Top-stable degenerations of finite dimensional representations I

TL;DR

This paper investigates the structure and classification of degenerations of finite dimensional algebra representations that preserve either the top or the radical layering, analyzing their poset hierarchies and bounds on chains.

Contribution

It introduces a detailed analysis of top-stable and layer-stable degenerations, including existence, size bounds, and classification within their natural orderings.

Findings

Characterization of posets of degenerations sharing the same top or radical layering.

Existence results for proper top-stable and layer-stable degenerations.

Bounds on the lengths of saturated chains in the posets.

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 $\bigl(J^lM/J^{l+1}M\bigr)_{l \ge 0}$ 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.