On exchange matrices from string diagrams

Abstract: In this talk, we will first recall the constructions of triangular extension and of source-sink extensio for skew-symmetrizable matrices and some invariants under these constructions. Secondly, we will recall the string diagrams introduced by Shen-Weng, which are very useful to describe many interesting skew-symmetrizable matrices closely related with Lie theory. Thirdly, we will sketch the proof of our main result: the skew-symmetrizable matrices from string diagrams are in the smallest class of skew-symmetrizable matrices containing the (1 times 1) zero matrix and closed under mutations and source-sink extensions. This result applies to the exchange matrices of cluster algebras from double Bruhat cells, unipotent cells, double Bott-Samelson cells among others. Finally, some immediate applications regarding reddening sequences and non-degenerate potentials for many quivers from Lie theory are given.

K-theory and homologyquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

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.