Title: Linearity of stability conditions,
Abstract: We study different concepts of stability for modules over a finite
dimensional algebra: linear stability, given by a "central charge", and
nonlinear stability given by the wall-crossing sequence of a "green path". Two
other concepts, finite Harder-Narasimhan stratification of the module category
and maximal forward hom-orthogonal sequences of Schurian modules, which are
always equivalent to each other, are shown to be equivalent to nonlinear
stability and to a maximal green sequence, defined using Fomin-Zelevinsky
quiver mutation, in the case the algebra is hereditary.
This is the first of a series of three papers whose purpose is to determine
all maximal green sequences of maximal length for quivers of affine type
$\tilde A$ and determine which are linear. The complete answer will be given in
the final paper [1].