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Hipolito Treffinger

Publications and source records attributed to Hipolito Treffinger.

At least 19 recordsLinked to original sources
arXiv2026

Scattering diagrams for Artin algebras

For an arbitrary Artin algebra $A$, we construct a minimal and consistent scattering diagram by approximating its module category $\mathrm{mod}\,A$ using the subcategories $(\mathrm{mod}\,A)_\ell$ of modules of length at most $\ell \in \mathbb{N}$. We prove that each subcategory $(\operatorname{mod}A)_\ell$ possesses a well-behaved lattice of torsion classes, a finite wall-and-chamber structure $\mathfrak{D}_\ell(A)$ and an associated picture group $G_\ell(A)$ with a natural categorical interpretation. Using these properties, we build a finite, minimal, consistent scattering diagram for each $\ell \in \mathbb{N}$. Passing to the inverse limit, we establish the existence of a minimal consistent scattering diagram for $A$. In particular, when $A$ is a finite-dimensional algebra over $\mathbb{C}$, our inverse limit construction is canonically isomorphic to Bridgeland's stability scattering diagram.

math.RT↗
arXiv2026

Higher torsion classes, $\tau_d$-tilting theory and silting complexes

Initiated in work by Adachi, Iyama and Reiten, the area known as $\tau$-tilting theory plays a fundamental role in contemporary representation theory. In this paper we explore a higher-dimensional analogue of this theory, formulated with respect to the higher Auslander-Reiten translation $\tau_d$. In particular, we associate to any functorially finite $d$-torsion class a maximal $\tau_d$-rigid pair and a $(d+1)$-term silting complex. In the case $d=1$, the notions of maximal $\tau_d$-rigid and support $\tau$-tilting pairs coincide, and our theory recovers the classical bijections. However, the proof strategies for $d>1$ differ significantly. As an intermediate step, we prove that a $d$-cluster tilting subcategory of a module category induces a $d$-cluster tilting subcategory of the category of $(d+1)$-term complexes, producing novel examples of $d$-exact categories. We introduce the notion of a $d$-torsion class in the exact setup, and use this to obtain the aforementioned $(d+1)$-term silting complex. We moreover apply our theory to study $d$-APR tilting modules and slices. To illustrate our results, we provide explicit combinatorial descriptions of maximal $\tau_d$-rigid pairs and $(d+1)$-term silting complexes for higher Auslander and higher Nakayama algebras.

math.RT↗
arXiv2025

Algebras determined by $τ$-slices

In this paper we revisit the notion of strict laura algebras through the lens of $τ$-tilting theory to define the family of algebras determined by $τ$-slices. We show that the representation dimension of every algebra determined by $τ$-slices satisfying mild conditions is at most three.

math.RA↗
arXiv2025

Deformation Theory for Finite Cluster Complexes

We study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In particular, we show that in the skew-symmetric case, these cluster complexes are unobstructed, generalizing a result of Ilten and Christophersen in the $A_n$ case. We also study the connection between cluster algebras with universal coefficients and cluster complexes. We show that for a full rank positively graded cluster algebra $\mathcal{A}$ of geometric and finite cluster type, the cluster algebra $\mathcal{A}^{\mathrm{univ}}$ with universal coefficients may be recovered as the universal family over a partial closure of a torus orbit in a multigraded Hilbert scheme. Likewise, we show that under suitable hypotheses, the cluster algebra $\mathcal{A}^{\mathrm{univ}}$ may be recovered as the coordinate ring for a certain torus-invariant semiuniversal deformation of the Stanley-Reisner ring of the cluster complex. We apply these results to show that for any cluster algebra $\mathcal{A}$ of geometric and finite cluster type, $\mathcal{A}$ is Gorenstein, and $\mathcal{A}$ is unobstructed if it is skew-symmetric. Moreover, if $\mathcal{A}$ has enough frozen variables then it has no non-trivial torus-invariant deformations. We also study the Gröbner theory of the ideal of relations among cluster and frozen variables of $\mathcal{A}$. As a byproduct we generalize previous results in this setting obtained by Bossinger, Mohammadi and Nájera Chávez for Grassmannians of planes and $\text{Gr}(3,6)$.

math.AG↗
arXiv2024

An algebraic approach to Harder-Narasimhan filtrations

In this article we study chains of torsion classes in an abelian category $\mathcal{A}$. We prove that each chain of torsion classes induce a Harder-Narasimhan filtration for every nonzero object $M$ in $\mathcal{A}$, generalising a well-known property of stability conditions. We also characterise the slicings of $\mathcal{A}$ in terms of chain of torsion classes. We finish the paper by showing that chains of torsion classes induce wall-crossing formulas in the completed Hall algebra of the category.

math.CT↗
arXiv2024

A characterisation of higher torsion classes

Let $\mathcal{A}$ be an abelian length category containing a $d$-cluster tilting subcategory $\mathcal{M}$. We prove that a subcategory of $\mathcal{M}$ is a $d$-torsion class if and only if it is closed under $d$-extensions and $d$-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the $d$-torsion classes in $\mathcal{M}$ form a complete lattice. Moreover, we use the characterisation to classify the $d$-torsion classes associated to higher Auslander algebras of type $\mathbb{A}$, and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras.

math.RT↗
arXiv2023

Weak stability conditions and the space of chains of torsion classes

In this paper we show an explicit relation between chains of torsion classes and weak stability conditions over an abelian category. In particular, up to a natural equivalence, they coincide. We investigate topological properties of the space of chains of torsion classes and its quotient given by this equivalence relation. In particular we show that this space is compact if and only if the abelian category has finitely many torsion classes.

math.RT↗
arXiv2022

Stability spaces of string and band modules

The stability space of a module is the cone of vectors which make the module semistable. These cones are defined in terms of inequalities; in this paper we draw insights from considering the dual description in terms of non-negative linear spans. We show how stability spaces of thin modules are related to order polytopes. In the case of non-thin modules, we show how the stability spaces of string and band modules are related to the stability spaces of the thin modules corresponding to the abstract string and band. We use this to analyse the way in which the stability space of a band module is the limit of stability spaces of string modules. Namely, the stability space of the band module is a union of cones, each of which is the limit of the stability spaces of a family of string modules.

math.RT↗
arXiv2022

On $τ$-tilting subcategories

The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent modules. If $\mathscr{A}={\rm{Mod\mbox{}}R}$, where $R$ is an unitary associative ring, we characterize all support $τ$-tilting, resp. all support $τ^-$-tilting, subcategories of ${\rm{Mod\mbox{}}R}$ in term of finendo quasitilting, resp. quasicotilting, modules. As a result, it will be shown that every silting module, respectively every cosilting module, induces a support $τ$-tilting, respectively support $τ^{-}$-tilting, subcategory of ${\rm{Mod\mbox{}}R}$. We also study the theory in ${\rm Rep}(Q, \mathscr{A})$, where $Q$ is a finite and acyclic quiver. In particular, we give an algorithm to construct support $τ$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from certain support $τ$-tilting subcategories of $\mathscr{A}$ and present a systematic way to construct $(n+1)$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from $n$-tilting subcategories in $\mathscr{A}$.

math.RT↗
arXiv2022

The size of a stratifying system can be arbitrarily large

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consists on stratifying systems of infinite size in the module category of an algebra $A$. In the second family of examples we show that the size of a finite stratifying system in the module category of a finite dimensional algebra $A$ can be arbitrarily large in comparison to the number of isomorphism classes of simple $A$-modules. We note that both families of examples are built using well-established results in higher homological algebra.

math.RT↗
arXiv2022

$τ$-tilting theory -- An introduction

The notion of $τ$-tilting theory was introduced by Adachi, Iyama and Reiten at the beginning of the last decade and quickly became one of the most active areas of research in the representation theory of finite dimensional algebras. The aim of these notes is two-fold. On the one hand, we want to give a friendly introduction to $τ$-tilting theory to anyone with a small background in representation theory. On the other, we want to fill the apparent gap for a survey on the subject by collecting in one place many of the most important results in $τ$-tilting theory.

math.RT↗
arXiv2022

Characterisations of trivial extensions

In this paper we give a characterisation of trivial extension algebras in terms of quivers with relations. This result is based on a explicit description of the ideal of relations of the trivial extension of an algebra, given by the first author in the appendix. We also give a new proof of Wakamatsu's theorem in terms of their quiver and relations, which determines when two given algebras have isomorphic trivial extensions.

math.RA↗
arXiv2022

On higher torsion classes

Building on the embedding of an $n$-abelian category $\mathscr{M}$ into an abelian category $\mathcal{A}$ as an $n$-cluster-tilting subcategory of $\mathcal{A}$, in this paper we relate the $n$-torsion classes of $\mathscr{M}$ with the torsion classes of $\mathcal{A}$. Indeed, we show that every $n$-torsion class in $\mathscr{M}$ is given by the intersection of a torsion class in $\mathcal{A}$ with $\mathscr{M}$. Moreover, we show that every chain of $n$-torsion classes in the $n$-abelian category $\mathscr{M}$ induces a Harder-Narasimhan filtration for every object of $\mathscr{M}$. We use the relation between $\mathscr{M}$ and $\mathcal{A}$ to show that every Harder-Narasimhan filtration induced by a chain of $n$-torsion classes in $\mathscr{M}$ can be induced by a chain of torsion classes in $\mathcal{A}$. Furthermore, we show that $n$-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) $n$-torsion classes.

math.RT↗
arXiv2021

Towards a categorification of scattering amplitudes

Categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential is reported. Amplitudes for cubic theories are directly written down in terms of projectives of hearts of intermediate $t$-structures restricted to the cluster category of quiver representations, without recourse to geometry. It is shown that for theories with $ϕ^{m+2}$ potentials those corresponding to $m$-cluster categories are to be used. The case of generic polynomial potentials is treated and our results suggest the existence of a generalization of higher cluster categories which we call pseudo-periodic categories. An algorithm to obtain the projectives of hearts of intermediate $t$-structures for these types is presented.

hep-th↗
arXiv2021

Stratifying systems and $g$-vectors

In this paper we study the Cartan matrix associated to the Ext-projective stratifying system induced by a basic and $τ$-rigid object $M$ in mod$(A)$ by means of the $g$-vectors of the indecomposable direct summands of $M$. In particular we show that the Cartan group of a stratifying system associated to a $τ$-rigid module can be calculated directly using these vectors. Moreover we characterise the stratifying systems coming from $τ$-rigid modules that have a diagonal Cartan matrix.

math.RT↗
arXiv2020

On band modules and $τ$-tilting finiteness

In this paper, motivated by a $τ$-tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $τ$-tilting finite if and only if no band module is a brick. We also recover a criterion for the $τ$-tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

math.RT↗