Algebraic Aspects of Abelian Sandpile Models

TL;DR

This paper explores the algebraic structure of abelian sandpile models, revealing their decomposition into cyclic groups, invariants for recurrent configurations, and symmetries related to Galois groups, providing new insights into their mathematical properties.

Contribution

It introduces a canonical decomposition of the sandpile group into cyclic components and constructs invariants that label recurrent configurations, highlighting symmetries beyond obvious geometric ones.

Findings

The sandpile group decomposes into cyclic groups with a number of generators equal to the lattice size.

Constructed scalar invariants that are linear in height variables and invariant under toppling.

Identified nontrivial symmetries related to Galois groups acting on eigenvalues of the toppling matrix.

Abstract

The abelian sandpile models feature a finite abelian group $G$ generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of $G$ as a product of cyclic groups $G = Z_{d_1} \times Z_{d_2} \times Z_{d_3} >... \times Z_{d_g}$ where $g$ is the least number of generators of $G$, and $d_i$ is a multiple of $d_{i+1}$. The structure of $G$ is determined in terms of the toppling matrix $\Delta$. We construct scalar functions, linear in height variables of the pile, that are invariant under toppling at any site. These invariants provide convenient coordinates to label the recurrent configurations of the sandpile. For an $L \times L$ square lattice, we show that $g = L$. In this case, we observe that the system has nontrivial symmetries, transcending the obvious symmetries of the square, viz. those coming from the action of the cyclotomic Galois group Gal$_L$ of the $2(L+1)$--th roots of unity (which operates on the set of eigenvalues of $\Delta$). We use Gal$_L$ to define other simpler, though under-complete, sets of toppling invariants.