Quivers with potentials and their representations I: Mutations

Electronic Research Archive

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.

Boletín de la Sociedad Matemática Mexicana, 2015

We show that Jacobian algebras arising from a sphere with n-punctures, with n ≥ 5, are finite dimensional algebras. We consider also a family of cyclically oriented quivers and we prove that, for any primitive potential, the associated Jacobian algebra is finite dimensional.

Communications in Mathematical Physics, 2019

We review the categorical approach to the BPS sector of a 4d N = 2 QFT, clarifying many tricky issues and presenting a few novel results. To a given N = 2 QFT one associates several triangle categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than on what we can learn from a single description. A second general theme is viewing the cluster category as a sort of 'categorification' of 't Hooft's theory of quantum phases for a 4d non-Abelian gauge theory. The S-duality group is best described as the auto-equivalences of the full web of categories. This viewpoint leads to a combinatorial algorithm to search for S-dualities of the given N = 2 theory. If the ranks of the gauge and flavor groups are not too big, the algorithm may be effectively run on a laptop. This viewpoint also leads to a clearer view of 3d mirror symmetry. For class S theories, all the relevant triangle categories may also be constructed in terms of geometric objects on the Gaiotto curve, and we present the dictionary between triangle categories and the WKB approach of GMN. We also review how the VEV's of UV line operators are related to cluster characters.

Journal of High Energy Physics, 2019

Argyres and co-workers started a program to classify all 4d N = 2 QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 N = 2 QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces. The classification of 4d N = 2 QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces. In the follow-up paper we apply the RT methods to higher rank 4d N = 2 SCFT.

Journal of Topology, 2020

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability conditions on the 3-Calabi-Yau category is simply connected in the sequel [19]. Contents 1. Introduction 1 2. Preliminaries 5 3. An alternative presentation of surface braid group 11 4. A finite presentation for braid twist group 14 5. Finite presentations for spherical twist groups 23 Appendix A. Proof of the relations (4.30)-(4.43) in Section 4 35 Appendix B. Calculations for the proof of Proposition 5.6 39 References 41 Key words and phrases. braid twist group, mapping class group, spherical twist, quiver with potential.

Association for Women in Mathematics Series, 2018

In this survey chapter, we explain the intricate links between Conway-Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton.

Journal of Algebra, 2021

Let X be a weighted projective line and C X the associated cluster category. It is known that C X can be realized as a generalized cluster category of quiver with potential. In this note, under the assumption that X has at most three weights or is of tubular type, we prove that if the generalized cluster category C (Q,W) of a Jacobi-finite non-degenerate quiver with potential (Q, W) shares a 2-CY tilted algebra with C X , then C (Q,W) is triangle equivalent to C X. As a byproduct, a 2-CY tilted algebra of C X is determined by its quiver provided that X has at most three weights. To this end, for any weighted projective line X with at most three weights, we also obtain a realization of C X via Buan-Iyama-Reiten-Scott's construction of 2-CY categories arising from preprojective algebras. Lenzing [BKL].

International Mathematics Research Notices, 2018

Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all p...

Advances in Mathematics

Dedicated to Idun Reiten on the occasion of her 75th birthday.

Algebra & Number Theory, 2010

We associate an algebra A() to a triangulation of a surface S with a set of boundary marking points. This algebra A() is gentle and Gorenstein of dimension one. We also prove that A() is cluster-tilted if and only if it is cluster-tilted of type ‫ށ‬ or‫,ށ‬ or if and only if the surface S is a disc or an annulus. Moreover all cluster-tilted algebras of type ‫ށ‬ or‫ށ‬ are obtained in this way.

Journal of Algebraic Combinatorics, 2010

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of nonlinear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations. We extend our construction to include recurrences with parameters, giving an explanation of some observations made by Gale. Finally, we point out a connection between quivers which arise in our classification and those arising in the context of quiver gauge theories.

Bulletin of the London Mathematical Society

We demonstrate that the blocks of a profinite group whose defect groups are cyclic have a Brauer tree algebra structure analogous to the case of finite groups. We show further that the Brauer tree of a block with defect group Zp is of star type.

Advances in Mathematics

We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra is 2n, where n is the number of crossing points in the knot diagram. We then construct 2n indecomposable modules T (i) over the Jacobian algebra of the quiver with potential. For each T (i), we show that the submodule lattice is isomorphic to the corresponding lattice of Kauffman states. We then give a realization of the Alexander polynomial of the knot as a specialization of the F-polynomial of T (i), for every i. Furthermore, we conjecture that the collection of the T (i) forms a cluster in the cluster algebra whose quiver is isomorphic to the opposite of the initial quiver, and that the resulting cluster automorphism is of order two.