## 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.## References

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## Bibliographic Information

**Roland Abuaf**- Affiliation: Paris, France
- MR Author ID: 962796
- Email: rabuaf@gmail.com
**Laurent Manivel**- Affiliation: Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France
- MR Author ID: 291751
- ORCID: 0000-0001-6235-454X
- Email: manivel@math.cnrs.fr
- Received by editor(s): April 17, 2020
- Received by editor(s) in revised form: April 2, 2021
- Published electronically: June 22, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Represent. Theory
**25**(2021), 527-542 - MSC (2020): Primary 15A66, 13A50, 17B45
- DOI: https://doi.org/10.1090/ert/573
- MathSciNet review: 4276499