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• W3173409451 abstract "We describe a remarkable rank <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=14> <mml:semantics> <mml:mn>14</mml:mn> <mml:annotation encoding=application/x-tex>14</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix factorization of the octic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper S normal p normal i normal n Subscript 14> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>S</mml:mi> <mml:mi mathvariant=normal>p</mml:mi> <mml:mi mathvariant=normal>i</mml:mi> <mml:mi mathvariant=normal>n</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>14</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>mathrm {Spin}_{14}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Z> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-grading of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=German e 8> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>e</mml:mi> </mml:mrow> <mml:mn>8</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>mathfrak {e}_8</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper S normal p normal i normal n Subscript 14> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>S</mml:mi> <mml:mi mathvariant=normal>p</mml:mi> <mml:mi mathvariant=normal>i</mml:mi> <mml:mi mathvariant=normal>n</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>14</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>mathrm {Spin}_{14}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical." @default.
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• W3173409451 date "2021-06-22" @default.
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• W3173409451 title "Gradings of Lie algebras, magical spin geometries and matrix factorizations" @default.
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