Understanding universal coefficients of Grassmannians through Groebner theory

Abstract: In this talk I will present recent results of a joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez. For a polarized weighted projective variety V(J) we introduce a flat family that combines all Groebner degenerations of V associated to a maximal cone in the Groebner fan of J. It turns out that this family can alternatively be obtained as a pull-back of a toric family (in the sense of Kaveh--Manon's classification of such). The most surprising application of this construction is its relation to cluster algebras with universal coefficients. To demonstrate this connection we analyze the cases of the Grassmannians Gr(2,n) and Gr(3,6) in depth. For Gr(2,n) we fix its Pluecker embedding and for Gr(3,6) what we call its "cluster embedding". In both cases we identify a specific maximal cone C in the Groebner fan of the defining ideal such that the algebra defining the flat family mentioned above is canonically isomorphic to the corresponding cluster algebra with universal coefficients.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)