# Symmetric shift-invariant subspaces and harmonic maps

## TL;DR

This paper explores symmetric shift-invariant subspaces in the Grassmannian model to better understand harmonic maps into symmetric spaces, providing new forms and insights into their structure.

## Contribution

It introduces a natural symmetry condition on shift-invariant subspaces, leading to new general forms for these subspaces and their extended solutions.

## Key findings

- New general forms for symmetric shift-invariant subspaces
- Characterization of harmonic maps into symmetric spaces
- Enhanced understanding of extended solutions

## Abstract

The Grassmannian model represents harmonic maps from Riemann surfaces by families of shift-invariant subspaces of a Hilbert space. We impose a natural symmetry condition on the shift-invariant subspaces that corresponds to considering an important class of harmonic maps into symmetric and $k$-symmetric spaces. In particular, we obtain new general forms for such symmetric shift-invariant subspaces and for the corresponding extended solutions.