Periodic staircase matrices and generalized cluster structures

TL;DR

This paper introduces a new construction for generalized cluster transformations, connecting determinant identities to cluster structures in various mathematical contexts, including Poisson-Lie brackets and difference operators.

Contribution

It provides a novel framework linking determinant identities with generalized cluster transformations and explores applications in Poisson-Lie structures and difference operators.

Findings

Established a construction for generalized cluster transformations

Applied the framework to GL_n with specific Poisson-Lie brackets

Connected cluster structures to spaces of periodic difference operators

Abstract

As is well-known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plucker relations, Desnanot--Jacobi identities and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL_n compatible with a certain subclass of Belavin--Drinfeld Poisson--Lie brackets, in the Drinfeld double of GL_n, and in spaces of periodic difference operators.