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Calin Chindris

Publications and source records attributed to Calin Chindris.

At least 19 recordsLinked to original sources
arXiv2026

Counting 3-way contingency tables via quiver semi-invariants

Let $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ be the number of $3$-way contingency tables of size $m \times n \times p$ with two of its three plane-sum margins fixed by $\mathbf{a}=(a_1, \ldots, a_m) \in \mathbb{N}^m$ and $\mathbf{b}=(b_1, \ldots, b_n) \in \mathbb{N}^n$. When $p=1$, this is the number of $m \times n$ non-negative integer matrices whose row and column sums are fixed by $\mathbf{a}$ and $\mathbf{b}$. In this paper, we study the numbers $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ through the lens of quiver invariant theory. Let $\mathcal{Q}^{p}_{m,n}$ be the $p$-complete bipartite quiver with $m$ source vertices, $n$ sink vertices, and $p$ arrows from each source to each sink. Let $\mathbf{1}$ denote the dimension vector of $\mathcal{Q}^{p}_{m,n}$ that takes value $1$ at every vertex of $\mathcal{Q}^{p}_{m,n}$, and let $\theta_{\mathbf{a}, \mathbf{b}}$ denote the integral weight that assigns $a_i$ to the $i^{th}$ source vertex and $-b_j$ to the $j^{th}$ sink vertex of $\mathcal{Q}^{p}_{m,n}$. We begin by realizing $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ as the dimension of the space of semi-invariants associated to $(\mathcal{Q}^{p}_{m,n}, \mathbf{1}, \theta_{\mathbf{a}, \mathbf{b}})$. Using this connection and methods from quiver invariant theory, we show that $\mathbf{T}_{\mathbf{a},\mathbf{b}}$ is a parabolic Kostka coefficient. In the case $p=1$, this recovers the formula for the number of the $m \times n$ contingency tables with row and column sums fixed by $\mathbf{a}$ and $\mathbf{b}$, which in the classical $2$-way setting can also be obtained via the Robinson-Schensted-Knuth correspondence.

math.CO↗
arXiv2026

Algebraicity of the Brascamp-Lieb constants

We show that the Brascamp-Lieb (BL) constant BL(-,p) is a semi-algebraic function on the set of feasible data. Consequently, it is algebraic in the sense that it satisfies a polynomial relation of the form P(V, BL(V,p))=0 for a non-zero polynomial P. In fact, we establish an analogous statement in the more general setting of quiver BL constants associated to representations of bipartite quivers.

math.RT↗
arXiv2025

The Jordan type of a multiparameter persistence module

Let $\mathscr{P}$ be a poset and $\mathcal{S}$ a sequence of $n$ finite substes of $\mathscr{P}$. The Jordan type of a $\mathscr{P}$-persistence module $M$ at $\mathcal{S}$, denoted by $\mathsf{J}_{\mathcal{S}}(M) \in \mathbb{N}^n$, is defined as the Jordan type of a nilpotent operator $\mathbf{T}_{M, \mathcal{S}}$, which is constructed from $M$ and $\mathcal{S}$. When $n=2$, we recover the notion of multirank previously introduced and studied in [Tho19]. We first prove that the multirank invariants are complete for persistence modules over finite zigzag posets. This proves a conjecture of Thomas in the zigzag case. The nilpotent operator $\mathbf{T}_{M, \mathcal{S}}$ is functorial in $M$. When $\mathscr{P}=\mathbb{Z}^d$ or $\mathbb{R}^d$, this functoriality allows us to define the Jordan filtered rank invariant of $M$ at $\mathscr{S}$. We demonstrate that these invariants are strictly finer than the classical rank invariants. We next prove that for any two $\mathscr{P}$-persistence modules $M$ and $N$, the landscape and erosion distances between their Jordan filtered rank invariants are bounded from above by the interleaving distance between $M$ and $N$.

math.RT↗
arXiv2023

The capacity of quiver representations and the Anantharam-Jog-Nair inequality

The Anantharam-Jog-Nair inequality [AJN22] in Information Theory provides a unifying approach to the information-theoretic form of the Brascamp-Lieb inequality [CCE09] and the Entropy Power inequality [ZF93]. In this paper, we use methods from Quiver Invariant Theory [CD21] to study Anantharam-Jog-Nair inequalities with integral exponents. For such an inequality, we first view its input datum as a quiver datum and show that the best constant that occurs in the Anantharam-Jog-Nair inequality is of the form $-{1\over 2}\log (\mathbf{cap}(V,\sigma))$ where $\mathbf{cap}(V, \sigma)$ is the capacity of a quiver datum $(V, \sigma)$ of a complete bipartite quiver. The general tools developed in [CD21], when applied to complete bipartite quivers, yield necessary and sufficient conditions for: (1) the finiteness of the Anantharam-Jog-Nair best constants; and (2) the existence of Gaussian extremizers. These results recover some of the main results in [AJN22] and [ACZ22]. In addition, we characterize gaussian-extremizable data in terms of semi-simple data, and provide a general character formula for the Anatharam-Jog-Nair constants. Furthermore, our quiver invariant theoretic methods lead to necessary and sufficient conditions for the uniqueness of Gaussian extremizers. This answers the third and last question left unanswered in [AJN22].

cs.IT↗
arXiv2022

Hive-type polytopes for quiver multiplicities and the membership problem for quiver moment cones

Let $Q$ be a bipartite quiver with vertex set $Q_0$ such that the number of arrows between any source vertex and any sink vertex is constant. Let $\beta=(\beta(x))_{x \in Q_0}$ be a dimension vector of $Q$ with positive integer coordinates. Let $rep(Q, \beta)$ be the representation space of $\beta$-dimensional representations of $Q$ and $GL(\beta)$ the base change group acting on $rep(Q, \beta)$ be simultaneous conjugation. Let $K^{\beta}_{\underline{\lambda}}$ be the multiplicity of the irreducible representation of $GL(\beta)$ of highest weight $\underline{\lambda}$ in the ring of polynomial functions on $rep(Q, \beta)$. We show that $K^{\beta}_{\underline{\lambda}}$ can be expressed as the number of lattice points of a polytope obtained by gluing together two Knutson-Tao hive polytopes. Furthermore, this polytopal description together with Derksen-Weyman's Saturation Theorem for quiver semi-invariants allows us to use Tardos' algorithm to solve the membership problem for the moment cone associated to $(Q,\beta)$ in strongly polynomial time.

math.CO↗
arXiv2021

A quiver invariant theoretic approach to Radial Isotropy and Paulsen's Problem for matrix frames

In this paper, we view matrix frames as representations of quivers and study them within the general framework of quiver invariant theory. We are thus led to consider the large class of semi-stable matrix frames. Within this class, we are particularly interested in radial isotropic and Parseval matrix frames. Using methods from quiver invariant theory [CD19], we first prove a far reaching generalization of Barthe's Radial Isotropy Theorem [Bar98] to matrix frames (see Theorems 1(3) and 29). With this tool at our disposal, we provide a quiver invariant theoretic approach to the Paulsen problem for matrix frames. We show in Theorem 2 that for any given $\varepsilon$-nearly equal-norm Parseval frame $\mathcal{F}$ of $n$ matrices with $d$ rows there exists an equal-norm Parseval frame $\mathcal{W}$ of $n$ matrices with $d$ rows such that $\mathsf{dist}^2(\mathcal{F},\mathcal{W})\leq 46 \epsilon d^2$.

math.FA↗
arXiv2020

Edmonds' problem and the membership problem for orbit semigroups of quiver representations

A central problem in algebraic complexity, posed by J. Edmonds, asks to decide if the span of a given $l$-tuple $\V=(\V_1, \ldots, \V_l)$ of $N \times N$ complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing $\V$ as a representation of the $l$-Kronecker quiver $\K_l$, Edmonds' problem can be rephrased as asking to decide if there exists a semi-invariant on the representation space $(\CC^{N\times N})^l$ of weight $(1,-1)$ that does not vanish at $\V$. In other words, Edmonds' problem is asking to decide if the weight $(1,-1)$ belongs to the orbit semigroup of $\V$. Let $Q$ be an arbitrary acyclic quiver and $\V$ a representation of $Q$. We study the membership problem for the orbit semi-group of $\V$ by focusing on the so-called $\V$-saturated weights. We first show that for any given $\V$-saturated weight $\sigma$, checking if $\sigma$ belongs to the orbit semigroup of $\V$ can be done in deterministic polynomial time. Next, let $(Q, \R)$ be an acyclic bound quiver with bound quiver algebra $A=KQ/\langle \R \rangle$ and assume that $\V$ satisfies the relations in $\R$. We show that if $A/\Ann_A(\V)$ is a tame algebra then any weight $\sigma$ in the weight semigroup of $\V$ is $\V$-saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.

math.RT↗
arXiv2020

Simultaneous robust subspace recovery and semi-stability of quiver representations

We consider the problem of simultaneously finding lower-dimensional subspace structures in a given $m$-tuple of possibly corrupted, high-dimensional data sets all 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.

math.RT↗
arXiv2019

The capacity of quiver representations and Brascamp-Lieb constants

Let $Q$ be a bipartite quiver, $V$ a real representation of $Q$, and $\sigma$ an integral weight of $Q$ orthogonal to the dimension vector of $V$. Guided by quiver invariant theoretic considerations, we introduce the Brascamp-Lieb operator $T_{V,\sigma}$ associated to $(V,\sigma)$ and study its capacity, denoted by $\mathbf{D}_Q(V, \sigma)$. 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 $\mathbf{D}_Q(V, \sigma)$ is equivalent to the $\sigma$-semi-stability of $V$. We also find a character formula for $\mathbf{D}_Q(V, \sigma)$ 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 $\sigma$ 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 show that the existence of gaussian extremals for $(V, \sigma)$ is equivalent to $V$ being $\sigma$-polystable, and that the uniqueness of gaussian extremals implies that $V$ is $\sigma$-stable. Finally, we explain how to find the gaussian extremals of a gaussian-extremisable datum $(V, \sigma)$ using the algebraic variety associated to $(V,\sigma)$.

math.RT↗
arXiv2018

Decomposing moduli of representations of finite-dimensional algebras

Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_θ$ of representations. We prove a decomposition theorem which relates any irreducible component of $\mathcal{M}(A,\mathbf{d})^{ss}_θ$ 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.

math.RT↗
arXiv2017

Moduli spaces of representations of special biserial algebras

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 irreducible components of varieties of representations of special biserial algebras are isomorphic to irreducible components of products of varieties of circular complexes, and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces.

math.RT↗
arXiv2016

GIT-Equivalence and Semi-Stable Subcategories 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 in [10]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.

math.RT↗
arXiv2015

On locally semi-simple representations of quivers

In this paper, we solve a problem raised by V. Kac in \cite{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.

math.RT↗
arXiv2014

Moduli spaces of modules of Schur-tame algebras

In this paper, 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. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and theta-stable irreducible component C of a module variety of A-modules, the moduli space of theta-semi-stable points of C is either a point or a rational projective curve.

math.RT↗
arXiv2014

Quiver representations of constant Jordan type and vector bundles

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 $\mathbb{Z}^L$, where $L$ is the Loewy length of the path algebra of $Q$, can be realized as the Jordan type of a virtual representation of $Q$ of relative constant Jordan type.

math.RT↗
arXiv2013

Module varieties and representation type of finite-dimensional algebras

In this paper we seek geometric and 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 preprojective component, each of these properties is equivalent to A being representation-finite. Next, we give an example of an algebra which is not representation-finite but still has 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.

math.RT↗
arXiv2013

On the invariant theory for acyclic gentle algebras

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 Schofield determinantal semi-invariants. We also show that the moduli space of modules over a regular irreducible component is just a product of projective spaces.

math.RT↗