Members of SC3A

Quick presentation of the project

Fomin and Zelevinsky invented cluster algebras in early 2000 in order to find a combinatorial approach to the study of Luzstig and Kashiwara’s canonical bases in quantum groups, and to total positivity in semisimple groups. The formalism they developed found many applications beyond the scope of their initial goals. A noticeable example is the fact that Fomin and Zelevinsky discovered a phenomenon of simplification of rational fractions, called “Laurent phenomenon”. In the study of rational sequences defined by recurrence relations (such as the Gale-Robinson sequence or the Somos sequences), the Laurent phenomenon implies that the sequences under consideration take integer values. A second remarkable example is the proof of Zamolodchikov’s periodicity conjecture by B. Keller.

Cluster algebras are defined in terms of generators and relations. Contrary to usual presentations, the set of generators and relations is not given a priori. The initial datum is that of an “initial seed” which contains a relatively small subset of the generators (the initial cluster) plus some matrix. That matrix contains all the necessary information in order to construct inductively the whole set of generators, starting from the initial cluster, by means of an operation called “mutation”. The theory of cluster algebras has had fast developments in many directions: Representation theory of quivers, Poisson geometry, integrable systems, Teichmüller spaces, combinatorial polyhedra, algebraic geometry (stability conditions, Calabi-Yau algebras, DT-invariants), Quantum Field Theory, operator algebras...

In the project SC3A, we focus on some connections between cluster algebras, algebraic and geometric combinatorics, representation theory, triangulated and monoidal categories and integrable systems.

Key words:

Publications and preprints

Journal

  1. C. Amiot : On the canonicity of the cluster category associated with a surface, appendix to Extensions in Jacobian algebras and cluster categories of marked surfaces, by I. Canakci, S. Schroll (to appear in Adv. Math.)
  2. L. Demonet, P.-G. Plamondon, D. Rupel, S. Stella & P. Tumarkin : SL_2-tilings do not exist in higher dimensions (mostly). Prépublication, 4pp.,arXiv:1604.02491, 2016 (to appear in Séminaire Lotharingien de Combinatoire).

Conference papers

  1. A. Dermenjian, C. Hohlweg & V. Pilaud. The facial weak order in finite Coxeter groups. FPSAC'16. 28th Intern. Conf. Formal Power Series and Algebraic Combinatorics, DMTCS Proc., 2016.
  2. T. Manneville & V. Pilaud. Compatibility fans realizing graphical nested complexes. FPSAC'16. 28th Intern. Conf. Formal Power Series and Algebraic Combinatorics, DMTCS Proc., 2016.
  3. V. Pilaud. Brick polytopes, lattices and Hopf algebras. FPSAC'16 (poster). 28th Intern. Conf. Formal Power Series and Algebraic Combinatorics, DMTCS Proc., 2016.

Preprints

  1. T. Manneville & V. Pilaud : Geometric realizations of the accordion complex of a dissection.
  2. C. Hohlweg, V. Pilaud & S. Stella : Polytopal realizations of finite type g-vector fans.
  3. G. Chatel, V. Pilaud & V. Pons : The weak order on integer posets.
  4. Pierre-Guy Plamondon : Cluster characters.
  5. D. Fuchs, A. Kirillov, S. Morier-Genoud & V. Ovsienko : On tangent cones of Schubert varieties, Prépublication, 14pp., arXiv:1606.07846, 2016.
  6. C. Amiot, D. Labardini-Fragoso & P.-G. Plamondon : Derived invariants for surface cut algebras II: the punctured case. Prépublication, 32pp., arXiv:1606.07364, 2016.
  7. F. Qin : Compare triangular bases of quantum cluster algebras. Prépublication, 18pp., arXiv:1606.05604, 2016.
  8. A. Dermenjian, C. Hohlweg & V. Pilaud : The facial weak order and its lattice quotients. Prépublication, 40 pp., arXiv:1602.03158, 2016.

Rencontres de l'ANR SC3A

Conférences organisées et co-organisées

Quelques exposés donnés par les membres de SC3A

Missions à l'étranger

Invitations pour collaboration