I. Applications of semistability of quiver representations
Background: The motivation in this project goes back to the celebrated BrascampLieb (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 semistability 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 msubspace quiver, the capacity of quiver data is intimately
related to the BL constants that occur in the mmultilinear BL inequalities.
Our first result states that cap(V, σ) > 0 if and only if V is σsemistable. This has important consequences for quiver theory since it yields a deterministic polynomial
time algorithm for checking semistability of quiver representations. The importance of the existence of such an algorithm stems from the fact that, in general, quiver semistability 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 KempfNess 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:
cap(V, σ)=χ(A)^{2}.
These results have already found applications to Edmonds problem in algebraic complexity and robust subspace recovery in machine learning.

Edmonds' problem and the membership problem for orbit semigroups of quiver representations
with
Dan Kline
Abstract 
pdf 
arxiv
A central problem in algebraic complexity, posed by J. Edmonds, asks to decide if the span of a given ltuple of NxN complex matrices
contains a nonsingular matrix.
In this paper, we provide a quiver invariant theoretic approach this problem. Viewing the ltuple of matrices as a representation W of
the lKronecker quiver, Edmonds problem can be rephrased as asking to decide if there exists a semiinvariant 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 semigroup
of W by focusing on the socalled Wsaturated weights. We first show that for any given Wsaturated 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 Wsaturated.
Our results provide a systematic way of producing families of tuples of matrices for which Edmonds problem can be solved effectively.
Last updated: 8/17/2020, 16 pages.

Simultaneous robust subspace recovery and semistability of quiver representations
with
Dan Kline
Abstract 
pdf 
arxiv
We consider the problem of simultaneously finding lowerdimensional subspace structures in a given tuple of possibly
corrupted, highdimensional 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 semistable (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 semistable. In this paper,
we show that SRSR and the more general quiver semistability problem can be solved effectively.
Last updated: 9/22/2020, 22 pages.

The capacity of quiver representations and BrascampLieb constants
with
Harm Derksen
Abstract 
pdf 
arxiv
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 BrascampLieb operator associated to (V,σ) and study its capacity, denoted by D(V,σ).
When Q is the msubspace quiver, the capacity of quiver data is intimately related to the BrascampLieb constants
that occur in the mmultilinear BrascampLieb inequality in analysis.
We show that the positivity of D(V,σ) is equivalent to the σsemistability 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 KempfNess 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 gaussianextremisable datum (V,σ) using the algebraic variety associated to (V,σ).
Last updated: 9/12/2019, 24 pages.
II. Moduli spaces for finitedimensional algebras and their representation type
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: representationfinite, 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 invarianttheoretic objects that I
have been studying in the papers listed below are:
 moduli spaces of representations , constructed by King via Geometric Invariant Theory (GIT)
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 HuisgenZimmermann, 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
 fields of rational invariants for bound quiver algebras; these are function fields of rational quotients (in the sense of Rosenlicht) that parametrize
generic representations.
The overarching goal of this project is to find characterizations of the GITtameness of an algebra in terms of its moduli spaces of representations. This research will
lead to an analog of Drozdâ€™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 GITtame algebras. To achieve this goal, several general reduction techniques for dealing with these objects have been developed. These reduction results enable us to:
 decompose a moduli space of representations or a field of rational invariants into ``smaller'' pieces which are easier to handle,
especially when the algebra in question is either tame or wild but GITtame;
 study the behavior of moduli spaces of representations under tilting functors; this allows one to study moduli spaces of representations
for an algebra by reducing the considerations to an algebra of smaller global dimension.
Workinprogress/Submitted papers/Preprints

Decomposing moduli of representations of finitedimensional algebras
with
Ryan Kinser
Mathematische Annalen, 372 (1), 555580, 2018
Abstract 
pdf 
arxiv
Consider a finitedimensional 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 Schurtame) algebras are rational varieties.
Last updated: 2/12/2018, 22 pages.

Moduli spaces of representations of special biserial algebras
with
Andrew Carroll, Ryan Kinser, and Jerzy Weyman
International Mathematics Research Notices 2018
Abstract 
pdf 
doi 
arxiv
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.

On locally semisimple quiver representations
with
Dan Kline
Journal of Algebra, 467 (2016), 284306.
Abstract 
pdf 
arxiv
In this paper, we solve a problem raised by V. Kac on locally semisimple
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 semisimple ring of endomorphisms
is locally semisimple.

Moduli spaces of modules of Schurtame algebras
with
Andrew Carroll
Algebras and Representation Theory, (2015) 18:961976.
Abstract 
pdf 
arxiv
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 semistable Amodules constructed by King, using methods from geometric invariant theory. It is wellknown that the closed points of a moduli space of Amodules correspond to direct sums of rather special Schur Amodules. 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 Schurtame, form a large class which goes beyond the class of tame algebras. Our objective is to describe the tameness, and more generally the Schurtameness, 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 Amodules 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.
Last updated: 3/6/2015.

On the invariant theory for acyclic gentle algebras
with
Andrew Carroll
Transactions of the American Mathematical Society, 367 (2015), no. 5, 34813508.
Abstract 
pdf 
arxiv
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 semiinvariants. 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.

Module varieties and representation type of finite dimensional algebras
with
Ryan Kinser and Jerzy Weyman
International Mathematics Research Notices 2015, no 3, 631650.
Abstract 
pdf 
arxiv
In this paper we seek invarianttheoretic characterizations of (Schur)representation finite algebras. To this end, we introduce two classes of finitedimensional algebras: those with the denseorbit property and those with the multiplicityfree property. We show first that when a connected algebra A admits a preprojective component, each of these properties is equivalent to A being representationfinite. Next, we give an example of a representationinfinite algebra with the denseorbit property. We also show that the string algebras with the dense orbitproperty are precisely the representationfinite ones. Finally, we show that a tame algebra has the multiplicityfree property if and only if it is Schurrepresentationfinite.
Last updated: 10/20/2013, 14 pages.

On the invariant theory for tame tilted algebras
Algebra & Number Theory, Vol. 7 (2013), No. 1, 193214.
Abstract 
pdf 
arxiv
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.

On the geometry of orbit closures for representationinfinite algebras
Glasgow Mathematical Journal, 54 (2012), 629636.
Abstract 
pdf 
arxiv
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 representationinfinite algebra A admitting a preprojective component. This result combined with another result of Zwara shows that a connected algebra with a preprojective component is representationfinite if and only if all of its orbit closures are unibranch.
It essentially follows from HappelVossieck's famous classification of minimal representationinfinite 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.

Geometric characterizations of the representation type of hereditary algebras and of canonical algebras
Advances in Mathematics, 228 (2011), no. 3, 14051434.
Abstract 
pdf 
arxiv
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.

Orbit semigroups and the representation type of quivers
Journal of Pure and Applied Algebra, 213 (2009), pp. 14181429.
Abstract 
pdf 
arxiv
It is an important and interesting task to find geometric characterizations of the representation type of a quiver (or more generally, of a finitedimensional 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.
III. Cones of effective weights for quivers, LittlewoodRichardson coefficients, and eigenvalue problems
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 ℜ^{n} whose lattice points are precisely those integral vectors θ (also called integral weights) for which the corresponding moduli space of θsemistable ddimensional representations of Q is nonempty. In the general context of GIT, these effective cones are some of the fundamental objects of study.
In the quiver setup, 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 semiinvariants SI(Q, d)_{θ} are nonzero. These tools turn out to be very powerful in the study of:
(i) LittlewoodRichardson coefficients (and of more general structure constants) since these coefficients can be realized as dimensions of weight spaces of semiinvariants of (starshaped) 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 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 semistable (sub)categories of quiver representations; (2) finite stability weights and cluster fans; (3) Okounkov's logconcavity conjecture for LittlewoodRichardson coefficients; and (4) generalized LittlewoodRichardson coefficients, long exact sequences of finite abelian pgroups, and eigenvalue problems.

GITequivalence and semistable subcategories of quiver representations
with
Valerie Granger
Journal of Pure and Applied Algebra, 223(8), 34993514, 2019
Abstract 
pdf 
arxiv
In this paper, we answer the question of when the subcategory of semistable 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.

Cluster fans, stability conditions, and domains of
semiinvariants
Transactions of the American Mathematical Society, 363 (2011), no. 4, 21712190.
Abstract 
pdf 
arxiv
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 semiinvariants of Q. Along the way, we give new proofs of Schofield's results on perpendicular categories and semiinvariants of quivers. We also explain how our results can be used to recover IgusaOrrTodorovWeyman theorem on cluster fans and doamins of semiinvariants for Dynkin quivers.

Quivers, long exact sequences and Horn type inequalities
II
Glasgow Mathematical Journal, 51 (2009), 117.
Abstract 
pdf 
arxiv
In this paper, we find necessary and sufficient Horn type inequalities for the existence of long exact sequences of finite abelian pgroups 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
Journal of Algebra, 320 (2008), no. 1, 128157.
Abstract 
pdf 
arxiv
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 nonvanishing of the LittlewoodRichardson coefficients and the existence of short exact sequences of finite abelian pgroups. 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 pgroups, using methods from quiver invariant theory. We explain how this result is related to some generalized LittlewoodRichardson coefficients and to eigenvalues of Hermitian matrices satisfying certain (in)equalities. Furthermore, these generalized LittlewoodRichardson coefficients can be used to describe decomposition numbers for qSchur algebras and tensor product multiplicities for extremal weight crystals.

Counterexamples to Okounkov's logconcavity
conjecture
with
Harm Derksen and Jerzy Weyman
Compositio Mathematica, 143 (2007), 15451557.
Abstract 
arxiv
Motivated by physical considerations, Okounkov conjectured that the LittlewoodRichardson coefficients are logconcave as a function of their highest weights. This conjecture, if true, would immediately imply the KnutsonTao saturation theorem, a conjecture of Fulton proved by Belkale, and the logconcavity theorem for skewSchur functions proved by LamPostnikovPylyavskyy. 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
International Mathematics Research Notices, 2006, Art. ID 59457, 27 pp.
Abstract 
pdf
Buch, answering a question raised by Barvinok, has showed that the set of the possible eigenvalues of Hermitan matrices with positive semidefinite 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 KnutsonTao saturation theorem for LittlewoodRichardson coefficients.

Notes on GIT fans for quivers
Abstract 
pdf 
arxiv
In this paper we go over the construction of the GITfans for quivers without oriented cycles. We follow closely the steps outlined by Ressayre in "The GITEquivalence for GLine Bundles" but we avoid DolgachevHu's finitness theorem. Our arguments are based on King semistability criterion for quiver representations and Schofield's theory of general representations.
IV. Quiver representations of constant Jordan type and vector bundles
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 KQmodules of (relative) constant Jordan type. On the algebraic
side, these modules have constant decomposition into indecomposable modules when pulledback to truncated polynomial rings in one variable.
On the geometric side, we set up a process that assigns to any KQmodule a sequence of coherent sheaves over moduli spaces of (thin)
KQmodules. Under this correspondence, a KQmodule has constant Jordan type if and only if its corresponding sheaves are locallyfree.
In our approach to building these KQmodules of constant Jordan type, we use GIT, more precisely moduli spaces of thin modules, instead of
the FriedlanderPevtsova's Πpoint schemes, which are not available in the context of modules over path algebras.

Quiver representations of constant Jordan type and vector bundles
with
Andrew Carroll and Zongzhu Lin
Abstract 
pdf 
arxiv
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 finitedimensional 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.