Abstract:We describe a remarkable rank $14$ matrix factorization of the octic $\mathrm {Spin}_{14}$-invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular $\mathbb {Z}$-grading of $\mathfrak {e}_8$. Intriguingly, the whole story can in fact be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the space of three-forms in several variables. As an application of our results on $\mathrm {Spin}_{14}$, we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.

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中文翻译：

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李代数的分级、神奇的自旋几何和矩阵分解

摘要：我们描述了 octic $\mathrm {Spin}_{14}$ 不变多项式在其任一半自旋表示上的显着秩 $14$ 矩阵分解。我们观察到，在适当的意义上，这种表示可以用两个八元代数的张量积来识别。此外，矩阵分解可以从 $\mathfrak {e}_8$ 的特定 $\mathbb {Z}$-grading 推导出来。有趣的是，整个故事实际上可以扩展到整个 Freudenthal-Tits 幻方，并在其他自旋表示上产生矩阵分解，以及在几个变量的三形式空间上的七次不变量。作为我们在 $\mathrm {Spin}_{14}$ 上的结果的应用，我们构造了一个特殊的 7 阶向量丛在一个双八进制三倍上，我们推测它是球形的。

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