Automorphisms of necklaces and sandpile groups

TL;DR

This paper explores the algebraic structure of aperiodic necklaces and their connection to sandpile groups, revealing new isomorphisms with circulant matrices and directed graph groups.

Contribution

It introduces a natural group action on necklaces, links it to circulant matrices, and shows these groups are isomorphic to sandpile groups for specific graphs.

Findings

Group acting on necklaces is isomorphic to a quotient of circulant matrices.

The introduced groups are isomorphic to sandpile groups of certain directed graphs.

Establishes a novel connection between combinatorics, linear algebra, and graph theory.

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 $\F_2$ 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.