Schur Rank, Compatibility Degree, and Canonical Decomposition

The notion of denominator vectors can be extended to all generic basis elements of upper cluster algebras in a natural way. Under a weakened version of generic pairing assumption, we provide a representation-theoretic interpretation for this extended notion. We derive several consequences in this generality. We present a counterexample to the conjecture that distinct cluster monomials have distinct denominator vectors. Utilizing a new rank function called the Schur rank, we extend the notion of compatibility degree. As an application, we find a tropical method to compute the multiplicity of a real component in the canonical decomposition of $\delta$-vectors.