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Representation Theory

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Gradings of Lie algebras, magical spin geometries and matrix factorizations


Authors: Roland Abuaf and Laurent Manivel
Journal: Represent. Theory 25 (2021), 527-542
MSC (2020): Primary 15A66, 13A50, 17B45
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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Additional 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
Article copyright: © Copyright 2021 American Mathematical Society