We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from su...

We use the quantum version of Chebyshev polynomials to explicitly construct recursive formulas for Kronecker cluster algebra with principal coefficients. As a byproduct, we obtain two bar-invariant positive ℤℙ-bases one being atomic basis.

In a series of works [18, 21, 19, 20, 23, 22], Geiß-Leclerc-Schröer defined the cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S∗. We give a set up for the quantization of their results and propose a conjecture which relates the quant...