The non-decreasing condition on g-vectors
Mohamad Haerizadeh, Siamak Yassemi
TL;DR
This work introduces the non-decreasing condition on g-vectors and proves it is necessary and sufficient for the generically indecomposable summands of a g-vector to be linearly independent, with this property preserved under natural dilations by $t$. It then connects generic decompositions of g-vectors to decompositions of generically $\tau$-reduced components, showing that under the non-decreasing condition the number of such components in a generically $\tau$-reduced component satisfies $|\mathcal{Z}| \le |\Lambda|$, thereby addressing the Cerulli-Labardini-Schröer conjecture in a broad setting. The paper also clarifies the role of $ au$-reduced components in relation to 2-term silting theory and wall-and-chamber structures, providing a precise criterion for when a component achieves maximal cardinality. Finally, it offers practical numerical criteria to determine tameness versus wildness of g-vectors via $e(g,h)$ and related bilinear forms, enhancing the toolkit for analyzing representation types and component decompositions.
Abstract
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 $Λ$, under the non-decreasing condition, the number of generically indecomposable irreducible components that appear in the decomposition of a given generically $τ$-reduced component is lower than or equal to $|Λ|$. 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.