Generalized cyclotomic mappings: Switching between polynomial, cyclotomic, and wreath product form

TL;DR

This paper explores generalized cyclotomic mappings over finite fields, providing methods to switch between polynomial, cyclotomic, and wreath product forms, with applications in permutation analysis and cycle structure classification.

Contribution

It introduces two rewriting procedures for generalized cyclotomic mappings and applies them to permutation classification, cycle structures, and inversion problems.

Findings

Rewriting procedures enable switching between different representations of cyclotomic mappings.

Applications include classification of permutations and cycle structures.

Methods facilitate analysis of involutions and inversion in finite fields.

Abstract

This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of $\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$ and pertain to cycle structures, the classification of $(q-1)$-cycles and involutions, as well as inversion.