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Reflection Groups with Clifford Algebras Dechant, Pierre-Philippe ORCID: https://orcid.org/0000-0002-4694-4010 (2017) A Clifford algebraic approach to reflection groups and root systems. In: Yau Institute Seminar in Geometry and Physics, 10th August 2017, Yau Mathematical Sciences Center (YMSC), Tsinghua, China. (Unpublished) Downloaded from: http://ray.yorksj.ac.uk/id/eprint/4028/ Research at York St John (RaY) is an institutional repository. It supports the principles of open access by making the research outputs of the University available in digital form. Copyright of the items stored in RaY reside with the authors and/or other copyright owners. Users may access full text items free of charge, and may download a copy for private study or non-commercial research. For further reuse terms, see licence terms governing individual outputs. Institutional Repository Policy Statement RaY Research at the University of York St John For more information please contact RaY at [email protected] Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions A Clifford algebraic approach to reflection groups and root systems Pierre-Philippe Dechant Mathematics Department, University of York Yau Institute Seminar in Geometry and Physics August 10th, 2017 Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Reflection groups: a new approach a2 (v a) sa (v) = v 2 j a = − (a a) a1 ava = ( a)jv( a) − − − − Work at the level of root systems (which define reflection groups) Interested in non-crystallographic root systems e.g. viruses, fullerenes etc. But: no Lie algebra, so conventionally less studied Clifford algebra is a uniquely suitable framework for reflection groups/root systems: reflection formula, spinor double covers, complex/quaternionic quantities arising as geometric objects Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Main results Framework for reflection, conformal, modular and braid groups New view on the geometry of the Coxeter plane Induction of exceptional root systems and ADE from Platonic symmetries Naturally defines a range of representations Described by Coxeter groups Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Platonic Solids Platonic Solid Group root system Tetrahedron A3 Cuboctahedron 3 A1 Octahedron Octahedron B3 Cuboctahedron Cube +Octahedron Icosahedron H3 Icosidodecahedron Dodecahedron Platonic Solids have been known for millennia Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Platonic Solids Platonic Solid Group root system Tetrahedron A3 Cuboctahedron 3 A1 Octahedron Octahedron B3 Cuboctahedron Cube +Octahedron Icosahedron H3 Icosidodecahedron Dodecahedron Platonic Solids have been known for millennia Described by Coxeter groups Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions 4D analogues of the Platonic Solids The 16-cell, 24-cell, 24-cell and dual 24-cell, the 600-cell and the 120-cell In higher dimensions there are only hypersimplices and hypercubes/octahedra (An and Bn) Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Platonic Solids Abundance of 4D root systems { exceptional Concatenating 3D reflections gives 4D Clifford spinors (binary polyhedral groups) These induce 4D root systems R = a0 + a1e2e3 + a2e3e1 + a3e1e2 ~ 2 2 2 2 ) RR = a0 + a1 + a2 + a3 This construction accidental to 3D perhaps explains the unusual abundance of 4D root systems Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Root systems Root system Φ: set of a2 a1 + a2 vectors a in a vector space with an inner product such that 1. Φ Ra = a;a a Φ a1 a1 \ {− g 8 2 − 2. sa Φ = Φ a Φ 8 2 reflection groups (a1 + a2) a2 (v a) − − sa : v sa (v) = v 2 j a ! − (a a) j Simple roots: express every element of Φ via a Z-linear combination with coefficients of the same sign. Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions The Icosahedron Rotational icosahedral group is I = A5 of order 60 Full icosahedral group is H3 of order 120 (including reflections/inversion); generated by the root system icosidodecahedron Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Non-crystallographic Coxeter groups H2 H3 H4 t(a1 + a2) ⊂ ⊂ a1 + ta2 ta + a a2 1 2 a1 a1 − (ta1 + a2) a2 − − (a1 + ta2) − t(a1 + a2) − H2 H3 H4: 10, 120, 14,400 elements, the only Coxeter groups ⊂ ⊂that generate rotational symmetries of order 5. Linear combinations now in the extended integer ring 1 p Z[t] = a + tb a;b Z golden ratio t = (1 + p5) = 2cos f j 2 g 2 5 2 1 2p x = x + 1 t0 = s = (1 p5) = 2cos t + s = 1;ts = 1 2 − 5 − Polyhedral groups, Platonic solids and root systems Reflection groups with Clifford algebras Conclusions Cartan-Dynkin diagrams Coxeter-Dynkin diagrams: node= simple root, no link = roots p p orthogonal i.e. angle 2 , simple link = roots at angle 3 , link with p label m = angle m . 5 n A2 H2 I2(n) 4 5 A3 B3 H3 4 5 D4 F4 H4 E8 Overview 1 Polyhedral groups, Platonic solids and root systems 2 Reflection groups with Clifford algebras A Clifford way of doing orthogonal transformations The geometry of the Coxeter plane Root system induction and ADE correspondences Representations from multivector groups Conformal, modular and braid groups 3 Conclusions A Clifford way of doing orthogonal transformations Polyhedral groups, Platonic solids and root systems The geometry of the Coxeter plane Reflection groups with Clifford algebras Root system induction and ADE correspondences Conclusions Representations from multivector groups Conformal, modular and braid groups Clifford Algebra and orthogonal transformations Geometric Product for two vectors ab a b + a b ≡ · ^ Inner product is symmetric part a b = 1 (ab + ba) · 2 Reflecting a in n is given by a0 = a 2(a n)n = nan (n and − · − n doubly cover the same reflection) − Via Cartan-Dieudonn´e theorem any orthogonal transformation can be written as successive reflections, which are doubly covered by Clifford versors/pinors A x0 = n1n2 :::nk xnk :::n2n1 =: AxA~ ± ± A Clifford way of doing orthogonal transformations Polyhedral groups, Platonic solids and root systems The geometry of the Coxeter plane Reflection groups with Clifford algebras Root system induction and ADE correspondences Conclusions Representations from multivector groups Conformal, modular and braid groups Clifford Algebra of 3D: the relation with 4D and 8D Clifford (Pauli) algebra in 3D is 1 e1;e2;e3 e1e2;e2e3;e3e1 I e1e2e3 |{z}f g |f {z g} |f {z g} |f ≡ {z g} 1 scalar 3 vectors 3 bivectors 1 trivector We can multiply together root vectors in this algebra ai aj ::: A general element has8 components: 8D even products (rotations/spinors) have four components: 2 2 2 2 R = a0 + a1e2e3 + a2e3e1 + a3e1e2 RR~ = a + a + a + a ) 0 1 2 3 So behaves as a 4D Euclidean object { inner product 1 (R ;R ) = (R R~ + R R~ ) 1 2 2 2 1 1 2 A Clifford way of doing orthogonal transformations Polyhedral groups, Platonic solids and root systems The geometry of the Coxeter plane Reflection groups with Clifford algebras Root system induction and ADE correspondences Conclusions Representations from multivector groups Conformal, modular and braid groups Spinors from reflections: easy example The 6 roots( 1;0;0) and permutations in A1 A1 A1 ± × × e1, e2, e3 generate group of 8 spinors ± ± ± 1; e1e2; e2e3; e3e1 ± ± ± ± This is a discrete spinor group isomorphic to the quaternion group Q. A Clifford way of doing orthogonal transformations Polyhedral groups, Platonic solids and root systems The geometry of the Coxeter plane Reflection groups with Clifford algebras Root system induction and ADE correspondences Conclusions Representations from multivector groups Conformal, modular and braid groups Pinors from reflections: easy example 1 e1; e2; e3 e1e2; e2e3; e3e1 I e1e2e3 |{±{z g} |{± ±{z ± g} |{± ±{z ± g} |{± ≡{z g} 1 scalar 3 vectors 3 bivectors 1 trivector The pin group also of course contains e1, e2, e3 and ± ± ± e1e2e3 ± So total pin group is a group of order 16 Since e1, e2, e3 generate the inversion e1e2e3, actually the 8 elements in the even subalgebra and the other 8 elements in the other 4D can be `Hodge' dualised So when the group contains the inversion Pin = Spin Z2 × A Clifford way of doing orthogonal transformations Polyhedral groups, Platonic solids and root systems The geometry of the Coxeter plane Reflection groups with Clifford algebras Root system induction and ADE correspondences Conclusions Representations from multivector groups Conformal, modular and braid groups Spinors from reflections: icosahedral case The H3 root system has 30 roots e.g. simple roots 1 a1 = e2;a2 = ((t 1)e1 + e2 + te3) and a3 = e3 : −2 − Subgroup of rotations A5 of order 60 is doubly covered by 120 1 spinors of the form a1a2 = (1 (t 1)e1e2 + te2e3) , −2 − − 1 a1a3 = e2e3 and a2a3 = (t (t 1)e3e1 + e2e3) : −2 − − The inclusion of the H3 inversion doubles this A Clifford way of doing orthogonal transformations Polyhedral groups, Platonic solids and root systems The geometry of the Coxeter plane Reflection groups with Clifford algebras Root system induction and ADE correspondences Conclusions Representations from multivector groups Conformal, modular and braid groups Polyhedral groups as multivector groups Group Discrete subgroup Order Action Mechanism SO(3) rotational (chiral) G x RxR~ j j ! O(3) reflection (full/Coxeter) 2 G x AxA~ j j ! ± Spin(3) binary 2 G (R1;R2) R1R2 j j ! Pin(3) pinory (?) 4 G (A1;A2) A1A2 j j ! e.g. the chiral icosahedral group has 60 elements, encoded by 120 spinors, which form the binary icosahedral group together with the inversion/pseudoscalar I this gives 60 rotations and 60 rotoinversions, i.e.
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