Bases of cluster algebras
Fan Qin (ICRA 2020 Award Winner) (Shanghai Jiao Tong University)
Abstract: One of Fomin and Zelevinsky’s main motivations for cluster algebras was to study the dual canonical bases. Correspondingly, it had been long conjectured that the quantum cluster monomials (certain monomials of generators) belong to the dual canonical bases up to scalar multiples. Geiss-Leclerc-Schröer proved an analogous statement that the cluster monomials belong to the dual semi-canonical bases, which are examples of generic bases.
In a geometric framework for cluster algebras, Fock and Goncharov expected that cluster algebras possess bases with good tropical properties.
In this talk, we consider a large class of quantum cluster algebras called injective-reachable (equivalently, there exists a green to red sequence). We study their tropical properties and obtain the existence of generic bases. Then we introduce the (common) triangular bases, which are Kazhdan-Lusztig type bases with good tropical properties. We verify the above motivational conjecture in full generality and, by similar arguments, a conjecture by Hernandez-Leclerc about monoidal categorification.
Mathematics
Audience: researchers in the discipline
Series comments: The Workshop and International Conference on Representations of Algebras (ICRA) will take place online between 9th November and 25th November 2020.
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Deadline for submitting research snapshots: November 1st, 2020
| Organizers: | Lidia Angeleri Hügel, Aslak Bakke Buan, Gustavo Jasso*, Henning Krause, Rosanna Laking, Øyvind Solberg |
| *contact for this listing |