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A Binary Encoding of Spinors and Applications Received April 22, 2020; Accepted July 27, 2020

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A Binary Encoding of Spinors and Applications Received April 22, 2020; Accepted July 27, 2020 Complex Manifolds 2020; 7:162–193 Communication Open Access Gerardo Arizmendi and Rafael Herrera* A binary encoding of spinors and applications https://doi.org/10.1515/coma-2020-0100 Received April 22, 2020; accepted July 27, 2020 Abstract: We present a binary code for spinors and Cliord multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit descriptions of the triality automorphism of Spin(8), explicit represen- tations of the Lie algebras spin(8), spin(7) and g2, etc. Keywords: Spin representation, Cliord algebras, Spinor representation, octonions, triality MSC: 15A66 15A69 53C27 17B10 17B25 20G41 1 Introduction Spinors were rst discovered by Cartan in 1913 [10], and have been of great relevance in Mathematics and Physics ever since. In this note, we introduce a binary code for spinors by using a suitable basis and setting up a correspondence between its elements and non-negative integers via their binary decompositions. Such a basis consists of weight vectors of the Spin representation as presented by Friedrich and Sulanke in 1979 [11] and which were originally described in a rather dierent and implicit manner by Brauer and Weyl in 1935 [6]. This basis is known to physiscits as a Fock basis [8]. The careful reader will notice that our (integer) en- coding uses half the number of bits used by any other binary code of Cliord algebras [7, 18], thus making it computationally more ecient. Furthermore, Cliord multiplication of vectors with spinors (using the termi-p nology of [12]) becomes a matter of ipping bits in binary expressions and keeping track of powers of −1. In order to show its usefulness, we develop very explicit descriptions of some well-known facts such as the triality automorphism of Spin(8) (avoiding entirely any reference to the octonions), the relationship between Cliord multiplication and the multiplication table of the octonions, and the construction of sets of linearly independent vector elds on spheres. The natural binary/integer code of spinors, as well as the three appli- cations, are the main contributions of the paper. The computer implementation of this code has allowed us to carry out many calculations in high dimensions which, otherwise, would have been impossible. Note that, although we only deal with the case of Cliord algebras and spinors dened by positive denite quadratic forms, the binary code can be easily extended to semi-denite quadratic forms. The paper is organized as follows. In Section 2, we recall standard facts about Cliord algebras, the Spin Lie groups and algebras, the spinor representations and the basis of weight spinors. In Section 3, we intro- duce the correspondence between these basic spinors and nonnegative integers, as well as the functions that encode the Cliord multiplication. In Section 4, we present the aforementioned descriptions/applications: triality in dimension 8, the relationship between Cliord multiplication and the multiplication table of the octonions and, nally, the construction of linearly independent vector elds on spheres. Gerardo Arizmendi: Department of Actuarial Sciences, Physics and Mathematics, UDLAP, San Andrés Cholula, Puebla, Méx- ico, E-mail: [email protected] *Corresponding Author: Rafael Herrera: Centro de Investigación en Matemáticas, A. P. 402, 36000 Guanajuato, Guanajuato, México, E-mail: [email protected] Open Access. © 2020 Gerardo Arizmendi and Rafael Herrera, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. A binary encoding of spinors and applications Ë 163 2 Preliminaries The details of the facts mentioned in this section can be found in [5, 12]. 2.1 Cliord algebra, spinors, Cliord multiplication and the Spin group n Let Cln denote the Cliord algebra generated by all the products of the canonical vectors e1, e2, ... , en 2 R subject to the relations ej ek + ek ej = − ej , ek for j ≠ k, ej ej = −1, n 0 where , denotes the standard inner product in R , and Cln the even Cliord subalgebra determined as the xed point of the involution of Cln induced by −IdRn . Let ⊗ 0 0 ⊗ Cln = Cln R C, and Cln = Cln R C 0 be the complexications of Cln and Cln. It is well known that ( 2k ∼ End(C ), if n = 2k, Cln = k k , End(C2 ) ⊕ End(C2 ), if n = 2k + 1, where k C2 = C2 ⊗ ... ⊗ C2 | {z } k times k n 2 is the tensor product of = [ 2 ] copies of C . Let us denote this space as k 2 ∆n = C , which is called the space of spinors. Consider the linear map 2k κ : Cln −! End(C ), which is the aforementioned isomorphism for n even, and the projection onto the rst summand for n odd. The Spin group Spin(n) ⊂ Cln is the subset n Spin(n) = fx1x2 ··· x2l−1x2l j xj 2 R , jxjj = 1, l 2 Ng, endowed with the product of the Cliord algebra. It is a Lie group and its Lie algebra is spin(n) = spanfei ej j 1 ≤ i < j ≤ ng. Recall that the Spin group Spin(n) is the universal double cover of SO(n), n ≥ 3. For n = 2 we consider Spin(2) to be the connected double cover of SO(2). The covering map will be denoted by λn : Spin(n) ! SO(n), where an element x1x2 ··· x2l−1x2l 2 Spin(n) is mapped to the orthogonal transformation n n λn : R −! R y 7! x1x2 ··· x2l−1x2l · y · x2l x2l−1 ··· x2x1. Its dierential is given by λn*(ei ej) = 2Eij , 164 Ë Gerardo Arizmendi and Rafael Herrera * * * where Eij = ei ⊗ ej − ej ⊗ ei is the standard basis of the skew-symmetric matrices and e denotes the metric V* n dual of the vector e. We will also denote by λn the induced representation on R . The restriction of κ to Spin(n) denes the Lie group representation κn : Spin(n) −! GL(∆n), which is special unitary. We have the corresponding Lie algebra representation κn* : spin(n) −! gl(∆n), which is, in fact, the restriction of the linear map κ : Cln −! End(∆n) to spin(n) ⊂ Cln. Note that SO(1) = f1g, Spin(1) = f±1g, and ∆1 = C the trivial 1-dimensional representation. The Cliord multiplication is dened by n µn : R ⊗ ∆n −! ∆n x ⊗ ψ 7! µn(x ⊗ ψ) = x · ψ := κ(x)(ψ). It is skew-symmetric with respect to the Hermitian product hx · ψ1, ψ2i = − hψ1, x · ψ2i , is Spin(n)-equivariant and can be extended to a Spin(n)-equivariant map V* n µn : (R ) ⊗ ∆n −! ∆n ω ⊗ ψ 7! ω · ψ. Let voln := e1e2 ··· en . When n is even, we dene the following involution ∆n −! ∆n n ψ 7! (−i) 2 voln · ψ. ± The ±1 eigenspaces of this involution are denoted by ∆n and called positive and negative spinors respectively. These spaces have equal dimension and are irreducible representations of Spin(n) ± ± κn : Spin(n) −! Aut(∆n). n Note that our denition diers from the one given in [12] by a factor (−1) 2 . There exist either real or quaternionic structures on the spin representations. A quaternionic structure α on C2 is given by ! ! z −z α 1 = 2 , z2 z1 and a real structure β on C2 is given by ! ! z z β 1 = 1 . z2 z2 Note that these structures satisfy α(v), w = v, α(w) , α(v), α(w) = hv, wi, β(v), w = v, β(w) , β(v), β(w) = hv, wi, A binary encoding of spinors and applications Ë 165 with respect to the standard hermitian product in C2, where v, w 2 C2. The real and quaternionic structures 2 ⊗[n/2] γn on ∆n = (C ) are built as follows ⊗2k γn = (α ⊗ β) if n = 8k, 8k + 1 (real), ⊗2k γn = (α ⊗ β) ⊗ α if n = 8k + 2, 8k + 3 (quaternionic), ⊗2k+1 γn = (α ⊗ β) if n = 8k + 4, 8k + 5 (quaternionic), ⊗2k+1 γn = (α ⊗ β) ⊗ α if n = 8k + 6, 8k + 7 (real). which also satisfy γn(v), w = v, γn(w) , γn(v), γn(w) = hv, wi, where v, w 2 ∆n. This means v + γn(v), w + γn(w) 2 R. 0 Now, we summarize some results about real representations of Clr in the next table (cf. [16]). Here dr 0 denotes the dimension of an irreducible representation of Clr and vr the number of distinct irreducible rep- resentations. 0 r (mod 8) Clr dr vr b r c 1 R(dr) 2 2 1 r 2 C(dr /2) 2 2 1 b r c+1 3 H(dr /4) 2 2 1 r 4 H(dr /4) ⊕ H(dr /4) 2 2 2 b r c+1 5 H(dr /4) 2 2 1 r 6 C(dr /2) 2 2 1 b r c 7 R(dr) 2 2 1 r −1 8 R(dr) ⊕ R(dr) 2 2 2 Table 1 0 ± Let ∆˜r denote the real irreducible representation of Clr for r ≢ 0 (mod 4) and ∆˜r denote the real irre- ducible representations for r ≡ 0 (mod 4). Note that the representations are complex for r ≡ 2, 6 (mod 8) and quaternionic for r ≡ 3, 4, 5 (mod 8). 2.2 A special basis of spinors and an explicit description of κ In this subsection, we recall the explicit descriptions of κ from [12] and the basis of spinors from [11], which were rst discussed by Brauer and Weyl in [6]. The vectors 1 1 u+1 = p (1, −i) and u−1 = p (1, i), 2 2 form a unitary basis of C2. Consequently, the vectors fu uε ⊗ ... ⊗ uε j ε j ... kg (ε1 ,...,εk) = 1 k j = ±1, = 1, , , (1) 2 ⊗[n/2] form a unitary basis of ∆n = (C ) , which are known to be weight vectors of the Spin representation (see below).
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