Perspectives in Tilting Theory and Related Topics

Let $\Bbbk$ be an algebraically closed field and $A$ be a finitely generated associative $\Bbbk$-algebra. The $A$-module structures on the vector space $\Bbbk^d$, $d\ge 1$, form an affine variety $\mathrm{mod}_A(d)$ called a module variety. The general linear group $\mathrm{GL}(d)$ acts regularly on $\mathrm{mod}_A(d)$ such that the orbits correspond bijectively to the isomorphism classes of $d$-dimensional $A$-modules. The orbits are locally closed subsets in $\mathrm{mod}_A(d)$ and their closures (in the Zariski topology) form an interesting class of affine varieties. One can ask if these varieties are nonsingular, regular in codimension 1 or 2, normal, unibranch, complete intersections, Cohen-Macaulay, etc. What is more interesting, we would like to know how these geometric properties are reflected in terms of $A$-modules. Instead of modules over a $\Bbbk$-algebra, we may consider finite dimensional representations of a quiver or a bound quiver. Here we get orbit closures as well. Bongartz showed that the orbit closures in module varieties and in the varieties of representations of the corresponding bound quivers are related by associated fibre bundles (in particular, the types of singularities are identical).

During the series of lectures we will present old and new results on local geometric properties of orbit closures in varieties of modules (or quiver representations), and their relationship with the spaces of homomorphisms or extensions between modules (or representations). We will also explain geometric relations between representations of quivers, and Schubert and affine Schubert varieties. In the last lecture, transversal slices in quiver varieties will be discussed.

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In this lecture series, I will discuss silting theory and various related topics. In the first lecture, I will provide an overview of the basic properties of partial order and mutation of silting complexes, as well as their connections to objects in the derived category. In the second lecture, I will focus on 2-term silting complexes and explore their relationships with objects in the module category, such as tau-tilting modules and semibricks. These topics are interesting in their own right and are still actively studied by many researchers. In the third lecture, I will address the properties of fans associated with the g-vectors of 2-term silting complexes. Specifically, I will discuss how the properties of silting complexes, discussed in the first and second lectures, are reflected in the characteristics of fans. Furthermore, I will explore how the properties of fans relate to those of representation theory, and examine their mutual relationships.

For a given quiver and a given dimension vector, we have a group acting on the representation space by base change. We will discuss polynomial invariants and semi-invariants for this group actions. Following A. King, semi-invariants and geometric invariant theory can be used to construct moduli spaces for quiver representations (and more generally modules over an algebra). Studying dimension vectors of representations leads to a very rich combinatorial structure. Some combinatorical topics that we will encounter are root systems of Kac-Moody Lie algebras, Littlewood-Richardson coefficients, the braid group action on exceptional sequences, Kac' canonical decomposition of dimension vectors, cluster algebras and simplicial complexes.

Slides

The McKay correspondence relates the geometry of nice resolutions of a quotient singularity and the representation theory of the corresponding finite group. It is usually considered for small subgroups $G$ of $GL(n, \mathbb{C})$, where $G$ is said to be small if $G$ contains no complex reflection. To consider the McKay correspondence for complex reflection groups, we consider not just the quotient variety (which is smooth) but the pair consisting of the quotient variety and the discriminant divisor with suitable coefficients. There is a conjectural semi-orthogonal decomposition of the $G$-equivariant derived category by Polishchuk and Van den Bergh, indexed by the conjugacy classes in $G$. In dimension two, the conjecture follows from a theorem of Kawamata. In this talk, we discuss some cases in dimension three by using the notion of the maximal $\mathbb{Q}$-factorial terminalization of the pair, which Kawamata used to study $GL(3)$-McKay correspondence.

Composition series is fundamental in the study of finite groups and finite dimensional modules. One of the most important properties of such composition series is the Jordan-Hölder property, and this implies the property (called the Jordan–Dedekind property) that all composition series have the same length. In this talk, I will introduce the notion of composition series for triangulated categories, and discuss composition series of derived categories of certain finite dimensional algebras and smooth projective varieties. In particular, I will explain that the Jordan–Dedekind property does not hold for derived categories of certain finite dimensional algebras of finite global dimension and certain smooth projective toric surfaces. This talk is based on joint work with Kalck and Ouchi.

Silting-discreteness of finite dimensional algebras has been actively studied in recent years. One of the motivations for studying silting-discreteness is that over silting-discrete algebras, any two silting complexes are connected by iterative irreducible silting mutations. In this talk, we examine when group algebras are silting-discrete. For a finite group $G$ and an algebraically closed field $k$ of positive characteristic $p$, we give a sufficient condition for a group algebra $kG$ to be silting-discrete in terms of a $p$-hyperfocal subgroup of $G$. Moreover, we see that this is also a necessary condition in some cases. This talk is based on a joint work with Yuta Kozakai.

Slides

We study singularity categories of Gorenstein algebras. There are many such rings whose singularity categories are Calabi-Yau, for example, preprojective algebras, their quotients by tilting ideals, cluster tilted algebras, and so on. These algebras and categories have played a prominent role in the development of cluster theory. We will discuss a lift of these Calabi-Yau properties to their dg enhancements. This is based on joint works with Bernhard Keller.

We investigate the set of pairwise Hom-orthogonal modules in the context of several open conjectures that have emerged in recent years, to which we refer as the brick-Brauer-Thrall (bBT) Conjectures. The bBT conjectures are closely connected to the study of bricks, and therefore to wide subcategories, torsion pairs, $\tau$-tilting theory, stability conditions, g-fan, and related subjects. In this talk, we first adopt a geometric perspective to see the significance of Hom-orthogonality in the context of a conjecture that I posed in 2019, now known as the Second Brick-Brauer-Thrall (2nd bBT) Conjecture. Then, we show that some of the more recent bBT conjectures actually follow from the 2nd bBT conjecture. This provides new insights into these challenging open problems. As a result, we are able to verify the validity of the bBT conjectures for some new families of algebras. This talk is primarily based on a recent joint work (arXiv:2407.20877) with Charles Paquette.

Hernandez-Leclerc studied certain Jacobian algebras of quivers with potentials called graded preprojective algebras. The generating functions of Euler characteristics of submodule Grassmannians of some modules over graded preprojective algebras give $q$-characters of certain class of representations of quantum affine algebras. In this talk, we will discuss the modules over graded preprojective algebras which are induced from some rigid modules over preprojective algebras with suitable grading, and give similar generating functions and equalities which they satisfy. This is a report on an ongoing joint work with Bernard Leclerc.

In this lecture, I will outline an introductory theory of deformation and degeneration of modules over rings. If time permits, I will also mention their generalization to differential graded modules.

This talk is based on joint work with Ueyama and Iyama. The existence of tilting or silting objects is a significant feature of algebraic triangulated categories, as it establishes an equivalence with the derived category of a ring. In this study, we focus on the existence of tilting objects in the stable category of Cohen–Macaulay modules over Artin–Schelter Gorenstein algebras. These algebras extend the concept of Gorenstein commutative rings from the perspective of noncommutative algebraic geometry. In the representation theory of Gorenstein commutative rings, the Gorenstein parameter plays a crucial role. This talk provides a characterization of the existence of tilting objects in stable categories using Gorenstein parameters. Our result is a noncommutative generalization of the results established by Buchweitz, Iyama and Yamaura.

In this talk, we generalize the algebraic McKay correspondence and the classification result of 2-dimensional rings of finite Cohen-Macaulay type to the case where the base field is non-algebraically closed. Moreover, to draw McKay quivers, we give a recipe to determine the irreducible representations of skew group algebras.

Based on results by Adachi-Iyama-Reiten, Marks-Šťovíček, Pauksztello-Zvonareva and Adachi-Tsukamoto, García successfully completed a commutative 'triangular prism' of bijections connecting the classes of support tau-tilting modules, functorially-finite torsion pairs and left finite wide subcategories in the category of finitely-generated $A$-modules — where $A$ is a finite-dimensional algebra over an algebraically closed field — to the classes of 'silting objects', complete cotorsion pairs and thick subcategories with enough injectives in the category of projective presentations of objects in $\mathrm{mod}(A)$ — which has many powerful properties. In this talk, we will present advances towards generalizing these results to the realm of infinite-dimensional modules, as well as their dualizations. It is based on joint work in progress with Lidia Angeleri Hügel.

Slides

By Koenig and Yang's result, there exists a bijection between basic silting objects of $K^b(proj\Lambda)$ and simple-minded collections of $D^b(mod\Lambda)$ for every finite dimensional algebra $\Lambda$. By taking the dg-End algebra, we obtain a non-positive dg algebra from silting objects and a positive dg algebra from simple-minded collection. In this talk, I will connect these two classes of dg algebras via Koszul duality and present applications to representation theory and triangulated category theory.

A conjecture by Ishii states that for a finite subgroup $G$ of $GL(2,\mathbb{C})$, a resolution $Y$ of $\mathbb{C}^2/G$ is isomorphic to a moduli space $\mathcal{M}_{\theta}$ of $G$-constellations for some generic stability parameter $\theta$ if and only if $Y$ is dominated by the maximal resolution. This paper affirms the conjecture in the case of dihedral groups as a class of complex reflection groups, and offers an extension of McKay correspondence.

Over complex numbers, McKay correspondence can be described as a relation between irreducible representations of finite groups and geometric properties of the associated quotient singularities, such as Euler characteristic of crepant resolutions. This relation is known as Batyrev's theorem. When the ground field is replaced by an algebraically closed field of prime characteristic, the naive analog of Batyrev's theorem fails for modular groups in general. In this talk, after giving a necessary introduction of background, I will introduce a series of specific modular groups for which the analogous McKay correspondence in positive characteristic holds, and discuss about a potential way to adjust the analog as a conjecture, such that it may hold for more modular groups.

I will survey how the idea of tilting can be used in various functor categories including the category of strict polynomial functors $P_d$, which is closely related to the category of representations of $GL_n$. More specifically I will discuss such topics as Koszul duality, de Rham complex and certain form of the Hilbert-Riemann correspondence, which can be studied in $P_d$.

Slides

Persistence modules are representations of a certain quiver with relation used for topological data analysis. We show that the set of irreducible components of moduli space of 2D persistence module has a structure of a Kashiwara crystal. In the $2 \times 2$ case, we give an explicit description of the crystal structure.

Classifying subcategories is an active subject in the representation theory of algebras. Especially, several subcategories of the module category of a commutative noetherian ring have been classified so far. In this talk, we will give a classification result of KE-closed subcategories (additive subcategories closed under extensions and kernels) for a commutative noetherian ring.

$k$-linear DG-categories have been understood as one of the models for $k$-linear $(\infty,1)$-categories. Consequently, certain special DG-categories can be regarded as $k$-linear $(n,1)$-categories. Inspired by this model, we introduce the notion of abelian $(n,1)$-categories as a higher-dimensional generalization of abelian categories. This concept recovers abelian categories when $n=1$ and stable DG-categories when $n=\infty$.

In this talk, we will explain this notion and their basic properties and show that their homotopy categories naturally acquire structures such as extriangulated categories and pretriangulated categories. Moreover, we show that an abelian $(n,1)$-category is a suitable candidate as a DG-enhancement of $n$-extended module category, that is, the subcategory of the derived category consisting of $n$-term complexes of modules. These categories serve as a natural setting for developing for higher $\tau$-tilting theory and their associated $(n+1)$-term silting theory.