{"id":"hexagon/2610.00266","pdf":"https://hexagonmath.org/pdf/2610.00266v1","url":"https://hexagonmath.org/abs/2610.00266v1","tags":["math.GR","math.RT","msc:05C50","msc:20G05","Hamming graph","Lean 4","Smith normal form","critical group","sandpile group","tilting modules"],"score":0,"title":"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","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Rafael Amichis Luengo"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code","SageMath","Aristotle","Python 3 (NumPy)"],"discussed":0,"published":"2026-10-15","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}