Algebraic Aspects of Abelian Sandpile Models

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 = Zd1 ×Zd2 ×Zd3 · · ·×Zdg where g is the least number of generators of G, and di is a multiple of di+1. The structure of G is determined in terms of the toppling matrix ∆. 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 × 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 GalL of the 2(L+ 1)–th roots of unity (which operates on the set of eigenvalues of ∆). These eigenvalues are algebraic integers, whose product is the order |G|. With the help of GalL we are able to group the eigenvalues into certain subsets whose products are separately integers, and thus obtain an explicit factorization of |G|. We also use GalL to define other simpler, though under-complete, sets of toppling invariants. PACS number: 05.40+j 1 Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India. 2 Departamento de Fı́sica Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain. 3 Chercheur qualifié FNRS. On leave from: Institut de Physique Théorique, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium. 4 School of Mathematics, Trinity College, Dublin, Ireland.