Hessian Defect, Compatibility Degree, and Canonical Decomposition

Abstract

We express the difference between dimension and denominator vectors as the rank defect of a Hessian differential. For a very general Jacobi-finite potential, this defect vanishes on a general representation in a principal component precisely when the positive-length cycles at the vertex act trivially. The associated Hessian vectors satisfy tropical $X$-mutation and Auslander--Reiten translation. We also give two distinct cluster variables with the same denominator vector but different Hessian vectors. Their Hom pairing extends ordered denominator compatibility, and its negative part computes the canonical multiplicity of any extended-reachable indecomposable class in an arbitrary presentation weight.

arXiv comment30 pages, comments are welcome. v5: Remove all imaginary-weight construction and skew-symmetrizable discussion, so we no longer need the generic pairing and 2-CY categorification. The length of the paper is reduced by 12 pages. Title change