Dual canonical bases and triangular bases of quantum cluster algebras

La exposición es parte del Online Cluster Algebra Seminar, en cuyo comité organizador está involucrado Daniel Labardini.

Dual canonical bases and triangular bases of quantum cluster algebras

One of the main motivations for cluster algebras was to create a combinatorial framework to study the dual canonical bases. Correspondingly, it has been long expected that the quantum cluster monomials (certain monomials of generators) belong to the dual canonical bases (of quantum unipotent subgroups) up to scalar multiples. We discuss how to use the triangular bases to show this conjecture in full generality. Moreover, we show that the (double) triangular bases verify an analog of Leclerc’s conjecture for dual canonical bases.