Quiver varieties and cluster algebras (I)


Yoshiyuki Kimura

17:20:00 - 18:20:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Cluster algebras were introduced by Fomin and Zelevinsky with an aim to provide concrete and combinatorial formalism for the study of Lusztig's dual canonical basis and total positivity. Inspired by a previous work of Nakajima, we consider a class of (equivariant) perverse sheaves on acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. In particular, we identifies the corresponding quantum Grothendieck ring and the acyclic quantum cluster algebra (with a specific coefficient) and show that the set of quantum cluster monomials is contained in the “dual canonical basis” of the quantum Grothendieck ring. In the first part, I will explain about the definition of cluster algebra (of geometric type) , its positivity conjectures and their motivations. In the second part, I will explain how the graded quiver varieties are related with cluster algebras and the positivity conjectures. This talk is based on a joint work with Fan Qin (Universite de Strasbourg, France).