arXiv subjects

Charles Paquette

Publications and source records attributed to Charles Paquette.

At least 19 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

Generalized permutation matrices and non-weight modules over $\mathfrak{sl}(m|1)$

We study the category $\mathcal{M}_{\mathfrak{sl}(m|1)}(k|k)$ of $\mathcal U(\mathfrak h)\text{-free}$ $\mathcal U(\mathfrak{sl}(m|1))$-modules of rank $k$ in each parity (rank $(k|k)$), where $k\in\mathbb{Z}_{\geq1}$. We construct an explicit family of such modules, provide an isomorphism theorem, and establish an indecomposability criterion.

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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

Inversion Sets and Quotient Root Systems

The main result of this paper is a recursive description of all decompositions \[ Δ^+ = Φ_1 \sqcup Φ_2 \sqcup \dots \sqcup Φ_k \] of the positive roots $Δ^+$ of an arbitrary root system $Δ$ into a disjoint union of inversion sets. Such decompositions play a central role in geometric invariant theory (GIT) in connection with studying the Littlewood-Richardson cone and related problems. This work can be considered as a continuation of the work of Dewji, Dimitrov, McCabe, Roth, Wehlau, and Wilson in which similar questions were studied for root systems of type $\mathbb{A}$. Their methods relied on properties of permutations and are not transferable to an arbitrary root system. In order to develop a type-independent approach, we go beyond root systems and consider quotient root systems (QRSs for short). We study subsets of positive roots in an arbitrary QRS $R$. We prove that every $Φ\subseteq R^+$ can be represented in a canonical way as an inflation and develop methods to study recursively properties of such subsets. We extend the notion of an inversion to subsets of any QRS, i.e., beyond the case where a Weyl group is associated with $R$. If $Φ\subseteq R^+$ is an inversion set, we introduce a graph $\text{G}(Φ)$ and endow the set Comp$(Φ)$ of connected components of $\text{G}(Φ)$ with a partial addition. The resulting monoid-like structure (Comp$(Φ),+)$ is a further generalization of root systems beyond QRSs. We study in detail the properties of (Comp$(Φ),+)$ and their applications to studying the properties of $Φ$. In particular, we investigate the relationship between $Φ$ being primitive and $Φ$ being irreducible. Apart from describing recursively all decompositions of $Δ^+$ into the disjoint union of inversion sets, we provide applications to GIT and derive enumerative results which may be of independent interest.

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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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arXiv2025

Categories of generalized thread quivers

We study the representation category of thread quivers and their quotients. A thread quiver is a quiver in which some arrows have been replaced by totally ordered sets. Pointwise finite-dimensional (pwf) representations of such a thread quiver admit a Krull-Remak-Schmidt-Azumaya decomposition. We show that an indecomposable representation is induced from an indecomposable representation of a quiver obtained from the original quiver by replacing some of its arrows by a finite linear $\mathbb{A}_n$ quiver. We study injective and projective pwf indecomposable representations and we fully classify them when the quiver satisfies a mild condition. We give a characterization of the indecomposable pwf representations of certain categories whose representation theory has similar properties to finite type or tame type. We further construct new hereditary abelian categories, including a Serre subcategory of pwf representations that includes every indecomposable representation.

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arXiv2024

Galois Coverings, $τ$-Rigidity and Mutations

For an algebraically closed field $\mathbb{K}$, we consider a Galois $G$-covering $\mathcal{B} \to \mathcal{A}$ between locally bounded $\mathbb{K}$-categories given by bound quivers, where $G$ is torsion-free and acts freely on the objects of $\mathcal{B}$. We define the notion of $(G,τ_{\mathcal{B}})$-rigid subcategory and of support $(G,τ_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. These are the analogues of the similar concepts in the context of a finite-dimensional algebra, where we additionally require that the subcategory be $G$-equivariant. When $\mathcal{A}$ is a finite-dimensional algebra, we show that the corresponding push-down functor $\mathcal{F}_λ: \mathcal{B}$-$\rm mod$ $\to \mathcal{A}$-$\rm mod$ sends $(G,τ_{\mathcal{B}})$-rigid subcategories (respectively support $(G,τ_{\mathcal{B}})$-tilting pairs) to $τ_{\mathcal{A}}$-rigid modules (respectively support $τ_{\mathcal{A}}$-tilting pairs). We further show that there is a notion of mutation for support $(G,τ_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. Mutations of support $τ_\mathcal{A}$-tilting pairs and of support $(G,τ_\mathcal{B})$-tilting pairs commute with the push-down functor. We derive some consequences of this, and in particular, we derive a $τ$-tilting analogue of the result of P. Gabriel that locally representation-finiteness is preserved under coverings. Finally, we prove that when the Galois group $G$ is finitely generated free, any rigid $\mathcal{A}$-module (and in particular $τ_\mathcal{A}$-rigid $\mathcal{A}$-modules) lies in the essential image of the push-down functor.

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arXiv2023

Idempotents in the group algebra of the infinite dihedral group

We prove that over an algebraically closed field $\mathbb{K}$ of characteristic different from $2$, the group algebra $R=\mathbb{K} D_\infty$ of the infinite dihedral group $D_\infty$ has exactly six conjugacy classes of involutions (equivalently, of idempotents). This allows us to recover the fact that $R$ admits exactly four non-isomorphic indecomposable projective modules of the form $eR$ where $e$ is an idempotent, a result that was first established by Berman and Buzási.

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arXiv2023

Semi-Invariant Rings: UFD and Codimension One Orbits

Let $A$ be a finite dimensional associative $\mathbb{K}$-algebra over an algebraically closed field $\mathbb{K}$ of characteristic zero. To $A$, we can associate its basic form that is given by a quiver $Q = (Q_0, Q_1)$ with an admissible ideal $R$. For a dimension vector $β$, we consider an irreducible component $\mathcal{C}$ of the module variety of $β$-dimensional representations of $A$. The reductive group ${\rm GL}_β(\mathbb{K}):= \prod_{i \in Q_0}{\rm GL}_{β_i}(\mathbb{K})$ acts on $\mathcal{C}$ by change of basis, and has a unique closed orbit. We consider the corresponding ring of semi-invariants ${\rm SI}(Q, \mathcal{C})$. We prove that if $\mathcal{C}$ is factorial and has maximal orbits of codimension one, then ${\rm SI}(Q, \mathcal{C})$ is a complete intersection and is not multiplicity free. If $\mathcal{C}$ is not factorial, then this conclusion does not necessarily hold. We present examples showing that the codimension of the complete intersection can be arbitrarily large. Finally, we interpret our results in the case of hereditary algebras.

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arXiv2022

Representations of free products of semisimple algebras via quivers

Let $\mathbb{K}$ denote an algebraically closed field and $A$ a free product of finitely many semisimple associative $\mathbb{K}$-algebras. We associate to $A$ a finite acyclic quiver $Γ$ and show that the category of finite dimensional $A$-modules is equivalent to a full subcategory of the category ${\rm rep}(Γ)$ of finite dimensional representations of $Γ$. Under this equivalence, the simple $A$-modules correspond exactly to the $θ$-stable representations of $Γ$ for some stability parameter $θ$. This gives us necessary conditions for an $A$-module to be simple, conditions which are also sufficient if the module is in general position. Even though there are indecomposable modules that are not simple, we prove that a module in general position is always semisimple. We also discuss the construction of arbitrary finite dimensional modules using nilpotent representations of quivers. Finally, we apply our results to the case of a free product of finite groups when $\mathbb{K}$ has characteristic zero.

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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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arXiv2021

Subregular $J$-rings of Coxeter systems via quiver path algebras

We study the subregular $J$-ring $J_C$ of a Coxeter system $(W,S)$, a subring of Lusztig's $J$-ring. We prove that $J_C$ is isomorphic to a quotient of the path algebra of the double quiver of $(W,S)$ by a suitable ideal that we associate to a family of Chebyshev polynomials. As applications, we use quiver representations to study the category mod-$A_K$ of finite dimensional right modules of the algebra $A_K=K\otimes_\Z J_C$ over an algebraically closed field $K$ of characteristic zero. Our results include classifications of Coxeter systems for which mod-$A_K$ is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules. Incidentally, we show that every group algebra of a free product of finite cyclic groups is Morita equivalent to the algebra $A_K$ for a suitable Coxeter system; this allows us to specialize the classifications to the module categories of such group algebras.

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arXiv2021

Completions of discrete cluster categories of type $\mathbb{A}$

We complete the discrete cluster categories of type $\mathbb{A}$ as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom-finite Krull-Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom-spaces in this new category can be described geometrically, even though the category is not $2$-Calabi-Yau and Ext-spaces are not always symmetric. We describe all cluster-tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite dimensional sub-representations of infinite dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi-Yau conditions.

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