Huisgen-Zimmermann, can be arbitrarily complicated in the sense that any projective variety can be realized as such a moduli space; in particular, no special features of moduli spaces of representations can be expected in general unless one imposes conditions on either the algebra in question or the classes of representations to be parametrized
Consider a finite-dimensional algebra A and any of its moduli spaces M of representations. We prove a decomposition theorem which relates any irreducible component of M to a product of simpler moduli spaces via a finite and birational map. Furthermore, this morphism is an isomorphism when the irreducible component is normal. As an example application, we show that the irreducible components of all moduli spaces associated to tame (or even Schur-tame) algebras are rational varieties.Last updated: 2/12/2018, 22 pages.
We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that certain varieties of representations of special biserial algebras are isomorphic to products of varieties of circular complexes, and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces.Last updated: 6/21/2017, 14 pages.
In this paper, we solve a problem raised by V. Kac on locally semi-simple quiver representations. Specifically, we show that an acyclic quiver Q is of tame representation type if and only if every representation of Q with a semi-simple ring of endomorphisms is locally semi-simple.
Our goal in this paper is to study the module category of an algebra A within the general framework of geometric invariant theory. The geometric objects that we are interested in are the moduli spaces of semi-stable A-modules constructed by King, using methods from geometric invariant theory. It is well-known that the closed points of a moduli space of A-modules correspond to direct sums of rather special Schur A-modules. Hence, from the point of view of invariant theory, one is naturally led to think of an algebra based on the complexity of its Schur modules. In this paper, we focus on those algebras whose Schur modules have a tame behavior. These algebras, called Schur-tame, form a large class which goes beyond the class of tame algebras. Our objective is to describe the tameness, and more generally the Schur-tameness, of an algebra in terms of invariant theory.
We first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. To prove these results, we rely on descriptions of the irreducible components of the module varieties of the algebras involved and a reduction technique for dealing with moduli spaces of modules.
In this paper we show that the fields of rational invariants over the irreducible components of the module varieties for an acyclic gentle algebra are purely transcendental extensions. Along the way, we exhibit for such fields of rational invariants a transcendence basis in terms of the Schofield's determinantal semi-invariants. We also show that certain moduli spaces of modules over regular irreducible components are just products of projective spaces.Last updated: 02/18/2013, 27 pages.
In this paper we seek invariant-theoretic characterizations of (Schur-)representation finite algebras. To this end, we introduce two classes of finite-dimensional algebras: those with the dense-orbit property and those with the multiplicity-free property. We show first that when a connected algebra A admits a pre-projective component, each of these properties is equivalent to A being representation-finite. Next, we give an example of a representation-infinite algebra with the dense-orbit property. We also show that the string algebras with the dense orbit-property are precisely the representation-finite ones. Finally, we show that a tame algebra has the multiplicity-free property if and only if it is Schur-representation-finite.Last updated: 10/20/2013, 14 pages.
The goal of this paper is to characterize the tameness of tilted algebras (more generally, quasitilted algebras) in terms of the invariant theory of the algebras in question. Along the way, we establish two general reduction results for dealing with moduli spaces of modules of arbitrary algebras. The first reduction result explains the behavior of moduli spaces of modules under tilting functors. This allows one to study moduli spaces of modules for an algebra by reducing the considerations to an algebra of smaller global dimension. The second reduction result allows one to decompose a moduli space of modules into smaller moduli spaces which are typically easier to handle.
Zwara has found a representation of the Kronecker quiver whose orbit closure is not unibranch. In this paper, we explain how to extend Zwara's example to any connected representation-infinite algebra A admitting a pre-projective component. This result combined with another result of Zwara shows that a connected algebra with a preprojective component is representation-finite if and only if all of its orbit closures are unibranch.
It essentially follows from Happel-Vossieck's famous classification of minimal representation-infinite algebras with a preprojective component that A has a tame concealed algebra as a factor. To deal with tame concealed algebras, we use orthogonal exceptional sequences to further reduce the considerations to Kronecker algebras at which point one can invoke the aforementioned result of Zwara.
The goal of this paper is to characterize the tameness of hereditary algebras and of canonical algebras in terms of the rational invariant theory of the algebras in question. Along the way, we establish a general reduction technique for studying fields of rational invariants on Schur irreducible components of representation varieties.
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). In 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.
Background: Let Q be an acyclic quiver with n vertices and d a dimension vector of Q. The cone Eff(Q,d) of effective weights associated to (Q,d) is the rational convex polyhedral cone in ℜn whose lattice points are precisely those integral vectors θ (also called integral weights) for which the corresponding moduli space of θ-semi-stable d-dimensional representations of Q is non-empty. In the general context of GIT, these effective cones are some of the fundamental objects of study.
(i) Littlewood-Richardson coefficients (and of more general structure constants) since these coefficients can be realized as dimensions of weight spaces of semi-invariants of (star-shaped) quivers; (ii) eigenvalue problems for Hermitian matrices since Hermitian matrices with prescribed spectra and satisfying certain matrix equations arise from the symplectic
description of moduli spaces of quiver representations.
In the quiver set-up, Derksen and Weyman found a beautiful description of the faces of Eff(Q,d) in terms of the root system of Q. Furthermore, they also showed that the lattice points of this cone are precisely those integral weights θ for which the corresponding weight spaces of semi-invariants SI(Q, d)θ are non-zero. These tools turn out to be very powerful in the study of:
In the papers listed below, we have made essential use of the rich combinatorics and geometry of quiver representations to address a series of questions on: (1) the possible interactions between semi-stable (sub)categories of quiver representations; (2) finite stability weights and cluster fans; (3) Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients; and (4) generalized Littlewood-Richardson coefficients, long exact sequences of finite abelian p-groups, and eigenvalue problems.
(i) Littlewood-Richardson coefficients (and of more general structure constants) since these coefficients can be realized as dimensions of weight spaces of semi-invariants of (star-shaped) quivers;
(ii) eigenvalue problems for Hermitian matrices since Hermitian matrices with prescribed spectra and satisfying certain matrix equations arise from the symplectic description of moduli spaces of quiver representations.
In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.Last updated: 3/29/2016, 17 pages.
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.
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.
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. Furthermore, these generalized Littlewood-Richardson coefficients can be used to describe decomposition numbers for q-Schur algebras and tensor product multiplicities for extremal weight crystals.
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.
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.
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.
Background: Let Q be an acyclic quiver and KQ its path algebra. In a series of papers, Benson, Carlson, Friedlander,
Pevtsova, and Suslin have introduced and studied the class of modules of constant Jordan type for finite group schemes. Inspired by their
work, Andrew Carroll, Zongzhu Lin, and myself have introduced the class of KQ-modules of (relative) constant Jordan type. On the algebraic
side, these modules have constant decomposition into indecomposable modules when pulled-back to truncated polynomial rings in one variable.
On the geometric side, we set up a process that assigns to any KQ-module a sequence of coherent sheaves over moduli spaces of (thin)
KQ-modules. Under this correspondence, a KQ-module has constant Jordan type if and only if its corresponding sheaves are locally-free.
In our approach to building these KQ-modules of constant Jordan type, we use GIT, more precisely moduli spaces of thin modules, instead of the Friedlander-Pevtsova's Π-point schemes, which are not available in the context of modules over path algebras.
Inspired by the work of Benson, Carlson, Friedlander, Pevtsova, and Suslin on modules of constant Jordan type for finite group schemes, we introduce in this paper the class of representations of constant Jordan type for an acyclic quiver Q. We do this by first assigning to an arbitrary finite-dimensional representation of Q a sequence of coherent sheaves on moduli spaces of thin representations. Next, we show that our quiver representations of constant Jordan type are precisely those representations for which the corresponding sheaves are locally free. We also construct representations of constant Jordan type with desirable homological properties. Finally, we show that any element of the Grothendieck group of Q can be realized as the Jordan type of a virtual representation of Q of relative constant Jordan type.Last updated: 2/9/2014, 23 pages.