A 3D Spinorial View of 4D Exceptional Phenomena

Dechant, Pierre-Philippe

doi:10.1007/978-3-319-30451-9_4

“…This subalgebra is in fact the 4D space that allows us to define 4D root systems from 3D root systems. We have used the full 8D algebra in other work [9,26], e.g. for constructing the root system E 8 from the icosahedron H 3 or for defining representations, but will only consider the even subalgebra from now in this work.…”

Section: (B) Induction Theoremmentioning

confidence: 99%

From the Trinity ( A ₃ , B ₃ , H ₃ ) to an ADE correspondence

Dechant

¹

2018

Proc. R. Soc. A.

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In this paper, we present novel ADE correspondences by combining an earlier induction theorem of ours with one of Arnold's observations concerning Trinities, and the McKay correspondence. We first extend Arnold's indirect link between the Trinity of symmetries of the Platonic solids ( A 3 , B 3 , H 3 ) and the Trinity of exceptional four-dimensional root systems ( D 4 , F 4 , H 4 ) to an explicit Clifford algebraic construction linking the two ADE sets of root systems ( I 2 ( n ), A 1 × I 2 ( n ), A 3 , B 3 , H 3 ) and ( I 2 ( n ), I 2 ( n )× I 2 ( n ), D 4 , F 4 , H 4 ). The latter are connected through the McKay correspondence with the ADE Lie algebras ( A n , D n , E 6 , E 7 , E 8 ). We show that there are also novel indirect as well as direct connections between these ADE root systems and the new ADE set of root systems ( I 2 ( n ), A 1 × I 2 ( n ), A 3 , B 3 , H 3 ), resulting in a web of three-way ADE correspondences between three ADE sets of root systems.

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“…Even before any of these examples were known, icosahedral symmetry had inspired Plato to formulate a 'unified theory of everything' in his dodecahedral 'ordering principle of the universe'. This pattern of (exceptional) symmetries inspiring 'grand unified theories' continues to this day, with A 4 = SU (5) in GUTs, and E 8 in string theory and GUTs, as well as D 4 = SO (8) and B 4 = SO (9) being critical in string and M theory.…”

Section: Introductionmentioning

confidence: 96%

“…For instance E 8 includes A 4 and H 4 , or H 4 includes H 3 , so that people seek to understand the smaller groups as subgroups of the larger ones. In recent work [6,8,10,11] the author has shown that instead there is also a 'bottomup' view, by which e.g. H 4 and even E 8 can be constructed from H 3 .…”

Section: Introductionmentioning

confidence: 99%

Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

Dechant

¹

2021

Adv. Appl. Clifford Algebras

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Recent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

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“…For instance E 8 includes A 4 and H 4 , or H 4 includes H 3 , so that people seek to understand the smaller groups as subgroups of the larger ones. In recent work [6,8,10,11] the author has shown that instead there is also a 'bottomup' view, by which e.g. H 4 and even E 8 can be constructed from H 3 .…”

Section: Introductionmentioning

confidence: 99%

Clifford spinors and root system induction: $H_4$ and the Grand Antiprism

Dechant

2021

Preprint

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Recent work has shown that every 3D root system allows the construction of a correponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem, giving a simple construction of the Pin and Spin covers. Using this connection with H3 via the induction theorem sheds light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of Cl(3), including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

show abstract

A 3D Spinorial View of 4D Exceptional Phenomena

Cited by 3 publications

References 33 publications

From the Trinity ( A ₃ , B ₃ , H ₃ ) to an ADE correspondence

From the Trinity ( A ₃ , B ₃ , H ₃ ) to an ADE correspondence

Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

Clifford spinors and root system induction: $H_4$ and the Grand Antiprism

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A 3D Spinorial View of 4D Exceptional Phenomena

Cited by 3 publications

References 33 publications

From the Trinity ( A 3 , B 3 , H 3 ) to an ADE correspondence

From the Trinity ( A 3 , B 3 , H 3 ) to an ADE correspondence

Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

Clifford spinors and root system induction: $H_4$ and the Grand Antiprism

Contact Info

Product

Resources

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From the Trinity ( A ₃ , B ₃ , H ₃ ) to an ADE correspondence

From the Trinity ( A ₃ , B ₃ , H ₃ ) to an ADE correspondence