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Summary: TOP-STABLE DEGENERATIONS OF FINITE
DIMENSIONAL REPRESENTATIONS I
Birge Huisgen-Zimmermann
Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday
Abstract. 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
`
JlM/Jl+1M
´
l0
with M. In particular, the article addresses 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.
1. Introduction
Let be a basic finite dimensional algebra over an algebraically closed field K. Our ob-
jective is to understand major portions of the degeneration theory of the finite dimensional
representations of . This line of inquiry was triggered by Gabriel's and Kac's initial ex-
ploitation of the affine variety Mod
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