# Geometric description of C-vectors and real L\"osungen

## TL;DR

This paper introduces real Loesungen as a geometric analogue of real roots, connecting mutation sequences, reflection families, and curves on Riemann surfaces to provide a new combinatorial description of c-vectors.

## Contribution

It defines real Loesungen and the L-matrix, linking them to C-matrices and proposing a geometric interpretation of c-vectors in cluster algebras.

## Key findings

- L-matrix arises from vectors associated with mutation sequences
- Conjecture that L-matrix depends only on the seed (up to signs)
- Curves can be drawn without self-intersections, offering a geometric view

## Abstract

We introduce real Loesungen as an analogue of real roots. For each mutation sequence of an arbitrary skew-symmetrizable matrix, we define a family of reflections along with associated vectors which are real Loesungen and a set of curves on a Riemann surface. The matrix consisting of these vectors is called L-matrix. We explain how the L-matrix naturally arises in connection with the C-matrix. Then we conjecture that the L-matrix depends (up to signs of row vectors) only on the seed, and that the curves can be drawn without self-intersections, providing a new combinatorial/geometric description of c-vectors.