The Dancing Sand Theorem: the 2-part of the sandpile group of the hypercube, for every n
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…
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