Gradings of Lie algebras, magical spin geometries and matrix factorizations

Abuaf, Roland; Manivel, Laurent

doi:10.1090/ert/573

Gradings of Lie algebras, magical spin geometries and matrix factorizations
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by Roland Abuaf and Laurent Manivel

Represent. Theory 25 (2021), 527-542

DOI: https://doi.org/10.1090/ert/573

Published electronically: June 22, 2021

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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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