A module-theoretic interpretation of quantum expansion formula

We provide a module-theoretic interpretation of the expansion formula given by Huang (2022), which defines a map on perfect matchings to compute the expansion of quantum cluster variables in quantum cluster algebras arising from unpunctured surfaces. In addition, we present a multiplication formula for string modules with one-dimensional extension space, derived using the skein relations. For the Kronecker type, an alternative expansion formula was given in Canakci and Lampe (2020), and we show that the two expansion formulas coincide.