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On The Mutations Loops of Valued Quivers

Ibrahim Saleh

Abstract

A mutation loop of a valued quiver $Q$, is a combination of quiver automorphisms (permutations of vertices and valuations) and mutations that sends $Q$ to itself. In this article we study what we called \emph{global mutations loops} which are sequences formed of mutations only that send each quiver $Q'$ in the mutation class $[Q]$ to $σ(Q')$ for some permutation $σ$. Based on the weight of $[Q]$ and the shape of $Q$, we identify which quivers have mutations global loops and we provide them for each case.

On The Mutations Loops of Valued Quivers

Abstract

A mutation loop of a valued quiver , is a combination of quiver automorphisms (permutations of vertices and valuations) and mutations that sends to itself. In this article we study what we called \emph{global mutations loops} which are sequences formed of mutations only that send each quiver in the mutation class to for some permutation . Based on the weight of and the shape of , we identify which quivers have mutations global loops and we provide them for each case.
Paper Structure (3 sections, 9 theorems, 7 equations)

This paper contains 3 sections, 9 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Valued Quivers Mutation
  3. Global Mutation loops

Key Result

Theorem 1.2

Let $Q$ be a valued quiver. Then we have the following

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.2: Valued quiver mutation
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.9
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • ...and 10 more