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New Hereditary and Mutation-Invariant Properties Arising from Forks

Tucker J. Ervin

Abstract

A hereditary property of quivers is a property preserved by restriction to any full subquiver. Similarly, a mutation-invariant property of quivers is a property preserved by mutation. Using forks, a class of quivers developed by M. Warkentin, we introduce a new hereditary and mutation-invariant property. We prove that a quiver being mutation-equivalent to a finite number of non-forks -- defined as having a finite forkless part -- is this new property, using only elementary methods. Additionally, we show that a more general property -- having a finite pre-forkless part -- is also a new hereditary and mutation-invariant property in much the same manner.

New Hereditary and Mutation-Invariant Properties Arising from Forks

Abstract

A hereditary property of quivers is a property preserved by restriction to any full subquiver. Similarly, a mutation-invariant property of quivers is a property preserved by mutation. Using forks, a class of quivers developed by M. Warkentin, we introduce a new hereditary and mutation-invariant property. We prove that a quiver being mutation-equivalent to a finite number of non-forks -- defined as having a finite forkless part -- is this new property, using only elementary methods. Additionally, we show that a more general property -- having a finite pre-forkless part -- is also a new hereditary and mutation-invariant property in much the same manner.
Paper Structure (7 sections, 41 theorems, 12 equations, 1 figure)

This paper contains 7 sections, 41 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Some Properties of Abundant Acyclic Quivers
  5. Finite Forkless Part
  6. Keys, Pre-forks, And Their Mutations
  7. Finite Pre-Forkless Part

Key Result

Theorem 1.1

Let $Q$ be a quiver that has a finite forkless part. Then any (possibly disconnected) full subquiver $Q'$ also has a finite forkless part. Furthermore, if $M$ is the number of non-forks mutation-equivalent to $Q$ and $m$ is the number of vertices in $Q'$, then the number of non-forks mutation-equiva

Figures (1)

  • Figure 1: Venn Diagram of Hereditary and Mutation-Invariant Properties, Represented as Classes of Quivers

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 60 more