Juggler's friezes

TL;DR

This paper extends the concept of SL(k)-friezes to include configurations with a ragged boundary, introducing new characterizations, properties, and construction methods for these generalized juggler's friezes.

Contribution

It generalizes SL(k)-friezes by incorporating ragged boundary rows, providing multiple equivalent definitions, and establishing connections to Grassmannians and matrix twists.

Findings

Established equivalences via determinants, recurrences, and duality.

Generalized periodicity and duality properties.

Developed a construction method using matrix twists.

Abstract

This note generalizes $\mathrm{SL}(k)$-friezes to configurations of numbers in which one of the boundary rows has been replaced by a ragged edge (described by a juggling function). We provide several equivalent definitions/characterizations of these juggler's friezes, in terms of determinants, linear recurrences, and a dual juggler's frieze. We generalize classic results, such as periodicity, duality, and a parametrization by part of a Grassmannian. We also provide a method of constructing such friezes from certain $k \times n$ matrices using the twist of a matrix.