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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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
Published electronically: June 22, 2021


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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Bibliographic Information
  • Roland Abuaf
  • Affiliation: Paris, France
  • MR Author ID: 962796
  • Email:
  • 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:
  • 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:
  • MathSciNet review: 4276499