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 $\Lambda$, 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 $|\Lambda|$. This solves the conjecture concerning the cardinality of component clusters by Cerulli-Labardini-Schr\"oer, in a reasonable generality. Lastly, we study numerical criteria to check the wildness of g-vectors.