arXiv subjects

Kaveh Mousavand

Publications and source records attributed to Kaveh Mousavand.

15 recordsLinked to original sources

arXiv2026

Brick infinite algebras admit infinitely many non-$τ$-rigid bricks

For finite dimensional algebras over algebraically closed fields, we settle a question previously known only for certain families of algebras. More specifically, motivated by some foundational interactions between bricks and $τ$-rigid modules, we show that a given algebra is brick infinite if and only if it admits infinitely many bricks which are not $τ$-rigid. This proves the $τ$-analogue of an open conjecture asserting that if (almost) all bricks over an algebra $A$ are rigid, then $A$ should be brick-finite. In retrospect, we strengthen some recent contributions to the study of a series of challenging open problems related to the $2$nd brick-Brauer-Thrall conjecture. Moreover, motivated by some of our arguments, we pose the question whether there exists any algebra that admits an infinite semibrick consisting of Ext-orthogonal rigid bricks. In connection with this, we present an algebra $A$ of rank $n$ that admits a semibrick of cardinality strictly greater than $n$ consisting of Ext-orthogonal rigid bricks.

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arXiv2026

Left modularity and extremality for (some) infinite lattices

For some important families of complete infinite lattices, we study some generalizations of two fundamental notions which are mostly treated for finite lattices. Specifically, for well-separated $κ$-lattices, and also for weakly atomic completely semidistributive lattices, we generalize the notions of left modularity and extremality. These two families of lattices coincide if restricted to finite lattices, but are distinct when infinite lattices are also included. For both families, we prove that extremality and left modularity imply each other. Furthermore, for weakly atomic completely semidistributive lattices, we give several conceptual characterizations of left modular elements, and show that the set of left modular elements form a complete distributive sublattice. Our results, combined with some recent work on finite lattices, imply that the weakly atomic completely semidistributive lattices that are left modular (or extremal) generalize the semidistributive trim lattices; from finite to infinite lattices. We then apply our results to the lattice of torsion classes of finite dimensional algebras, which are known to fall in the intersection of the two families treated in our work. For an algebra $A$, we obtain that the lattice of torsion classes is left modular (equivalently, extremal) if and only if $A$ is brick-directed. This leads to an abundance of concrete examples and non-examples.

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arXiv2025

Geometric interactions between bricks and $τ$-rigidity

For finite-dimensional algebras over algebraically closed fields, we consider two fundamental classes of modules and their geometric counterparts: bricks and $τ$-rigid modules, as well as brick components and $τ$-regular components. We then apply our results in the study of some open conjectures. First, we investigate the situation where every brick is $τ$-rigid. We prove that this occurs exactly when the algebra is locally representation-directed; a family of algebras introduced by Dräxler in the 1990s, which are always representation-finite. Then, we adopt a geometric perspective and analyze the brick and $τ$-regular components of module varieties. In this greater generality, we establish new properties of such components. Inspired by some recent conjectures, we apply our results to the study of minimal brick-infinite algebras. Along the way, we construct some limits of rigid $g$-vectors, under a condition that we call the $τ$-convergence property. This construction is novel and, in certain cases, yields an integral $g$-vector lying outside the $τ$-tilting fan (a.k.a. $g$-vector fan). We show how our results provide new tools to the study of some open conjectures and particularly illustrate that for $E$-tame algebras.

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arXiv2025

Infinite string bricks and Sturmian words over some gentle algebras

We study infinite string modules that are bricks over some gentle algebras. In particular, we first give a complete classification of these modules over the double-Kronecker gentle algebra and prove that each family is in bijection with a family of Sturmian (binary) words. We then generalize some of our results to a larger family of gentle algebras.

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arXiv2025

Brick-splitting Torsion Pairs and Left Modularity

We introduce the notion of brick-splitting torsion pairs as a modern analogue and generalization of the classical notion of splitting torsion pairs. A torsion pair is called brick-splitting if any given brick is either torsion or torsion-free with respect to that torsion pair. After giving some properties of these pairs, we fully characterize them in terms of some lattice-theoretical properties, including left modularity. This leads to the notion of brick-directed algebras, which are those for which there does not exist any cycle of non-zero non-isomorphisms between bricks. This class of algebras is a novel generalization of representation-directed algebras. We show that brick-directed algebras have many interesting properties and give several characterizations of them. In particular, we prove that a brick-finite algebra is brick-directed if and only if the lattice of torsion classes is left modular (or equivalently, extremal). We also give a characterization of brick-directed algebras in terms of their wall-and-chamber structure, as well as of a certain Newton polytope associated to them. Moreover, we introduce an explicit construction of an abundance of brick-directed algebras, both of the tame and wild representation types.

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arXiv2025

On the bricks (Schur representations) of finite dimensional algebras

This manuscript treats the diverse applications of bricks within modern representation theory and several related domains, and reviews the recent developments and new results on bricks (a.k.a Schur representations). The current survey is an extended version of a mini-course by the second-named author, delivered in the research school on ``New Developments in Representation Theory of Algebras", held in November of 2024, at Okinawa Institute of Science and Technology (OIST), Japan. The review is mainly oriented towards the direction of research developed by the authors, which has evolved around the algebraic and geometric properties of bricks. More specifically, we discuss the emergence of bricks in $τ$-tilting theory, torsion theory, geometric representation theory and invariant theory, while providing some links between those. Although we review the applications and properties of bricks from many different areas, the article is not meant to be an exhaustive survey on bricks in representation theory. In the setting of finite dimensional algebras over an algebraically closed field, this manuscript (and many of the recent works of the authors) is strongly motivated by an open conjecture originally posed by the first-named author in 2019, the so-called \emph{second brick Brauer-Thrall conjecture}. In the later sections, where the main focus is on the tame algebras and some other new notions of tameness, we prove some new results on the aforementioned conjecture.

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arXiv2025

A continuous associahedron of type A

Taking a representation-theoretic viewpoint, we construct a continuous associahedron motivated by the realization of the generalized associahedron in the physical setting. We show that our associahedron shares important properties with the generalized associahedron of type A. Our continuous associahedron is convex and manifests a cluster theory: the points which correspond to the clusters are on its boundary, and the edges that correspond to mutations are given by intersections of hyperplanes. This requires development of several methods that are continuous analogues of discrete methods. We conclude the paper by showing that there is a sequence of embeddings of type A generalized associahedra into our continuous associahedron.

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arXiv2025

Biserial algebras and generic bricks

We consider generic bricks and use them in the study of arbitrary biserial algebras over algebraically closed fields. For a biserial algebra $Λ$, we show that $Λ$ is brick-infinite if and only if it admits a generic brick, that is, there exists a generic $Λ$-module $G$ with $End_Λ(G)=k(x)$. Furthermore, we give an explicit numerical condition for brick-infiniteness of biserial algebras: If $Λ$ is of rank $n$, then $Λ$ is brick-infinite if and only if there exists an infinite family of bricks of length $d$, for some $2\leq d\leq 2n$. This also results in an algebro-geometric realization of $τ$-tilting finiteness of this family: $Λ$ is $τ$-tilting finite if and only if $Λ$ is brick-discrete, meaning that in every representation variety $mod(Λ, \underline{d})$, there are only finitely many orbits of bricks. Our results rely on our full classification of minimal brick-infinite biserial algebras in terms of quivers and relations. This is the modern analogue of the recent classification of minimal representation-infinite (special) biserial algebras, given by Ringel. In particular, we show that every minimal brick-infinite biserial algebra is gentle and admits exactly one generic brick. Furthermore, we describe the spectrum of such algebras, which is very similar to that of a tame hereditary algebra. In other words, $Brick(Λ)$ is the disjoint union of a unique generic brick with a countable infinite set of bricks of finite length, and a family of bricks of the same finite length parametrized by the ground field.

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arXiv2025

Hom-orthogonal modules and brick-Brauer-Thrall conjectures

For finite dimensional algebras over algebraically closed fields, we study the sets of pairwise Hom-orthogonal modules and obtain new results on some open conjectures on the behaviour of bricks and several related problems, which we generally refer to as brick-Brauer-Thrall (bBT) conjectures. Using some algebraic and geometric tools, and in terms of the notion of Hom-orthogonality, we find necessary and sufficient conditions for the existence of infinite families of bricks of the same dimension. This sheds new light on the bBT conjectures and we prove some of them for new families of algebras. Our results imply some interesting algebraic and geometric characterizations of brick-finite algebras as conceptual generalizations of local algebras. We also verify the bBT conjectures for any algebra whose Auslander-Reiten quiver has a generalized standard component, which particularly extends some results of Chindris-Kinser-Weyman on the algebras with preprojective components.

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arXiv2021

ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the $F$-polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the g-vector fan has the property that its nef cone is simplicial.

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arXiv2021

Minimal ($τ$-)tilting infinite algebras

Motivated by a new conjecture on the behavior of bricks, we start a systematic study of minimal $τ$-tilting infinite algebras. In particular, we treat minimal $τ$-tilting infinite algebras as a modern counterpart of minimal representation infinite algebras and show some of the fundamental similarities and differences between these families. We then relate our studies to the classical tilting theory and observe that this modern approach can provide fresh impetus to the study of some old problems. We further show that in order to verify the conjecture it is sufficient to treat those minimal $τ$-tilting infinite algebras where almost all bricks are faithful. Finally, we also prove that minimal extending bricks have open orbits, and consequently obtain a simple proof of the brick analogue of the First Brauer-Thrall Conjecture, recently shown by Schroll and Treffinger using some different techniques.

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arXiv2019

$τ$-Tilting Finiteness of Non-distributive Algebras and their Module Varieties

We treat the $τ$-tilting finiteness of those minimal representation-infinite (min-rep-infinite) algebras which are non-distributive. Building upon the new results of Bongartz, we fully determine which algebras in this family are $τ$-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the min-rep-infinite biserial algebras. Consequently, we obtain nontrivial explicit sufficient conditions for $τ$-tilting infiniteness of a large family of algebras. This also produces concrete families of "minimal $τ$-tilting infinite algebras"-- the modern counterpart of min-rep-infinite algebras, independently introduced by the author and Wang. We further use our results on the family of non-distributive algebras to establish a conjectural connection between the $τ$-tilting theory and two geometric notions in the study of module varieties introduced by Chindris, Kinser and Weyman. We verify the conjectures for the algebras studied in this note: For the min-rep-infinite algebras which are non-distributive or biserial, we show that if $Λ$ has the dense orbit property, then it must be $τ$-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is $τ$-tilting finite. The latter result gives a categorical interpretation of Schur-representation-finiteness over this family of min-rep-infinite algebras.

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arXiv2019

$τ$-tilting finiteness of biserial algebras

In this paper we treat the $τ$-tilting finiteness of biserial (respectively special biserial) algebras over algebraically closed (respectively arbitrary) fields. Inside these families, to compare the notions of representation-finiteness and $τ$-tilting finiteness, we reduce the problem to the $τ$-tilting finiteness of minimal representation-infinite (special) biserial algebras. Building upon the classification of minimal representation-infinite algebras, we fully determine which minimal representation-infinite (special) biserial algebras are $τ$-tilting finite and which ones are not. To do so, we use the brick-$τ$-rigid correspondence of Demonet, Iyama and Jasso, and the classification of minimal representation-infinite special biserial algebras due to Ringel. Furthermore, we introduce the notion of minimal $τ$-tilting infinite algebras, analogous to the notion of minimal representation infinite algebras, and prove that a minimal representation-infinite (special) biserial algebra is minimal $τ$-tilting infinite if and only if it is a gentle algebra. As a consequence, we conclude that a gentle algebra is $τ$-tilting infinite if and only if it is representation infinite. We also show that for every minimal representation-infinite (special) biserial algebra, the notions of tilting finiteness and $τ$-tilting finiteness are equivalent. This implies that a mild (special) biserial algebra is tilting finite if and only if it is brick finite.

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arXiv2018

A categorification of biclosed sets of strings

We consider the closure space on the set of strings of a gentle algebra of finite representation type. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows. Finite congruence-uniform lattices admit an alternate partial order known as the shard intersection order. In many cases, the shard intersection order of a congruence-uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra. Analogous to torsion shadows, we introduce wide shadows, and prove that the shard intersection order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows.

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arXiv2018

On the combinatorics of gentle algebras

For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for Hom($X,τY$). We use this to describe support $τ$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $τ$-tilting modules and functorially finite torsion classes. We give an explicit basis of Ext$^1(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

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