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Rencontre finale du projet SC3A: Conférence "Algèbres amassées, théorie des représentations et combinatoire" au LAMFA (Amiens) du 4 au 6 septembre 2019.

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. Y. Palu, V. Pilaud & P.-G. Plamondon: Non-kissing and non-crossing complexes for locally gentle algebras. Preprint, 27pp. Accepted for publication in Journal of Combinatorial Algebra.
  2. Y. Palu & H. Nakaoka : Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories. Preprint 69pp, 2016. Accepted for publication in Cahiers de Topologie et Géométrie Différentielle Catégoriques.
  3. P.-G. Plamondon Tau-tilting finite gentle algebras are representation finite. Preprint, 7pp. To appear in Pacific Journal of Mathematics.
  4. Y. Palu, V. Pilaud & P.-G. Plamondon: Non-kissing complexes and tau-tilting for gentle algebras. Preprint, 69pp., 2017. To appear in Memoirs of the AMS.
  5. S. Morier-Genoud & V. Ovsienko: Farey boat I. Continued fractions and triangulations, modular group and polygon dissections. Jahresber. Dtsch. Math. Ver. (2019) 121:91–136.
  6. V. Pilaud: Cambrian triangulations and their tropical realizations. Preprint, 16pp. Accepted in European Journal of Combinatorics.
  7. J. Gay & V. Pilaud: The weak order on Weyl posets. Preprint, 23pp. Accepted in Canadian Journal of Mathematics.
  8. V. Pilaud & F. Santos: Quotientopes. Bulletin of the London Mathematical Society, vol. 51(3), pp. 406 – 420, 2019.
  9. F. Qin: Compare triangular bases of quantum cluster algebras. Transactions of the American Mathematical Society, Volume 372, Number 1 (2019), 485-501, DOI:10.1090/tran/7610.
  10. T. Manneville & V. Pilaud: Geometric realizations of the accordion complex of a dissection. Preprint, 25pp. To appear in Discrete & Computational Geometry, 2018.
  11. V. Pilaud: Hopf algebras on decorated noncrossing arc diagrams. Journal of Combinatorial Theory, Series A, vol. 161, p. 486--507, 2019.
  12. A. Dermenjian, C. Hohlweg & V. Pilaud: The facial weak order and its lattice quotients. Transactions of the AMS, vol. 370(2), pp. 1469--1507, 2018.
  13. V. Pilaud, P.-G. Plamondon & S. Stella: A tau-tilting approach to dissections of polygons. SIGMA, vol. 14, 045, 8pp, 2018.
  14. V. Pilaud & V. Pons: Permutrees. Algebraic Combinatorics, vol. 1(2), pp. 173--224, 2018.
  15. C. Hohlweg, V. Pilaud & S. Stella: Polytopal realizations of finite type g-vector fans. Advances in Mathematics, vol. 328, pp. 713--749, 2018.
  16. V. Pilaud: Brick polytopes, lattice quotients, and Hopf algebras. Journal of Combinatorial Theory, Series A, vol. 155, pp. 418--457, pp., 2018.
  17. A. Dermenjian, C. Hohlweg & V. Pilaud: The facial weak order and its lattice quotients. Transactions of the American Mathematical Society, vol. 370(2), pp. 1469--1507, 2018.
  18. 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.)
  19. T. Manneville & V. Pilaud: Compatibility fans for graphical nested complexes. Journal of Combinatorial Theory, Series A, vol. 150C, pp. 36--107, 2017.
  20. D. Fuchs, A. Kirillov, S. Morier-Genoud et V. Ovsienko : On tangent cones of Schubert varieties, Arnold Math J. (2017) 3: 451.
  21. F. Qin, Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Mathematical Journal, Volume 166, Number 12 (2017), 2337--2442.
  22. G. Chatel & V. Pilaud: Cambrian Hopf Algebras. Advances in Mathematics, vol. 311, pp. 598--633, 2017.
  23. P.-G. Plamondon: Cluster characters. In Homological methods, representation theory, and cluster algebras.
  24. L. Demonet, P.-G. Plamondon, D. Rupel, S. Stella & P. Tumarkin : SL_2-tilings do not exist in higher dimensions (mostly). Prépublication, 4pp. 2016 (to appear in Séminaire Lotharingien de Combinatoire).

Preprints

  1. A. Padrol, Y. Palu, V. Pilaud & P.-G. Plamondon : Associahedra for finite type cluster algebras and minimal relations between g-vectors. Preprint, 69pp.
  2. S. Morier-Genoud & V. Ovsienko: Farey boat II. Q-deformations: q-deformed rationals and q-continued fractions. Preprint, 39pp.
  3. M. Cuntz & P.-G. Plamondon: Counting friezes in type E_6, 4pp., 2018, appendice de la prépublication: K. Baur, E. Faber, S. Gratz, K. Serhiyenko & G. Todorov: Friezes satisfying higher SL_k-determinants. Préprint, 46pp.
  4. V. Bazier-Matte & P.-G. Plamondon: Unistructurality of cluster algebras from surfaces without punctures. Preprint, 13pp.
  5. N. Bergeron, C. Ceballos & V. Pilaud: Hopf dreams. Preprint, 40pp.
  6. O. Iyama, H. Nakaoka & Y. Palu: Auslander--Reiten theory for extriangulated categories. Preprint, 28pp.
  7. S. Morier-Genoud: Symplectic frieze patterns. Preprint, 33pp.
  8. S. Opper, P.-G. Plamondon & S. Schroll: A geometric model for the derived category of gentle algebras. Preprint, 39pp.
  9. C. Amiot & P.-G. Plamondon: The cluster category of a surface with punctures via group actions.
  10. D. Fuchs, A. Kirillov, S. Morier-Genoud & V. Ovsienko: On tangent cones of Schubert varieties, Preprint, 14pp.
  11. C. Amiot, D. Labardini-Fragoso & P.-G. Plamondon: Derived invariants for surface cut algebras II: the punctured case. Preprint, 32pp.

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