The UC Berkeley Combinatorics Seminar
Mondays 2:10pm - 3:00pm
939 Evans Hall
Organizers: Florian Block, Max Glick, and Lauren Williams
Tropicalization method in cluster algebras
Speaker: Tomoki Nakanishi
In cluster algebras, after making several mutations of sends, you may sometimes end up with the initial seed. That is the periodicity phenomenon in cluster algebras. Periodicity is a rare event, but once you have it, you can also get the associated dilogarithm identity, plus its quantum version, for free!
There are two basic questions for periodicity: How to find it and how to prove it? The answer to the second question is given by the tropicalization method, which I explain in this talk by several examples.
The first question is more difficult, and I do not know the answer. However, we are lucky to have several (infinitely many) conjectured periodicities from the Bethe ansatz method in 90\'s, even before the birth of cluster algebras, and they are recently proved by the tropicalization method. There is always some root system behind the scene.
The talk is based on the work with R. Inoue, O. Iyama, B. Keller, and A. Kuniba.