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We introduce real Loesungen as an analogue of real roots. For each mutationsequence of an arbitrary skew-symmetrizable matrix, we define a family ofreflections along with associated vectors which are real Loesungen and a set ofcurves on a Riemann surface. The matrix consisting of these vectors is calledL-matrix. We explain how the L-matrix naturally arises in connection with theC-matrix. Then we conjecture that the L-matrix depends (up to signs of rowvectors) only on the seed, and that the curves can be drawn withoutself-intersections, providing a new combinatorial/geometric description ofc-vectors.

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