Algebraic Aspects of Abelian Sandpile Models

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 $Δ$. 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 $Δ$). We use Gal$_L$ to define other simpler, though under-complete, sets of toppling invariants.