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Positivity in Quantum Cluster Algebras and Flags of Valued Quiver Representations

In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of adapted cluster monomials. In particular, we describe all cluster monomials in cluster algebras and quantum cluster algebras of rank 2. One may obtain similar formulas for all finite type cluster monomials. The above results are achieved by computing explicit set-theoretic decompositions of Grassmannians of subrepresentations in adapted valued quiver representations into a disjoint union of products of standard vector space Grassmannians. We actually prove a more general result which should be of independent interest: we compute these decompositions for arbitrary flags of subrepresentations in adapted valued quiver representations. This implies the existence of counting polynomials for the number of points in these sets over different finite fields. Using this we extend the results of \cite{rupel} to quantum cluster algebras $\Acal_q(Q,\bfd)$, where $q$ is an indeterminate.