2025/03/16 by Fei, Jiarui · 1 citation
#16G10 #Combinatorics (math.CO) #FOS: Mathematics #Primary 13F60 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary 14T99
paper · doi:10.48550/arxiv.2503.12700
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 δ-vectors.