Infinite constructions: Grassmannian categories of infinite rank and triangulations of an infinity-gon

Abstract: The homogeneous coordinate ring of the Grassmannian of $k$-dimensional subspaces in $n$-space carries a natural structure of a cluster algebra. There is an additive categorification of this coordinate ring into a so-called Grassmannian cluster category $C(k,n)$, as shown by Jensen, King, and Su in 2016. In particular, the cluster category $C(2,n)$ models triangulations of a regular $n$-gon. A natural question is, if there is some kind of limit construction, i.e., the category ``$C(2,\infty)$'' and how to model triangulations of a regular ``$\infty$-gon''. This talk is about a categorification of the homogeneous coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules/matrix factorizations over a hypersurface singularity. This gives an infinite rank analogue of the categories of Jensen, King, and Su. We show that there is a structure preserving bijection between the generically free rank one modules in a Grassmannian category of infinite rank and the Plücker coordinates in a Grassmannian cluster algebra of infinite rank. In a special case, when the hypersurface singularity is a curve of countable Cohen-Macaulay type, our category has a combinatorial model by an ``infinity-gon'' and we can determine triangulations of this infinity-gon. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

mathematical physicsalgebraic geometryalgebraic topologyK-theory and homologyoperator algebrasquantum algebrarepresentation theory

Audience: researchers in the topic

Online GAPT Seminar

Series comments: Description: Seminar of the GAPT group at Cardiff University