Maximum antichains in posets of quiver representations

TL;DR

This paper investigates maximum antichains in posets related to quiver representations, constructing them for certain Dynkin diagrams and analyzing Sperner properties in subrepresentation posets.

Contribution

It introduces methods to construct maximum antichains in posets of indecomposable quiver representations and proves Sperner properties in specific subrepresentation posets.

Findings

Constructed maximum antichains for Dynkin type A orientations.

Proved Sperner property for certain subrepresentation posets.

Analyzed posets over finite fields and pointed sets.

Abstract

We study maximum antichains in two posets related to quiver representations. Firstly, we consider the set of isomorphism classes of indecomposable representations ordered by inclusion. For various orientations of the Dynkin diagram of type A we construct a maximum antichain in the poset. Secondly, we consider the set of subrepresentations of a given quiver representation, again ordered by inclusion. It is a finite set if we restrict to linear representations over finite fields or to representations with values in the category of pointed sets. For particular situations we prove that this poset is Sperner.