Top‐stable degenerations of finite‐dimensional representations I

Given a finite‐dimensional representation M of a finite‐dimensional algebra, two hierarchies of degenerations of M are analyzed in the context of their natural orders: the poset of those degenerations of M which share the top M / JM with M (here J denotes the radical of the algebra) and the sub‐poset of those which share the full radical layering gl ( J l M / J l +1 M gr ) l ⩾0 with M . In particular, the article addresses the existence of proper top‐stable or layer‐stable degenerations — more generally, it addresses the sizes of the corresponding posets including bounds on the lengths of saturated chains — as well as structure and classification.