# Séminaires : Séminaire d'Algèbre

 Equipe(s) : gr, Responsables : J. Alev, D. Hernandez, B. Keller, Th. Levasseur, et S. Morier-Genoud. Email des responsables : Jacques Alev , David Hernandez , Bernhard Keller , Thierry Levasseur , Sophie Morier-Genoud Salle : 001 Adresse : IHP Description

 Orateur(s) Toshiki NAKASHIMA - Tokyo, Titre Geometric crystals on cluster varietie Date 05/02/2018 Horaire 14:00 à 15:00 Résume The notion of geometric crystal was initiated by A.Berenstein and D.Kazhdan to consider certain geometric analogue to the Kashiwara's crystal base theory. Their structures are described by rational maps and rational functions. If all these rational maps are positive'', such geometric crystals are called positive'' and they can be tranfered to the Langlands dual crystal bases'' by tropicalization/ultra-discretization procedure. V.Fock and A.Goncharov defined certain pair of varieties (A,X), called cluster ensemble'' which is obtained by glueing algebraic tori using the A-mutations and X-mutations'' respectively. They gave the conjectures on tropical duality'' between cluster ensemble A-variety and X-variety (called Fock-Goncharov conjectures). We shall define the positive geometric crystal structure on cluster varieties and then obtain the resulting tropicalized crystals, which will be a guide to understand the Fock-Goncharov conjectures in terms of crystal base theory. Finally, we shall show some compatibility of geometric crystal structures on A-variety and X-variety in the classical type A case. Salle 001 Adresse IHP