Séminaires : Séminaire d'Algèbre

The Lambda-invariant and the d-invariant are two integer-valued invariants introduced by Kang-Kashiwara-Kim-Oh and by Kashiwara-Kim-Oh-Park in their study of monoidal categorification of cluster algebras using finite-dimensional modules over quiver Hecke algebras and quantum affine algebras. The Lambda-invariant can be viewed as a monoidal categorification of the compatible Poisson structure on those cluster algebras. The d-invariant is defined as half the symmetrized sum of Lambda-invariants, and it can be used to characterize the strong commutativity between real simples.

In this talk, we introduce the tropical invariant and the F-invariant in cluster algebras. The tropical invariant is defined for any cluster algebra with a compatible Poisson structure and it generalizes the Lambda-invariant. The F-invariant is defined as the symmetrized sum of the tropical invariants and it simultaneously generalizes all of the following invariants: the d-invariant, Derksen-Weyman-Zelevinsky’s E-invariant, Fu-Gyoda’s f-compatibility degree, Fomin-Zelevinsky’s compatibility degree, and Qiu-Zhou’s f-intersection number on marked surfaces.

This talk will take place on Zoom only.