Abstract
We introduce real Lösungen 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 Lösungen 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.
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