{"id":"hexagon/2610.00205","pdf":"https://hexagonmath.org/pdf/2610.00205v2","url":"https://hexagonmath.org/abs/2610.00205v2","tags":["math.GR","math.RT","msc:05C50","msc:20K01","msc:20G05","Lean","Smith normal form","critical group","folded cube","hypercube"],"score":0,"title":"The Dancing Sand Theorem: the 2-part of the sandpile group of the hypercube, for every n","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Rafael Amichis Luengo"],"reviews":0,"subject":"math.CO","summary":"Let $K (Q_{n})$ be the sandpile group (critical group, Jacobian) of the $n$-dimensional hypercube graph $Q_{n}$. Its odd part was determined by Bai in 2003; its $2$-part has been open since then. Gao, Marx-Kuo, McDonald and Yuen determined its $n - 1$ largest cyclic factors, and Betzer the $n$-th, through a remark of theirs stated without proof. We determine the whole 2-part of $K (Q_{n})$ for every $n$, by an explicit rule. The group splits over the indecomposable tilting modules $T (\\lambda)$ of $\\mathrm{SL}_{2}$ in characteristic $2$ that occur in $V^{\\otimes n}$, with their…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code","Aristotle","Mathlib","Python"],"discussed":0,"published":"2026-10-12","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}