Notes on Pl¨ucker's Relations in Geometric Algebra Garret Sobczyk Universidad de las Am´ericas-Puebla Departamento de F´ısico-Matem´aticas 72820 Puebla, Pue., M´exico Email: [email protected] January 21, 2019 Abstract Grassmannians are of fundamental importance in projective geometry, algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher mathematics, such as multilinear and tensor algebra, matroid theory, and Lie groups and Lie algebras. Here we explore the basic idea of the Pl¨ucker relations in Clifford’s geometric algebra. We discover that the Pl¨ucker relations can be fully characterized in terms of the geometric product, without the need for a confusing hodgepodge of many different formalisms and mathematical traditions found in the literature. AMS Subject Classification: 14M15, 14A25, 15A66, 15A75 Keywords: Clifford geometric algebras, Grassmannians, matroids, Pauli matrices, Pl¨uckerrelations 0 Introduction The standard treatments of the Pl¨ucker relations rely heavily on the concept of a vector space, its dual vector space, and generalizations. More elementary treatments express the basic ideas in terms of familiar matrices and determi- nants. Here we exploit a powerful formalism that was invented 140 years ago by William Kingdon Clifford. Clifford himself dubbed this formalism, a gener- alization of the works of H. Grassmann and W. Hamilton, \geometric algebra", but it is often referred to today as “Clifford algebra". It is well known that Clifford’s geometric algebras are algebraically isomor- phic to matrix algebras over the real or complex numbers. However, what is lacking is the recognition that whereas geometric algebras give a comprehensive geometrical interpretation to matrices, they also provide a way of unifying vec- tors and their duals into a rich algebraic structure that is missed in the standard 1 treatment of vector spaces and their dual spaces as separate objects. Our study of the famous Pl¨ucker relations utilizes the powerful higher dimensional vector analysis of geometric algebra. When the Pl¨ucker relations are satisfied, an ex- plicit factorization of an r-vector into the outer product of r vectors is found. Also, a new characterization of the Pl¨ucker relations is given. Let a and b be two non-zero null vector generators of a real or complex associative algebra, R(a; b) or C(a; b), respectively, with the identity 1, and satisfying the rules 1) a2 = 0 = b2; and 2) ab + ba = 1: (1) From these rules, a multiplication table for these elements is easily constructed, given in Table 1. Table 1: Multiplication Table. a b ab ba a 0 ab 0 a b ba 0 b 0 ab a 0 ab 0 ba 0 b 0 ba The null vectors a and b are dual to one another in the sense that the symmetric inner product of a and b, defined by 1 1 a · b := (ab + ba) = : 2 2 In addition, the outer product of a and b, gives the bivector 1 a ^ b := (ab − ba); 2 with the property 1 1 1 1 (a ^ b)2 = (ab − ba)2 = (ab + ba) = so ja ^ bj := : 4 4 4 2 Taking the sum of the above inner and outer products shows that the geomet- ric (Clifford) product of the reciprocal null vectors a and b is the primitive idempotent ab = a · b + a ^ b; already identified in Table 1. More details for the interested reader of the construction of basic geometric algebras, obtained from R(a; b) and C(a; b), and by taking the Kronecker products of such algebras, is deferred to the Appendix. Geometric algebras are considered here to be the natural geometrization of the real and complex number sytems; an overview is provided in [9]. We use exclusively the language of geometric algebra, developed in the books [6, 12]. 2 Pl¨ucker's coordinates are briefly touched upon on page 30 of [6], but Pl¨ucker's relations are not discussed. These relations characterize when an r-vector B in n n the geometric algebra Gn := G(R ), or Gn(C) := G(C ), of an n-dimensional (real or complex) Euclidean vector space Rn, or Cn, respectively, can be ex- pressed as an outer product of r-vectors. Several references to the relevant literature, and its more advanced applications, are [1, 2], [3, pp.227-231], [5], [7, pp.144-148], and [13]. 1 Pl¨ucker's relations in geometric algebra A general geometric number g 2 Gn is the sum of its k-vector parts, n X g = hgik; k=0 k k where the k-vector hgik 2 Gn is in the linear subspace Gn of k-vectors in Gn, 0 r and the 0-vector part hgi0 2 Gn ≡ R. A nonzero r-vector B 2 Gn is said to be 1 an r-blade if B = v1 ^ · · · ^ vr for vectors v1; : : : ; vr 2 Gn, Let B = P B e be an r-vector expanded in the standard orthonormal Jm Jm Jm basis of r-vector blades r eJm := ej1 ej2 ··· ejr 2 Gn; ordered lexicographically, with Jm = fj1; ··· ; jrg for 1 ≤ j1 < ··· < jr ≤ n, and let #(Jp \ Jm) be the number of integers that the sequences Jp and Jm have in common. For example, if Jp = f12345g and Jm = f45678g, then #(Jp\Jm) = 2. y The reverse g of a geometric number g 2 Gn is obtained by reversing the order of all of the geometric products of the vectors which define g. n Lemma 1 In the standard orthonormal basis feigi=1 of Gn, an r-vector B 2 r Gn, expanded in this basis, is given by X B = BJm eJm ; (2) m where B = B · ey . Jk Jk r Proof: In the orthonormal r-vector basis feJm g, the r-vector B 2 Gn is given by n ! r X B = BJk eJk k=1 Dotting both sides of this equations with ey , the reverse of e , shows that Jm Jm B = B · ey , since e · ey = δ . Jm Jm Jk Jm k;m 3 r Definition 1 A non-zero r-vector B 2 Gn is said to be divisible by a non-zero k k-blade K 2 Gn, where k ≤ r, if KB = hKBir−k. If k = r, B is said to be totally decomposable by K. If B is divisible by the k-blade L, then L(LB) L · B B = = L : (3) L2 L2 L·B If B is totally decomposable by L, then B = βL for β = L2 2 R. Clearly, a totally decomposable r-vector is an r-blade. P r A classical result is that an r-vector B = m BJm eJm 2 Gn is an r-blade iff B satisfies the Pl¨uckerrelations. The Pl¨ucker relations, [4, Thm1(1)], are (A · B) ^ B = 0 (4) r−1 for all (r − 1)-vectors A 2 Gn . In particular, for all coordinate (r − 1)-vectors r−1 eKp 2 Gn , (eKp · B) ^ B = 0: Since an r-blade trivially satisfies the Pl¨ucker relations, it follows that if there is a coordinate (r − 1)-blade eK , such that (eK · B) ^ B 6= 0, then B is not an r-blade. For every coordinate (r − 1)-blade eKp , there is a coordinate r-blade eJp , generally not unique, and a coordinate vector ejp , satisfying eJp = ±ejp eKp () ejp eJp = ±eKp : (5) Indeed, when eKp ·B 6= 0, we always pick the coordinate vector ejp , and rearrange the order of K = ±K , in such a way that B = (e ^ e )y · B 6= 0. It p jp Jjp jp Kjp follows that the Pl¨ucker relations (4) are equivalent to h i h i e · e · B ^ B = B B − e · B ^ e · B = 0; jp Kjp Jjp Kjp jp or equivalently, −1 h i B = B eK · B ^ ej · B ; (6) Jjp jp p r−1 for all coordinate (r − 1)-blades eKp 2 Gn for which eKp · B 6= 0. It follows that when the Pl¨ucker relation (4) is satisfied, then the vector eKp · B divides B. P r For a given non-zero r-vector B = m BJm eJm 2 Gn, let #maxS be the maximal number of linearly independent vectors in the set r−1 SB = feKj · Bj eKj 2 Gn g: (7) Lemma 2 Given an r-vector B 6= 0, and the set SB defined in (7). Then r ≤ #maxS ≤ n, and #max = r only if B is an r-blade. 4 Proof: Since B 6= 0, it follows that BJm 6= 0 for some n 1 ≤ m ≤ : r For each such eJm , there are r possible choices for eKj obtained by picking Kj ⊂ Jm, and the subset of SB generated by them are linearly independent. It follows that r ≤ #maxSB ≤ n. Suppose now that r < #maxSB. To complete the proof of the Lemma, we show that there is a Pl¨ucker relation that is not satisfied for B, so it cannot be an r-blade. Let L = (eK1 · B) ^ · · · ^ (eKk · B) 6= 0; be the outer product of the maximal number of linearly independent vectors from SB. Then k > r and LB = L · B 6= 0. Letting 1 w = (eK1 · B) ^ · · · ^ (eKr+1 · B) · B 2 Gn; it follows that w 6= 0, and 2 w = (eK1 · B) ^ · · · ^ (eKr+1 · B) · (B ^ w) 6= 0: Since w2 6= 0, it follows that w ^ B 6= 0. Using the identity w = (eK1 · B) ^ · · · ^ (eKr+1 · B) · B = (eK1 · B) (eK2 · B) ^ · · · ^ (eKr+1 · B) · B 2 −(eK2 · B) (eK1 · B) ^ · · · _ · · · ^ (eKr+1 · B) · B r + ··· + (−1) (eKr+1 · B) (eK1 · B) ^ · · · ^ (eKr · B) · B; i where _ means we are omitting the term eKi · B in the exterior product.