Notes on Pl¨ucker’s Relations in

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 are of fundamental importance in projective , algebraic geometry, and representation theory. A vast literature has grown up utilizing using many different languages of higher , such as multilinear and 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 , 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 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 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 1 a ∧ b := (ab − ba), 2 with the property 1 1 1 1 (a ∧ b)2 = (ab − ba)2 = (ab + ba) = so |a ∧ b| := . 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) 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 ∈ Gn is the sum of its k-vector parts, n X g = hgik, k=0

k k where the k-vector hgik ∈ Gn is in the Gn of k-vectors in Gn, 0 r and the 0-vector part hgi0 ∈ Gn ≡ R. A nonzero r-vector B ∈ Gn is said to be 1 an r-blade if B = v1 ∧ · · · ∧ vr for vectors v1, . . . , vr ∈ Gn, Let B = P B e be an r-vector expanded in the standard orthonormal Jm Jm Jm of r-vector blades

r eJm := ej1 ej2 ··· ejr ∈ Gn, ordered lexicographically, with Jm = {j1, ··· , jr} for 1 ≤ j1 < ··· < jr ≤ n, and let #(Jp ∩ Jm) be the number of integers that the Jp and Jm have in common. For example, if Jp = {12345} and Jm = {45678}, then #(Jp∩Jm) = 2. † The reverse g of a geometric number g ∈ 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 {ei}i=1 of Gn, an r-vector B ∈ r Gn, expanded in this basis, is given by X B = BJm eJm , (2) m where B = B · e† . Jk Jk r Proof: In the orthonormal r-vector basis {eJm }, the r-vector B ∈ Gn is given by n ! r X B = BJk eJk k=1 Dotting both sides of this equations with e† , the reverse of e , shows that Jm Jm B = B · e† , since e · e† = δ . Jm Jm Jk Jm k,m 

3 r Definition 1 A non-zero r-vector B ∈ Gn is said to be divisible by a non-zero k k-blade K ∈ 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 ∈ 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 ∈ 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 ∈ Gn . In particular, for all coordinate (r − 1)-vectors r−1 eKp ∈ 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 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 )† · 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 ∈ 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 ∈ Gn, let #maxS be the maximal number of linearly independent vectors in the

r−1 SB = {eKj · B| eKj ∈ Gn }. (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 ∈ 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. It follows that r X i+1 w ∧ B = (−1) αi(eKi · B) ∧ B 6= 0, i=1  i  where αi := (eK1 · B) ∧ · · · ∨ · · · ∧ (eKr+1 · B) · B ∈ R. But this implies that

αj(eKj · B) ∧ B 6= 0 for some j, so B is not an r-blade.

On the other hand, if #maxS = r, then for L = (eK1 · B) ∧ · · · ∧ (eKr · B), LB = L · B ∈ R, and using (3), L · B B = L L2 is an r-blade.  As a direct consequence of Lemma 2, we have

r Theorem 1 For 2 ≤ r ≤ n, a non-zero r-vector B ∈ Gn is an r-blade iff the Pl¨uckerrelations (4) are satisfied. When the Pl¨uckerrelations are satified, then L·B B = L2 L for the r-blade L defined in the last paragraph of the proof of Lemma 2.

5 2 Examples

Everybodies’ favorite example is when r = 2 and n = 4. A general non-zero 2 2-vector B ∈ G4 has the form

6 X 2 B = BJm eJm ∈ G4. m=1

Since B 6= 0, at least one component BJp eij 6= 0. Without loss of generality, we can assume eij = e12. It follows that the Pl¨ucker relation (6) for this component is −1h  i B = B12 e1 · B ∧ e2 · B

−1h i = B12 (B12e2 + B13e3 + B14e4) ∧ (−B12e1 + B23e3 + B24e4) . (8) Since

2(ei · B) ∧ B = ei · (B ∧ B) = (B12B34 − B13B24 + B14B23)eie1234, it follows that B is a 2-blade iff B ∧ B = 0, or equivalently,

B12B34 − B13B24 + B14B23 = 0. (9) This occurs when B is totally decomposable, given in (8). More generally, a similar argument holds true for r = 2 and any n > 4, but with more Pl¨ucker relations (9) to be satisfied. P r To get more insight into what is going on, let B = m BJm eJm ∈ Gn be a general r-vector. Following [8], we calculate

2 X X X B = BJm BJp eJm eJp = BJm BJp eJm eJp + BJm BJp eJm eJp m,p m=p m6=p

X X  r−k = BJm BJp eJm eJp + BJm BJp 1 + (−1) eJm eJp , (10) m=p m

6 r Theorem 2 A non-zero r-vector B ∈ Gn is an r-blade iff both the conditions 2 1 1 B = B · B, and BvB ∈ Gn for all v = eK · B ∈ Gn, are satisfied.

r Proof: It is easy to show that if B ∈ Gn is an r-blade, then both of the conditions in the Theorem are true. To complete the proof we must only show that if both conditions are not satisfied, then B is not an r-blade. If B2 6= B ·B, then B is not an r-blade, so it is only left to show that if B2 = B · B and n BvB 6∈ Gn, then B is not an r-blade. The following identity is needed: BvB = B(v · B + v ∧ B) = B(v · B) + B(v ∧ B)

= B · (v · B) + B · (v ∧ B) + hB(vB)i3,5,··· (12) 2 1 Suppose now that B = B · B and BvB 6∈ Gn for some v = eK · B. The identity (12) implies that hBvBi3,5,··· 6= 0. We also know that 2B ∧ v = Bv + (−1)rvB ⇐⇒ Bv = 2B ∧ v − (−1)rvB, so that

r   hBvBi3,5,··· = h 2B ∧ v − (−1) vB Bi3,5,··· = 2h B ∧ v Bi3,5,··· 6= 0,

2 since B = B · B. But this implies that B ∧ (eK · B) 6= 0, so by (4), B is not an r-blade.  Sm ooooxxxxx xxxxxoooo Sp

Figure 1: Shows Sm and Sp in the case for odd r = 9, with an odd overlap k = 5. In this case we choose eK = ej2···j9 , the j2, ··· j9 are the last 8 digits of Sm. A similar construction applies when r and k are even.

Note that (4) is fully equivalent to the conditions in the Theorem 2. If 2 r−1 B 6= B · B then there is at least one eK ∈ G1 such that (eK · B) ∧ B 6= 0. This follows from the coordinate expansion of B given in (10), because the parity condition allows us to find at least one non-zero BJm BJp eJm eJp , with overlap k = #(SJm ∩ SJp ), such that (eK · B) ∧ B 6= 0. We simply pick eK in such a way that K ⊂ SJm , but #(K ∩ Jp) ≤ r − 2, see Figure 1. In the example (11), B is not a 3-blade because even though B2 = B · B, for

5 e12 · B = −e3, Be3B = 2e12456 ∈ G6.

7 It is also interesting to note that just as B2 = B·B whenever B is an r-vector r+1 satisfying the Pl¨ucker relations (4), BvB = (−1) (B · B)v for all v = eK · B whenever B is an r-vector satisfying the Pl¨ucker Relations (4). This follows from the identity BvB = B[2v ∧ B + (−1)r+1Bv] = (−1)r+1B2v ⇐⇒ v ∧ B = 0 for all v = eK · B, when B satisfies the Pl¨ucker relations (4). 3 Let us consider now some other examples. For the 3-vector B ∈ G5,

B := −2e134 + e145 − 2e234 + e245 + e315 + e325 + e345, it is easily verified that the Pl¨ucker relations

(e41 · B) ∧ B = (2e3 + e5) ∧ B = 0, e54 · B = (e1 + e2 + e3) ∧ B = 0,

e13 · B = (2e4 + e5) ∧ B = 0, and that the vectors (e41·B) = 2e3+e5, (e54·B) = (e1+e2+e3), e13·B = 2e4+e5 are linearly independent. Applying Theorem 1, with

L = (e1 + e2 + e3) ∧ (2e3 + e5) ∧ (2e4 + e5) L · B 1 B = L = − L. L2 2 For

3 B = e123 + e124 + e125 + e126 + e345 + e346 + e356 + e456 ∈ G6, we find that

(e21 · B) ∧ B = (B123B456 − B124B356 + B125B346 − B126B345)e3456

+(B123B145 − B124B135 + B125B134)e1345 + ···

+(B124B156 − B125B146 + B126B145)e1456 = 0, but (e13 · B) ∧ B = e2 ∧ B = e2456 + e2356 + e2346 + e2345 6= 0.

Since (e21 · B) ∧ B = 0, it is divisible by e12 · B. Using (6), we find that

B = (e21 · B) ∧ (e3 · B) = (e3 + e4 + e5 + e6) ∧ (e12 + e56 + e46 + e45). But since it does not satisfy the second condition, it is not totally decomposable. 3 As a final example, consider B = (e1 + e2 + e3 + e4) ∧ (e23 + e34 + e46) ∈ G6. There are no non-zero vectors of the form eij · B such that

(eij · B) ∧ B = 0.

However, the 2-blade e36 − e46, satisfying

(e36 − e46) · B = e1 + e2 + e3 + e4 does divide B, since (S · B) ∧ B = 0.

8 Appendix: Geometric matrices   g11 g12 Let [g] := be a real matrix in Mat2(R). Since R(a, b) is an asso- g21 g22  ciative algebra, the module Mat2 R(a, b) is well-defined. We use the matrix [g] to define the unique element g ∈ R(a, b), specified by     g11 g12 ba g := ( ba a ) = g11ba + g12b + g21a + g22ab. g21 g22 b

We say that [g] is the matrix of g ∈ R(a, b) with respect to the Witt basis of null vectors, {a, b}, over the real numbers R. It is easy to show that this establishes an algebraic isomorphism between R(a, b) and Mat2(R), [10]. The standard orthonormal basis {e1, f1} of the geometric algebra

G1,1 := R(e1, f1) ≡ R(a, b) is e1 := a + b, and f1 := a − b, and satisfies the rules

2 2 1) e1 = 1 = −f1 , and 2) e1f1 = −f1e1. (13)

The geometric algebra G1,1 := R(e1, f1), defined by the rules (13), is nothing more than a change of basis, from the Witt basis of null vectors (1) to standard orthonormal basis (13). As a linear space over R,

G1,1 = spanR{1, e1, f1, e1f1}.

In terms of the null basis, the bivector e1f1 = (a+b)(a−b) = ba−ab = 2b∧a. 1 Letting u1 = ba = 2 (1 + e1f1), the matrices of the vectors, are  ba   0 1   1  e1 = ( ba a )[e1] = ( 1 e1 ) u1 , a 1 0 e1 and  ba   0 −1   1  f1 = ( ba a )[e1] = ( 1 e1 ) u1 . a 1 0 e1

The matrix of the bivector e1f1, is then easily found to be

 0 1   0 −1   1 0  [e f ] = [e ][f ] = = . 1 1 1 1 1 0 1 0 0 −1

By complexifying the basis vector f1 ∈ G1,1, we obtain the geometric algebra G3, a geometric interpretation of the√ famous Pauli matrices, and a geometric interpretation of the imaginary i = −1. We define

G3 := R(e1, e2, e3),

9 where e2 := if1 and e3 := e1f1. It then follows that e123 = e1if1e1f1 = i, giving 3 imaginary unit i the geometric interpretation of the unit trivector e123 ∈ G3. 3 Note also the unit vector e3, along the z-axis of R is identified with the bivector 2 e1f1 ∈ G1,1. The Pauli matrices [ek], satisfying

 ba  e = ( ba a )[e ] , k k b are given by

 0 1   0 −i   1 0  [e ] = , [e ] = , [e ] = . 1 1 0 2 i 0 3 0 −1

Geometric matrices of real or complex higher dimensional geometric algebras n,n n Gn,n := G(R ), or Gn,n(C) := G(C ), are obtained by taking directed Kro- necker products of blocks of pairs of null vectors {ak, bk}, each block satisfying the rules (1), in addition to being anti-commutative. That is

aiaj = −ajai, bibj = −bjbi, aibj = −bjai, for all 1 ≤ i, j ≤ n and i 6= j. For example, the geometric matrix [g] ∈ Mat4(R) defines the geometric number g ∈ G2,2, given by g :=  1      −→ 1 ←− 1 b1 ( 1 a1 ) ⊗ ( 1 a2 ) u12[g] ⊗ = ( 1 a1 a2 a12 ) u12[g]   , b2 b1  b2  b21 where u12 = u1u2 = b1a1b2a2. Details of the general construction can be found in [11].

Acknowledgement

I thank Dr. Dung B. Nguyen for calling my attention to the problem of express- ing and proving the Pl¨ucker relations in geometric algebra. This work is the result of many email exchanges with him, false starts, and trying to understand the roots of the vast literature on the subject and its generalizations. The re- viewer’s suggestions for improvements were extremely valuable to the author in making the final revision of the manuscript.

References

[1] F. Ardila, The Geometry of Matroids, Notices of the AMS, Vol. 65, No. 8, Sept. 2018. [2] A. Dress and W. Wenzel, Grassmann-Pl¨uckerRelations and Matroids with Coefficients, Advances in Mathematics 86, 68-110 (1991).

10 [3] W. Fulton, J. Harris, Representation Theory: A First Course, Springer- Verlag 1991. [4] M.G. Eastwood and P.W. Michor, Some Remarks on the Pl¨uckerRelations, Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 63 (2000), 85-88. https://www.mat.univie.ac.at/˜ michor/plue-lon.pdf

[5] J. Harris, Algebraic Geometry: A First Course, Spring, p.63-71, 1992. http://math.rice.edu/˜evanmb/math465spring11/ Harris-Grassmannians.pdf [6] D. Hestenes and G. Sobczyk. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, 2nd edition, Kluwer 1992.

[7] J. Milne, Algebraic Geometry, Version 6.06, March 2017, pp. 144-148. http://www.jmilne.org/math/CourseNotes/AG.pdf [8] D. B. Nguyen, Pl¨ucker’sRelations, Geometric Algebra, and the Electromag- netic , unpublished version: Sept. 3, 2018.

[9] G. Sobczyk, Geometrization of the System, July, 2017. http://www.garretstar.com/geonum2017.pdf [10] G. Sobczyk, Hyperbolic Numbers Revisited, preprint 2017. http://www.garretstar.com/hyprevisited12-17-2017.pdf [11] G. Sobczyk, Geometric Matrices and the Symmetric Group, 2018. https://arxiv.org/pdf/1808.02363.pdf [12] G. Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number, Birkh¨auser,New York 2013. [13] C.T.C. Wall, Pl¨uckerformulae for curves in high dimensions, University of Liverpool, 2008. https://www.liverpool.ac.uk/˜ ctcw/pluckD.pdf

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