Laurent Demonet

**Professional address :** Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602 Japan.

**Office :** A-329.

**Professional (japanese) phone :** 0081 52 789 2812

**Personal (french) phone :** 0033 970 44 970 8

- Cluster algebras;
- Representations of finite dimensional algebras;
- Combinatorics of representation varieties;
- Cohen-Macaulay modules;
- Categorifications of cluster algebras.

I defended my PhD thesis "Categorification of skew-symmetrizable cluster algebras" under the direction of Bernard Leclerc the 18th of November 2008.

**Abstract :** The aim of this thesis is to categorify some skew-symmetrizable cluster algebras. A lot of skew-symmetric cases have been handled for example by Keller, Caldero-Keller, Geiß-Leclerc-Schröer, Dehy-Keller, Fu-Keller, Palu. In order to do that, one uses stably 2-Calabi-Yau exact categories. For the skew-symmetrizable case, we consider an action of a finite group on such a category and we introduce an equivariant category which is also stably $2$-Calabi-Yau. We develop a theory of mutations for its invariant rigid objects. A large family of examples is given by categories of representations of preprojective algebras : for instance, the category of representations of the preprojective algebra of type $A_{2n-1}$ with its automorphism of order $2$ gives rise to the cluster algebra of functions over the unipotent Lie group of type $C_n$. In a similar way, we can get all the cluster algebras of functions over unipotent maximal subgroups of semi-simple Lie groups. Moreover, we obtain all the cluster algebras of finite type. All these categorifications lead to a proof, for the corresponding cluster algebras, of a conjecture of Fomin and Zelevinsky which states that the cluster monomials are linearly independent.

**Thesis :** Click here (french).

**Defense slides :** Click here (french).

I defended my Habilitation à diriger des recherches "Combinatorics of Mutations in Representation Theory" the 8th of November 2017.

**Abstract :** This memoir is a survey about my research topics and my own results. I discuss
certain problems situated at the boundary between representation theory of finite dimensional
algebras and combinatorics. The first part is an introduction about representations of finite
dimensional algebras and Cohen-Macaulay modules, with a focus on Auslander-Reiten theory.
These techniques are then exploited to develop certain topics of my research. First, a technique
to understand the lattice structure of the set of torsion classes on a finite dimensional algebra,
in particular the lattice quotients of these lattices. Secondly, several categorifications of cluster
algebras via Cohen-Macaulay modules: using triangulations of polygons on the one hand, and
more abstract results which permit in particular to categorify all multihomogeneous coordinate
rings on partial flag varieties, on the other hand. Lastly, a family of algebras constructed
combinatorially from partial triangulations of surfaces and certain of their most noticeable properties:
the existence of a mutation giving rise to derived equivalences, and their tameness (in other
terms, the fact that the module categories of these algebras can theoretically be understood
more easily).

**Thesis :** Click here.

**Cluster algebras and preprojective algebras : the non simply-laced case**, C. R. Math. Acad. Sci. Paris 346 (2008), no. 7-8, 379—384. We generalize to the non simply-laced case results of Geiß, Leclerc and Schröer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non simply-laced case can be seen as projections of cluster structures in the simply-laced case. This allows us to prove that cluster monomials are linearly independent in the non simply-laced case. (arXiv ; journal)

**Skew group algebras of path algebras and preprojective algebras**, J. Algebra 323, Issue 4 (2010), 1052—1059. We compute explicitly up to Morita-equivalence the skew group algebra of a finite group acting on the path algebra of a quiver and the skew group algebra of a finite group acting on a preprojective algebra. These results generalize previous results of Reiten and Riedtmann for a cyclic group acting on the path algebra of a quiver and of Reiten and Van den Bergh for a finite subgroup of $\operatorname{SL}(\mathbb{C} X + \mathbb{C} Y)$ acting on $\mathbb{C}(X,Y)$. (arXiv ; journal)

**Categorification of skew-symmetrizable cluster algebras**, Algebr. Represent. Theory 14 (2011), no. 6, 1087—1162. We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably $2$-Calabi-Yau category $\mathcal{C}$ endowed with the action of a finite group $G$, we construct a $G$-equivariant mutation on the set of maximal rigid $G$-invariant objects of $\mathcal{C}$. Using an appropriate cluster character, we can then attach to these data an explicit skew-symmetrizable cluster algebra. As an application we prove the linear independence of the cluster monomials in this setting. Finally, we illustrate our construction with examples associated with partial flag varieties and unipotent subgroups of Kac-Moody groups, generalizing to the non simply-laced case several results of Geiß-Leclerc-Schröer. (arXiv ; journal)

**Mutations of group species with potentials and their representations. Applications to cluster algebras**. This article tries to generalize former works of Derksen, Weyman and Zelevinsky about skew-symmetric cluster algebras to the skew-symmetrizable case. We introduce the notion of group species with potentials and their decorated representations. In good cases, we can define mutations of these objects in such a way that these mutations mimic the mutations of seeds defined by Fomin and Zelevinsky for a skew-symmetrizable exchange matrix defined from the group species. These good cases are called non-degenerate. Thus, when an exchange matrix can be associated to a non-degenerate group species with potential, we give an interpretation of the $F$-polynomials and the $\mathbf{g}$-vectors of Fomin and Zelevinsky in terms of the mutation of group species with potentials and their decorated representations. Hence, we can deduce a proof of a serie of combinatorial conjectures of Fomin and Zelevinsky in these cases. Moreover, we give, for certain skew-symmetrizable matrices a proof of the existance of a non-degenerate group species with potential realizing this matrix. On the other hand, we prove that certain skew-symmetrizable matrices can not be realized in this way. (arXiv)

**Example of a categorification of a cluster algebra**, Proceedings of the 44th Symposium on Ring Theory and Representation Theory, 30—42, Symp. Ring Theory Represent. Theory Organ. Comm., Nagoya, 2012. We present two detailed examples of additive categorifications of the cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a simple Lie group. The first one is of simply-laced type ($A_3$) and relies on an article by Geiß, Leclerc and Schrôoer. The second is of non simply-laced type ($C_2$) and relies on an article by the author of this note. This is aimed to be accessible, specially for people who are not familiar with this subject. (article)

**Quotients of exact categories by cluster tilting subcategories as module categories** (with Y. Liu), J. Pure Appl. Algebra 217 (2013), no. 12, 2282—2297. We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category $\mathcal{B}$ with enough projectives and injectives has a cluster tilting subcategory $\mathcal{M}$, then $\mathcal{B}/\mathcal{M}$ is abelian. More precisely, it is equivalent to the category of finitely presented modules over $\underline{\mathcal{M}}$. (arXiv ; journal)

**Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders** (with X. Luo), Trans. Amer. Math. Soc. 368 (2016), no. 6, 4257—4293. Given a triangulation of a polygon $P$ with $n$ vertices, we associate an ice quiver with potential such that the associated Jacobian algebra has the structure of a Gorenstein tiled $K[x]$-order $\Lambda$.
Then we show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is equivalent to the cluster category $\mathcal{C}$ of type $A_{n-3}$.
It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal{C}$ by triangulations of $P$. Moreover, it extends naturally the triangulated categorification by $\mathcal{C}$ of the cluster algebra of type $A_{n-3}$ to an exact categorification by adding coefficients corresponding to the sides of $P$. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay $\Lambda$-modules and the bounded derived category of modules over a path algebra of type $A_{n-3}$. (arXiv ; journal)

**Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules** (with X. Luo), Publ. Res. Inst. Math. Sci. 52 (2016), no. 2, 141—205. Given a tagged triangulation of a once-punctured polygon $P^*$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated frozen Jacobian algebra has the structure of a Gorenstein $K[X]$-order $\Lambda$.
Then we show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is equivalent to the cluster category $\mathcal{C}$ of type $D_n$.
It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal{C}$ by tagged triangulations of $P^*$. Moreover, it extends naturally the triangulated categorification by $\mathcal{C}$ of the cluster algebra of type $D_n$ to an exact categorification by adding coefficients corresponding to the sides of $P$. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay $\Lambda$-modules and the bounded derived category of modules over a path algebra of type $D_n$. (arXiv ; journal)

**Lifting preprojective algebras to orders and categorifying partial flag varieties** (with O. Iyama), Algebra & Number Theory 10 (2016), no. 7, 1527—1580. In this article, we describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of type $A$ and $D$ using Cohen-Macaulay modules over orders. To achieve this, we construct several equivalences of categories, relating Cohen-Macaulay modules over an order $A$ to finitely generated modules over certain finite length algebras obtained as quotient of $A$ by an idempotent. (arXiv ; journal)

**$\tau$-rigid finite algebras and $g$-vectors** (with O. Iyama and G. Jasso), Int. Math. Res. Not. (2017), 1—41. The class of support $\tau$-tilting modules was introduced recently by Adachi-Iyama-Reiten so as to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\tau$-rigid finite algebras, i.e. algebras with finitely many isomorphism classes of indecomposable $\tau$-rigid modules. We show that a finite dimensional algebra $A$ is $\tau$-rigid finite if and only if every torsion class in $\operatorname{mod} A$ is functorially finite. We also study combinatorial properties of $g$-vectors associated with $\tau$-tilting modules. Given a finite dimensional algebra $A$ with $n$ simple modules we construct an $(n-1)$-dimensional simplicial complex $\Delta(A)$ whose maximal faces are in bijection with the isomorphism classes of basic support $\tau$-tilting $A$-modules. We show that $\Delta(A)$ can be realized in the Grothendieck group of $\operatorname{mod} A$ using $g$-vectors. We show that if $A$ is a $\tau$-rigid finite algebra, then the geometric realization of $\Delta(A)$ is homeomorphic to an $(n-1)$-dimensional sphere. (arXiv ; journal)

**Introduction to algebras of partial triangulations**, Proceedings of the 49th Symposium on Ring and Representation Theory (2017). The aim of this note is to give a gentle introduction to algebras of partial triangulations of marked surfaces, following the structure of a talk given during the 49th symposium on ring theory and representation theory, held in Osaka. This class of algebras, which always have finite rank, contains classical Jacobian algebras of triangulations of marked surfaces and Brauer graph algebras. We discuss representation theoretical properties and derived equivalences. All results are proven in arXiv:1602.01592, under slightly milder hypotheses. (arXiv)

**$\operatorname{SL}_2$-tilings do not exist in higher dimensions (mostly)** (with P.-G. Plamondon, D. Rupel, S. Stella, P. Tumarkin). We define a family of generalizations of $\operatorname{SL}_2$-tilings to higher dimensions called $\varepsilon-\operatorname{SL}_2$-tilings. We show that, in each dimension 3 or greater, $\varepsilon-\operatorname{SL}_2$-tilings exist only for certain choices of $\varepsilon$. In the case that they exist, we show that they are essentially unique and have a concrete description in terms of odd Fibonacci numbers. (arXiv)

**Algebras of partial triangulations**. We introduce two classes of algebras coming from partial triangulations of marked surfaces. The first one, called frozen, is generally of infinite rank and contains frozen Jacobian algebras of triangulations of marked surfaces. The second one, called non-frozen, is always of (explicit) finite rank and contains non-frozen Jacobian algebras of triangulations of marked surfaces and Brauer graph algebras. We classify the partial triangulations, the frozen algebras of which are lattices over a formal power series ring. For non-frozen algebras, we prove that they are symmetric when the surface has no boundary. From a more representation theoretical point of view, we prove that these non-frozen algebras of partial triangulations are at most tame and we define a combinatorial operation on partial triangulation, generalizing Kauer moves of Brauer graphs and flips of triangulations, which give derived equivalences of the corresponding non-frozen algebras. (arXiv)

**Lattice theory of torsion classes** (with O. Iyama, N. Reading, I. Reiten, H. Thomas). For a finite-dimensional algebra $A$ over a field $k$, we consider the complete lattice $\operatorname{\mathsf{tors}} A$ of torsion classes. We introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of $\operatorname{\mathsf{tors}} A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}} (A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras $\Pi$, for which $\operatorname{\mathsf{tors}} \Pi$ is the Weyl group endowed with the weak order. In particular, we give a new proof of the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian lattice when $Q$ is a Dynkin quiver, which is more representation theoretical. We also prove that, in type $A$, the algebraic quotients of $\operatorname{\mathsf{tors}} \Pi$ are exactly its Hasse-regular lattice quotients. (arXiv)

**Action of finite groups on 2-Calabi-Yau categories and categorification of skew-symmetrizable
cluster algebras**, International Conference on Cluster Algebras and Related Topics, Mexico, 18/12/2008. (slides)

**Catégorification d'algèbres amassées antisymétrisables**, Séminaire d'algèbre, Saint-Étienne, 13/01/2009. (slides)

**Catégorification d'algèbres amassées antisymétrisables**, Séminaire Algèbre et gémétrie, Versailles, 20/01/2009. (slides)

**Catégorification d'algèbres amassées antisymétrisables**, Séminaire d'algèbre, I.H.P., Paris, 02/02/2009.

**Catégorification d'algèbres amassées antisymétrisables**, Séminaire d'algèbre, Besançon, 12/02/2009. (slides)

**Algèbres de groupe tordues des algèbres de chemins et des algèbres préprojectives**, Séminaire d'algèbre, Lyon, 19/02/2009.

**Catégorification d'algèbres amassées antisymétrisables**, Séminaire d'algèbre, Amiens, 08/04/2009.

**Algèbres amassées**, 7th Pan African Congress of Mathematicians, Yamoussoukro, 08/2009.

**Categorification of skew-symmetrizable cluster algebras**, Oberseminar Darstellungstheorie, Bonn, 11/11/2009.

**Group actions on generalized cluster categories**, Advanced School and Conference on Homological and Geometrical Methods in Representation Theory, ICTP, Trieste, 01/02/2010.

**Positivité totale, algèbres amassées et catégorification**, Séminaire d'algèbre, Université de Versailles, 09/02/2010.

**Positivité totale, algèbres amassées et catégorification**, Séminaire d'algèbre, Université de Strasbourg, 15/03/2010.

**Total positivity, cluster algebras and categorification** and **Categorification of skew-symmetrizable cluster algebras and application
to Kac-Moody groups**, Algebra-Topology Seminar, ETH Zürich, 2010.

**Total positivity, cluster algebras and categorification**, Oberseminar, MPIM, Bonn, 2010.

**Group species with potential and their applications to cluster algebras**, Algebra seminar, Bonn University, 2010.

**Mutations of group species with potentials and their representations. Applications to cluster algebras**, XIV International Conference on Representations of Algebras, Tokyo, 2010.

**Categorification of some skew-symmetric and skew-symmetrizable cluster algebras by categories of representations of preprojective algebras** - 2 exposés, Representation theory seminar, Nagoya University, 2010.

**Catégorification des algèbres amassées antisymétrisables**, Séminaire de topologie algébrique, Villetaneuse - Paris 13 University, 2011.

**Catégorification des algèbres amassées antisymétrisables**, Colloque tournant, Poitiers University, 2011.

**Categorification of cluster algebra structures of coordinate rings of simple Lie groups**, Symposium on Ring and Representation Theory, Okayama, 2011.

**Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups**, Lie groups and representation theory seminar, Tokyo University, 2011.

**Categorification of skew-symetrizable cluster algebras** - 2 talks, 10th Shizuoka Seminar on Algebra, Shizuoka, 2011.

**Quiver of skew-group algebras. Application to hereditary and preprojective algebras**, Conference on resolution of singularities and the McKay correspondence, Nagoya, 2012.

**Mutation of quiver with potential at several vertices**, XV International Conference on Representations of Algebras, Bielefeld, 2012.

**Mutation of quiver with potential at several vertices**, Symposium on Ring Theory and Representation Theory, Matsumoto, 2012.

**Cohen-Macaulay modules over orders associated with triangulations and cluster categories (type A and D)**, Perspectives of Representation Theory of Algebras, Nagoya, 2013.

**From categories of Cohen-Macaulay over orders to subcategories of modules categories**, XVI International Conference on Representations of Algebras, Sanya, 2014.

**From categories of Cohen-Macaulay modules over orders to subcategories of module categories, application to cluster algebras of homogeneous coordinate rings of partial flag varieties**, Cluster Algebras
and Representation Theory, Séoul, 2014.

**Cohen-Macaulay modules over orders associated with triangulations and cluster categories (type A and D)**, Cluster Algebras in Combinatorics and Topology, Séoul, 2014.

**Orders categorifying cluster algebras structures of partial flag varieties**, Workshop on Homological Interactions between Representation Theory and Singularity Theory, Edinburgh, 2014.

**Categorification of cluster algebras structures coming from Lie theory**, Series of three lectures, Winter School on Representation Theory, Tokyo (january 2016).

**Algèbres de triangulations partielles**, Séminaire d'algèbre et de géométrie, Université de Caen (march 2016).

**Algebras of partial triangulations**, Séminaire, Universität Bielefeld (march 2016).

**Orders categorifying cluster algebras structures of partial flag varieties**, Workshop on Cluster Algebras and Geometry, Universität Münster (march 2016).

**Algebras of partial triangulations**, Workshop on Brauer Graph Algebras, Universität Stuttgart (march 2016).

**Algèbres de triangulations partielles**, Séminaire d'algèbre, Université de Bourgogne (march 2016).

**Algèbres de triangulations partielles**, Séminaire Groupes, Représentations et Géométrie, Université Paris 7 (march 2016).

**Algebras of partial triangulations**, Seminario di Algebra e Geometria, Sapienza Universita di Roma (march 2016).

**Orders categorifying cluster algebras structures of partial flag varieties**, Algebra seminar, Bonn Universität (may 2016).

**Algebras of partial triangulations**, XVII International Conference on Representations of Algebras, Syracuse (august 2016).

**Algebras of partial triangulations**, Symposium on Ring Theory and Representation Theory, Osaka (september 2016).

**Algèbres de triangulations partielles**, Séminaire Groupes, Représentations et Géométrie, Université Paris 7 (march 2016).

**Treillis des classes de torsions**, Séminaire d’algèbre, IHP, Paris (may 2017).

**Lattices of torsion classes**, International Workshop on Cluster Algebras and Related Topics, Chern Institut of Mathematics, Tianjin (july 2017).

**Lattices of torsion classes**, Symposium on Ring Theory and Representation Theory, Yamanashi (october 2017).

Referee for Algebra and Number Theory, Archiv der Mathematik, Boletín de la Sociedad Matemática Mexicana, Communication in Algebra, Compositio Mathematica, Journal of Algebra, Journal of Algebra and its Applications, Journal of Algebraic Combinatorics, Journal of Pure and Applied Algebra, Mathematische Zeitschrift, Monatshefte für Mathematik, Proceedings of London Mathematical Society, Symmetry, Integrability and Geometry : Methods and Applications, Transaction of the American Mathematical Society.

Evaluator (quick opinion) for Advances in Mathematics, Bulletin of London Mathematical Society, Journal of the American Mathematical Society, Nagoya Mathematical Journal.

Member of the jury and the administration council of the French Federation of Mathematical Games.

Member of the administration council of the International Comity of Mathematical Games.