Perturbation Effects on Word Lengths in Three-Reflection Symmetric Presentations of Dihedral Groups

2025/06/24 by Michael P. Allocca, Allocca, Michael, Max Trimmer +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Psychology · #DNA and Biological Computing #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Phonetics and Phonology Research

paper · pdf · doi:10.48550/arxiv.2506.19216

openalex publication_date 2025/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the properties of word lengths of elements from a three-reflection symmetric generating set of the dihedral group D_n. Specifically, we provide the upper bound λ₁(D_n,S) ≤ \lfloor(n)/(2)\rfloor + 1 for a quantity λ₁ defined in arXiv:1104.5044, which measures the stability of a finitely presented group under perturbations in the words corresponding to certain elements with respect to specific presentations. This quantity has been of recent interest due to its role in the application of group theory to computational genomics, and we aim to introduce techniques in additive combinatorics to its discourse.

Citations

• Word Length Perturbations in Certain Symmetric Presentations of Dihedral Groups
• Group-theoretic models of the inversion process in bacterial genomes
• Group-theoretic models of the inversion process in bacterial genomes

Related

• Word Length Perturbations in Certain Symmetric Presentations of Dihedral Groups
• On the expected number of commutations in reduced words
• The depth of a permutation
• Diameter of reduced words
• Stable reflection length in Coxeter groups