The non-decreasing condition on g-vectors

The non-decreasing condition on g-vectors is introduced. Our study shows that this condition is both necessary and sufficient to ensure that the generically indecomposable direct summands of a given g-vector are linearly independent. Additionally, we prove that for any finite dimensional algebra $\mathrm{\Lambda }$, under the non-decreasing condition, the number of generically indecomposable irreducible components that appear in the decomposition of a given generically $\tau$-reduced component is lower than or equal to $\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$. This solves the conjecture concerning the cardinality of component clusters by Cerulli-Labardini-Schröer, in a reasonable generality. Lastly, we study numerical criteria to check the wildness of g-vectors.