Revisiting Derksen-Weyman-Zelevinsky's mutations

Abstract: The mutation theory of quivers with potential and their representations, developed around 15 years ago by Derksen-Weyman-Zelevinsky, has had a profound impact both inside and outside the theory of cluster algebras. In this talk I will present results obtained in joint works with Geiss and Schröer, and with de Laporte, about some interesting behaviors of DWZ's mutations of representations. Namely, despite needing several non-canonical choices of linear-algebraic data in order to be performed, they can always be arranged so as to become regular maps on dense open subsets of representation spaces rep(Q,S,d). As a consequence, one obtains the invariance of Geiss-Leclerc-Schröer's 'generic basis' under mutations even in the Jacobi-infinite case, thus generalizing a result of Plamondon. Furthermore, given two distinct vertices k, \ell of a quiver with potential (Q,S), the k-th mutation of representations takes the \ell-th indecomposable projective over (Q,S) to the \ell-th indecomposable projective over \mu_k(Q,S). When a certain 'optimization' condition is satisfied by \ell, this allows to compute certain 'Landau-Ginzburg potentials' as F-polynomials of projective representations.

K-theory and homologyquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Comments: In-person talk at the room 01 of the Institut Henri Poincaré, Paris, France

Paris algebra seminar

Series comments: For the Zoom links and passwords, please subscribe to the mailing list (link and password will be emailed shortly before each talk) or contact one of the organizers. The slides and notes are available here. For recordings of talks, please contact Bernhard Keller.