The formula for the permutation of mutation sequences in $A_n$ straight orientation

TL;DR

This paper derives a formula for permutations linked to mutation sequences in $A_n$ straight orientation, extending to arbitrary sequences and introducing standard matrices for canonical ordering.

Contribution

It provides a new formula for permutations in mutation sequences of $A_n$ and introduces standard matrices for ordering cluster-tilting components.

Findings

Formula for permutations in $A_n$ straight orientation proven

Extension of permutation formula to arbitrary mutation sequences

Introduction of standard matrices for canonical ordering

Abstract

In this paper we state and prove a formula for the permutations associated to reddening and loop sequences in $A_n$ straight orientation using the picture group. In particular this applies to maximal green sequences in $A_n$ straight orientation. Furthermore we extend the definition and formula of the associated permutation to arbitrary mutation sequences based on our results. We introduce the concept of standard matrices which gives a canonical order on indecomposable components of cluster-tilting objects. Preservation of standardness of C-matrices by a combination of a mutation and its associated permutation gives the formula.