Schur Rank, Compatibility Degree, and Canonical Decomposition
Abstract
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 -vectors.
Cite
@article{arxiv.2503.12700,
title = {Schur Rank, Compatibility Degree, and Canonical Decomposition},
author = {Jiarui Fei},
journal= {arXiv preprint arXiv:2503.12700},
year = {2025}
}
Comments
41 pages, comments are welcome. text overlap with arXiv:2303.10591; v2 two typos in the introduction are corrected; v3 minor correction in the proof of Theorem 7.17; v4 reference update, add Corollary 7.18