Spaces of quasiperiodic sequences

Abstract: A "quasiperiodic space" is a vector space of sequences which are periodic up to a constant factor. The moduli of such vector spaces are 1-dimensional extensions of Grassmannians, and there are analogous positroid stratifications of the former. I will demonstrate that these "quasiperiodic positroid varieties" have a Y-type cluster structure that is mirror dual to the X-type cluster structure on (the Plucker cone over) the corresponding positroid variety. This structure is defined by extending a version of Postnikov's boundary measurement map to the quasiperiodic case. Time permitting, I will discuss an alternative construction of this boundary measurement map, which uses the twist to construct a linear recurrence whose solutions are the space in question. This provides a generalization of MGOST's connection between linear recurrences, friezes, and the Gale transform. A motivating goal of this project is to understand the tropical points of these quasiperiodic positroid varieties, as they parametrize the canonical basis of theta functions on (the Plucker cone over) the corresponding positroid variety.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)