The Coloured Sand Theorem: the p-part of the sandpile group of the Hamming graph, for every prime p dividing q and every dimension d
Let $H(d,q)=K_q^{d}$ be the Hamming graph and $K(H(d,q))$ its sandpile group (critical group, Jacobian). For primes $p\nmid q$ its $p$-part was determined by Jacobson, Niedermaier and Reiner in 2003, who wrote that the other primes appear "to be difficult, due to bad behavior of its $p$-primary structure". We determine the Sylow $p$-subgroup of $K(H(d,q))$ for every prime $p$ dividing $q$ and every $d$, by an explicit rule. After the splitting of Jacobson, Niedermaier and Reiner, the group is a sum of cokernels of operators $q(c+H)+U$ on the Boolean lattice (Theorem A). These split over the…
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