Automorphisms of necklaces and sandpile groups

2013/04/09 by S. Duzhin, Duzhin, S., Dmitrii V. Ṗasechnik +2
Computer Science · Mathematics · #20K01 12E20 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:12E20 #msc:20K01 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1304.2563

12 pages, several tables, no pictures

arxiv created 2013/04/09 · openalex publication_date 2013/04/09 · arxiv updated 2013/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a group naturally acting on aperiodic necklaces of length n with two colours using the 1--1 correspondences between aperiodic necklaces and irreducible polynomials over the field \F₂ of two elements. We notice that this group is isomorphic to the quotient group of non-degenerate circulant matrices of size n over that field modulo a natural cyclic subgroup. Our groups turn out to be isomorphic to the sandpile groups for a special sequence of directed graphs.

Related

• Automorphism groups of circulant graphs -- a survey
• Automorphisms of circulants that respect partitions
• Automorphism groups of graphs
• Sandpile Groups of Cayley Graphs of \mathbbF₂^r
• New classes of groups related to algebraic combinatorics with applications to isomorphism problems