Sandpile Groups of Cayley Graphs of ${\mathbb{F}}_2^r$

Abstract

The sandpile group of a connected graph $G$, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of ${\mathbb{F}}_2^r$, focusing on their poorly understood Sylow-$2$ component. We find the number of Sylow-$2$ cyclic factors for "generic" Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-$2$ cyclic factors. In the case of hypercubes, we give exact formulae for the largest $n-1$ Sylow-$2$ cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the $2$-adic valuations of binomial sums via the combinatorics of carries.

v2: 19 pages, 1 figure, 1 table; proved Conjecture 6.1 in older version (now Theorem 2.9), change of authorship. v3: 20 pages, 1 figure, 1 table; minor revision from v2