Doctor of Philosophy. at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY. February 1995

Transcription

1 Geometric Identities in Invariant Theory by Michael John Hawrylycz B.A. Colby College (1981) M.A. Wesleyan University (1984) Submitted to the Department of Mathematics in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the MASSACHUSTTS INSTITUT OF TCHNOLOGY February 1995 ( 1995 Massachusetts Institute All rights reserved of Technology Signature of Author..., Department of Mathematics 26 September, 1994 Certified by Gian-Carlo Rota Professor of Mathematics Accepted by David Vogan Chairman, Departmental Graduate Committee Department of Mathematics Scier~, MASSACHUSTTS INSTITUT ()F *rrr-"!1yjnn',/ MAY

2 Geometric Identities in Invariant Theory by Michael John Hawrylycz Submitted to the Department of Mathematics on 26 September, 1994, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract The Grassmann-Cayley (GC) algebra has proven to be a useful setting for proving and verifying geometric propositions in projective space. A GC algebra is essentially the exterior algebra of a vector space, endowed with the natural dual to the wedge product, an operation which is called the meet. A geometric identity in a GC algebra is an identity between expressions P(A, V, A) and Q(B, V, A) where A and B are sets of anti-symettric tensors, and P and Q contain no summations. The idea of a geometric identity is due to Barnabei, Brini and Rota. We show how the classic theorems of projective geometry such as the theorems of Desargues, Pappus, Mobius, as well as well as several higher dimensional analogs, can be realized as identities in this algebra. By exploiting properties of bipartite matchings in graphs, a class of expressions, called Desarguean Polynonials, is shown to yield a set of dimension independent identities in a GC algebra, representing the higher Arguesian laws, and a variety of theorems of arbitrary complexity in projective space. The class of Desarguean polynomials is also shown to be sufficiently rich to yield representations of the general projective conic and cubic. Thesis Supervisor: Gian-Carlo Rota Title: Professor of Mathematics

3 Acknowledgements I would like to thank foremost my thesis advisor Professor Gian-Carlo Rota without whom this thesis would not have been written. He contributed in ideas, inspiration, and time far more than could ever be expected of an advisor. I would like to thank Professors Kleitman, Propp, and Stanley for their teaching during my stay at M.I.T. I am particularly grateful that Professors Propp and Stanley were able to serve on my thesis committee. Several other people who contributed technically to the thesis were Professors Neil White of the University of Florida, Andrea Brini of the University of Bologna, and Rosa Huang of Virginia Polytechnic Institute, and Dr. manuel Knill of the Los Alamos National Laboratory. A substantial portion of the work was done as a member of the Computer Research and Applications Group of the Los Alamos National Laboratory. The group is directed my two of the most generous and interesting people I have known, group leader Dr. Vance Faber, and deputy group leader Ms. Bonnie Yantis. I am very indebted to both of them. The opportunity to come to the laboratory is due to my friend Professor William Y.C. Chen of the Nankai Institute and LANL. I especially thank Ms. Phyllis Ruby of M.I.T. for many years of assistance and advice. I would also like to express my sincere gratitude to my very supportive family and friends. Three special friends are John MacCuish, Martin Muller, and Alain Isaac Saias.

4

5 Contents 1 The Grassmann-Cayley Algebra Introduction The xterior Algebra of a Peano Space Bracket methods in Projective Geometry Duality and Step Identities Alternative Laws Geometric Identities Arguesian Polynomials The Alternative xpansion The Theory of Arguesian Polynomials Classification of Planar Identities Arguesian Lattice Identities A Decomposition Theorem Arguesian Identities Arguesian Identities Projective Geometry The Transposition Lemma nlargement of Identities 105

6 CONTNTS The nlargement Theorem... xam ples Geom etry The Linear Construction of Plane Curves 5.1 The Planar Conic The Planar Cubic The Spacial Quadric and Planar Quartic

7 List of Figures 1.1 The Theorem of Pappus The Theorem of Desargues The graphs Bp, for i = 1,..., The Theorem of Bricard The Theorem of the third identity The First Higher Arguesian Identity Bp for P = (av BC) A (bv AC) A (cv BCD) A (d V CD) and B2p The matrix representation of two polynomials P Q A non-zero term of an identity P - Q Linear Construction of the Conic Linear Construction of the Cubic

8 8 LIST OF FIGURS

9 Chapter 1 The Grassmann-Cayley Algebra Malgrd les dimensions restreintes de ce livre, on y trouvera, je l'espe're, un expose assez complet de la G6omitrie descriptive. Raoul Bricard, Geometrie Descriptive, Introduction The Peano space of an exterior algebra, especially when endowed with the additional structure of the join and meet of extensors, has proven to be a useful setting for proving and verifying geometric propositions in projective space. The meet, which is closely related to the regressive product defined by Grassmann, was recognized as the natural dual operation to the exterior product, or join, by Doubilet, Rota, and Stein [PD76]. Recently several researchers including Barnabei, Brini, Crapo, Kung, Huang, Rota, Stein, Sturmfels, White, Whitely and others have studied the bracket ring of the exterior algebra of a Peano space, showing that this structure is a natural structure for geometric theorem proving, from an algebraic standpoint. Their work has largely focused on the bracket ring itself, and less upon the Grassmann-Cayley algebra, the algebra of antisymmetric tensors endowed with the two operations of

10 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA the wedge product, join, and its natural dual meet. The primary goal of this thesis, is to develop tools for generating identities in the Grassmann-Cayley algebra. In his Calculus of xtensions, Forder [For60], using precursors to this method, develops thoroughly the geometry of the projective plane, with some attention to projective three space. The work of Forder contains implicitly, although not stated as such, the idea of a geometric identity, a concept first made precise in the work of Barnabei, Brini, and Rota [MB85]. Informally, a geometric identity is an identity between expressions P(A, V, A) and Q(B, V, A), involving the join and meet, where A and B are sets of extensors, and each expression is multiplied by possible scalar factors. The characteristic distinguishing geometric identities in a Grassmann-Cayley algebra from expressions in the Peano space of a vector space is that in the former no summands appear in either expression. Such identities are inherently algebraic encodings of theorems valid in projective space by propositions which interpret the join and meet geometrically. One problem in constructing Grassmann-Cayley algebra identities is that the usual expansion of the meet combinatorially or via alternative laws, leaves summations over terms which are not easily interpreted. While the work of Sturmfels and Whitely [BS91] is remarkable, in showing that any bracket polynomial can be "factored" into a Grassmann-Cayley algebra expression by multiplication by a suitable bracket factor, their work does not provide a direct means for constructing interesting identities. Furthermore, because of the inherent restrictions in forming the join and meet based on rank, natural generalizations of certain basic propositions in projective geometry, do not seem to have analogs as identities in this algebra. The thesis is organized into chapters as follows: The first chapter develops the basic notions of the Grassmann-Cayley algebra, within the context of the exterior algebra of a Peano space, following the presentation of Barnabei, Brini, and Rota [MB85]. We define the notion of an extensor polynomial as an expression in extensors, join and meet and prove several elementary properties about extensor polynomials which will be useful in the sequel. Next we demonstrate how bracket ring methods are useful in geometry by giving a new result for an n-dimensional version of Desargues' Theorem, as well as several results about higher-dimensional projective configurations. This chapter concludes by defining precisely the notion of geometric identity in the Grassmann-Cayley algebra, and giving several examples of geometric identities, including identities for theorems of Bricard [Haw93], M6bius, and Pappus, [Haw94]. In Chapter 2 we identify a class of expressions, which we call Arguesian polynomials, so named because they yield geometric identities most closely related to the theorem of Desargues in the projective plane and its many generalizations to higher-

11 1.1. INTRODUCTION dimensional projective space. The notion of equivalence between two Arguesian polynomials is made precise by -equivalence. In essence, two Arguesian polynomi- als P and Q are -equivalent, written P = Q if P and Q reduce to the same bracket polynomial in the monomial basis of column tableaux, in vectors and covectors, via a certain expansion, called the alternative expansion (P). The alternative expansion is a recursive evaluation of P subject to the application of alternative laws for vectors and covectors, as presented in Chapter 1. After presenting several technical lemmas necessary in the subsequent chapters, Chapter 2 explores the structure of Arguesian polynomials, by classifying the planar Arguesian identities. Surprisingly, there are only three distinct theorems up to -equivalence in the plane, the theorem of Desargues, a theorem attributable to Raoul Bricard, and a third lesser known theorem of plane projective geometry. In addition, a particularly simple subclass of Arguesian identities are characterized which yield geometric identities for the higher Arguesian lattice laws, justifying our choice of terminology. The characterization results of chapter 2 rely on a decomposition theorem for Arguesian polynomials. The proof of this theorem is given in the final section of this chapter. Identities between Arguesian polynomials are closely related to properties of perfect matchings in bipartite graphs. ach perfect matching in a certain associated graph Bp corresponds to a non-zero term of the given polynomial P. The theory of Arguesian identities is more complex than the theory of bipartite matchings because of a sign associated with each such matching. In Chapter 3 we present a general construction, from which all Arguesian identities follow, enabling a variety of identities in any dimension. The construction may be seen as a kind of alternative law for Arguesian polynomials in the sense of Barnabei, Brini, and Rota [MB85]. Ideally, our identities would be proven in the context of superalgebras [RQH89, GCR89], thereby eliminating the need to consider the details of sign considerations. To this date, however, the meet as an operation in supersymmetric algebra has not been rigorously defined, and such attempts have led to contradictory results, or results which are difficult to interpret. A recent announcement by Brini [Bri94] indicates that the theory of Capelli operators and Lie superalgebras may provide the required setting. The fourth chapter proves a dimension independence theorem for Arguesian identities, called the enlargement theorem. Specifically, given any identity P = Q between two Arguesian polynomials P(a, X) and Q(a, X), both in step n, we may formally substitute for each vector a a, (and each covector X X), the join (or meet) of distinct vectors al V... Vak = a(k) a(k) (or covectors X 1 A... A Xk = X(k) X(k)) of steps k (and n - k), to yield Arguesian polynomials p(k) and Q(k) which then satisfy p(k) f Q(k). This theorem suggests that Arguesian identities are in fact con-

12 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA sequences of underlying lattical identities, which we conjecture. The enlargement theorem strongly suggests that indeed Arguesian identities are a class of identities valid in supersymmetric algebra [GCR89], in terms of positive variables, an idea suggested by Rota. Indeed, the enlargement theorem itself was first intuited by Rota as an effort to understand when Grassmann-Cayley algebra identities are actually identities in supersymmetric algebra. In the fifth and final chapter we conclude with another give another application of vector/covector methods to the study of projective plane curves or surfaces. The vanishing of an Arguesian polynomial, in step 3 with certain vectors (or equivalently covectors ) replaced by common variable vectors represents the locus of a projective plane curve of given order. This addresses an old problem of algebraic geometry, dating to even Newton, the linear construction of plane curves. This idea is due to Grassmann and sees a considerable simplification in the language of Grassmann- Cayley algebra. We show how the forms for Arguesian polynomials in the plane yield symmetric and elegant expressions for the conic, cubic, and a partial solution to the quartic. As a final result, a generalization of Pascal's Theorem for the planar cubic is given. A Maple V program was written which reduces any Arguesian polynomial to its canonical monomial basis. This code was extremely useful in obtaining and verifying many of the results of the thesis, and undoubtably the code is useful for further work. The author will gladly supply this code upon request. 1.2 The xterior Algebra of a Peano Space A Peano space is a vector space equipped with the additional structure provided by a form with values in a field. The definition of a Peano space and its basic properties were first developed by Doubilet, Rota, and Stein [PD76] and later Barnabei, Brini, and Rota [MB85]. We will state and prove only some of their results, for completeness, and the reader is referred to these papers for a more complete treatment. Let K be an arbitrary field, whose values will be called scalars, and let V be a vector space of dimension n over K, which will remain fixed throughout. Definition. A bracket of step n over the vector space V is a non-degenerate alternating n-linear form defined over the vector space V, in symbols, a function X1, X,..., x, -4 [Xl,X2,... X] K defined as the vectors xl, x2,..., x,, range over the vector space V, with the following properties:

13 1.2. TH XTRIOR ALGBRA OF A PANO SPAC [l, 22...,,n] = 0 if any two of the xi coincide. 2. For every x, y V, a, 3 K the bracket is multilinear [Xi,..., Xi-l,ax + 3py, Xi+1,...,IXn] = a[xi,...,zi-1,x,xi+1,...,xn] + P[xx,x i -,y, xi+ l,..., n ]. 3. There exists a basis bl,b 2,...,b, of V such that [bl,b 2,...,bn] $ 0. Definition A Peano space of step n is defined to be a pair (V, [.]), where V denotes a vector space of dimension n and [-] is a bracket of step n over V. A Peano space will be denoted by a single letter V leaving the bracket understood when no confusion is possible. A non-degenerate multilinear alternating n-form is uniquely determined to within a non-zero multiplicative constant, however the choice of this constant will determine the structure of the Peano space. A Peano space can be viewed geometrically as a vector space in which an oriented volume element is specified. The bracket [x 1, x. 2,..., xn] gives the volume of the parellelpiped those sides are the vectors xi. If V is a vector space of dimension n, a bracket on V of step n can be defined in several ways. The usual way is simply to take a basis el, e2,..., en of V and then given vectors; xj = ijej i = 1, 2,... to set [XI, 2,..., ] = det(xij). Although a bracket can always be computed as a determinant, it will prove more interesting in this context to view the bracket as an operation not unlike a norm in the theory of Hilbert space, rather than as a determinant in a specific basis. The exterior algebra of a vector space is a special case of a Peano space and can be developed in this context. To construct the exterior algebra of a Peano space V of step n over the field K, let S(V) be the free associative algebra with unity over K generated by the elements of V. For every integer k, 1 _< k < 'n consider the subspace Nk(V) of S(V) of all vectors f = i aiewi with wi = () 1 2) (i) such that xi V, a K and such that for every z1, z2,..., z 7 n-k V, a[ i),,..z2,]= 0. O[xl,... 2 Xk, Z1,Z2..,Z. n-] "- 0.

14 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA For k > n we denote by Nk(V) the subspace of S(V) spanned by all words of length k, and let N(V) = N 1 e N 2 (V) e*-e.. Nn(V). It is easy to see that N(V) is an ideal of the algebra S(V). The quotient algebra G(V) = S(V) \ N(V) is called the exterior algebra of the Peano space V. It is readily seen that G(V) decomposes as where n G(V) = Gk(V), k=o dimlgk(v) =(. The above construction may be equivalently performed by imposing an equivalence relation on sequences of vectors. Given two sequences of vectors of length k we write al,a2,..., ak ", bib 2,..., bk when for every choice of vectors Xk+1,. - --,n we have [ai,a 2,...,ak, xk+l,..,x,,] = [blb 2,...,bk, k+1,..., ]. An equivalence class under this relation will be called an extensor. More precisely, let 0 : S(V) -- G(V) denote the canonical projection of S(V) onto G(V). If Xl,X2,.. -,k is a word in S(V) with X1, 2,..., k V, for k > 0 we denote its image under 0 by 0(x1,X2,...,Xk) = Xl V x2 V... V Xk, and provided step k. (x1, x2,..., xk) : 0 the element is called the extensor 1XX2 "... k of The product in the exterior algebra of a Peano space is called the join, in order to emphasize its geometric significance, and is denoted by the symbol V. We note that this usage differs from the ordinary usage where exterior multiplication is denoted as the wedge product A. It is clear that the join of two extensors, since non-zero by definition, is an extensor. When an extensor A is written as A = al Va2 V... Vak, we say that the linearly ordered set {al, a2,' extensor A., ak} is a representation of the It is not always possible to write a sum of two or more extensors of step k as another extensor of step k, and hence one also has indecomposable k-vectors in

15 1.2. TH XTRIOR ALGBRA OF A PANO SPAC 15 Gk(V). For example, if a, b, c, and d are linearly independent in V, then ab + cd is an indecomposable 2-vector in G 2 (V). Let B = b 1 b 2... bj be an extensor of step j. Then A V B = al V a 2 V... V ak V bl V... bj = ala2.. akbl.. bj is an extensor of step j+k. In particular, AVB is nonzero if and only if al, a2,..., ak bl,..., bj are distinct and linearly independent. Consider any k vectors al,..., ak V and their expansions ai = 'U 1 ai, j ej in terms of the given basis {el,..., e,}. By multilinearity and antisymmetry, the expansion of their join equals al Va 2 V **a= l<il<...<ikl L al,i 1 ali 2 " aljk 12,i a2,i 2 a1,jk ak,i, ak,i2 " " kjk but in general we will avoid passing to the coordinate level. ei, Vei,... eik Summarized below are some of the properties of the exterior algebra of a Peano space. These results are well-known in the context of an exterior algebra. Proposition 1.1 Let V be a Peano space of step n over the field K and let G(V) be its exterior algebra. Choose a basis {al,a2,... I a,, } of V and let S(al, a 2,... a n) be the free associative algebra with unity over K in the variables al, a2,..., an. The algebra G(V) is isomorphic to the quotient of the algebra S(a, a2,..., an) by the ideal generated by the following elements of S(al, a2,..., a n) aiaj + ajai a? i = 1,2,...,n i,j= 1,2,...,n Proposition 1.2 For every i = 1,2,...,k, Xl,X2,... i-1, Xi+l,...,k, y V and for every a, 8 K we have 1. X 1 V xz2" V... V xi- 1 V (ax+ -y) V xi+ 1 V... VXk = a(xl V... V i-1 V V Xi+l V V X k) ) +P(XiV... V Xi- 1 V y V Xi+V V xk). 2. For every permutation a of {1,2,...,k},,(1)VX,(2)V. *"VXa(k) = sgn(a)x V X2 V " X k where sgn(a) is the signature of the permutation a. Proposition 1.3 Let A Gh,(V) and B Gk(V), then BVA = (-1)hkA V B (1.1)

16 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA Proposition 1.4 Let A be a subspace of V of dimension k > 0; if {x1, X2,..., xk} and {yl, Y2,...,Yk} are two bases of A then X1 V X2 V.V k = Cyl V y2 V... V Yk for some non-zero scalar C. By Proposition 1.4 every non-trivial subspace of V is uniquely represented modulo a non-zero scalar by a non-zero extensor, and vice-versa. The zero subspace is represented by scalars. We say that the extensor x1... Xk is associated to the subspace generated by the vectors of V corresponding to {x 1,..., k }. We also remark that the join al V... V ak is non-zero if and only if the set of associated vectors is a linearly independent set. The following proposition is fundamental. Proposition 1.5 Let A, B be two subspaces of V with associated extensors F and G respectively. Then 1. F V G = O if and only if A nb l {0}. 2. If A n B = {0}, then the extensor F V G is the extensor associated to the subspace generated by A U B. Let W be a subspace of a Peano space V, and let wz, w2,..., wn be a basis of V such that Wi, w2,..., Wk is a basis of W. We define the restriction of a Peano space V to W to be the Peano space obtained by giving W the bracket [xlx2,...,xk]w = [X1, 2,...,k, Wk+1,..., W]. The bracket [-]w depends to within a multiplicative constant on a choice of the basis elements wk+lwk+2,..., wn. Let W' be the subspace spanned by Wk+l,..., wn. We define the Peano space on the quotient space V \ W' by setting [Vi!,V2 -...,VUn-k\W' ] = [ZiX 2?...,Xk, Wk+l,.-..,Wn, where vi are vectors in V \ W' and xi is any vector that is mapped to vi by the canonical map of V into V \ W'. The bracket in the quotient Peano space depends on a choice of a basis of W', again to within a multiplicative constant. These constructions suggest there might be a relationship between matroids and the bracket algebra, and this connection is fully explored in White [Whi75].

17 1.2. TH XTRIOR ALGBRA OF A PANO SPAC Proposition 1.6 Let W be a subspace of the Peano space V endowed with a restriction of the bracket, and let V \ W be endowed with the quotient bracket. Then the exterior algebra of the Peano space W is naturally isomorphic to the restriction to W of the exterior algebra of the Peano space V, and the exterior algebra of the Peano space V \ W is naturally isomorphic to the quotient of the exterior algebra of V by the ideal generated by the exterior algebra of W. A second operation in the exterior algebra of a vector space is the meet. A precursor to this operation was originally recognized by Hermann Grasssman in his famous Ausdehnungslehre [Grall], whose intention was to develop a calculus for the geometry of linear varieties. The equivalent of the meet was called the regressive product, unfortunately denoted by Grassrnmann by the same notation as the join or wedge product. While this operation was used by later authors such as Whitehead [Whi97] and Fordor [For60], the realization that the exterior algebra of a Peano space, with its two operations of join V and meet A, is the natural structure for the study of projective invariant theory under the special linear group was not made until Rota [PD76]. Given a representation of an extensor A = al V a2 V... V ak and an ordered r-tuple of non-negative integers hi, h 2,..., h1t such that hi + h 2 + ".. + hr = k, a split of type (hi, h2,..., h,.) of the representation A = al V a2 V... ak is an ordered r-tuple of extensors (A 1, A 2,..., A,) such that 1. Ai = Aif hi = 0 and Ai = ai, V aiv... V a if hi Ai V Aj A1VA 2 V... v A,. = ±A. In what follows we shall denote by S(a, a2,..., ak; 1, h 2,..., hr) the finite set of all splits of type (hi, h 2,..., h,.) of the extensor A relative to the representation A = al V... V ak. One can easily extend the definition of the signature of A to the signature of the split A = A 1 V A 2 V... V Ar as follows; sgn(a1, A2,..., A.) = I 1 ifai VA2V... V A = A -1 ifaiva V...VA r=-a 1 f A, v A 2 V... V A, = -A The bracket notation can be extended to include the case where its entries are extensors instead of just vectors. Furthermore, this definition is easily verified to be independent of the choice of representation of the extensors A 1, A 2,..., A,

18 18 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA Definition. Let A 1, A 2,..., A, be extensors in a Peano space of step n. Choose representations A 1 = allva12v---val=s A 2 = a 2 1 Va 22 V V a 2 2 Ak = akl V ak2 V... V akek and define [A 1, A 2,..., A,] = all, a12,.. * a ls,,, a 2 a22,., a, 2,2,., a 2,2, an,1,..., n,s, when sl + s sn = n and [A, A 2,..., A,,] = 0 otherwise. A proposition which easily follows from this definition is Proposition 1.7 Let A and B be two extensors of step k and step n - k. Then [B, A] = (-1)k(n-k)[A, B]. The definition of the meet of two extensors is based on the following fundamental property of Peano spaces. Proposition 1.8 Let al,a2, -..., ak and bl, b 2,..., bp be vectors of a Peano space V of step n with k + p > n. If A = al V a 2 V... V ak and B = bl V b 2 V... V bp then the following identity holds: sgn(a 1, A 2 ) [A1, B] A 2 = (A,,A 2 )S(A;n-p,k+p-n) sgn(b 1, B 2 ) [A, B 2 ] B 1. (B1,B 2 )S(B;k+p-n,n-k) PROOF. We consider the functions defined as f, 9g: Vk+p --+ Gk+p-n(V) f(a 1,a 2,...,akl,b2,...,b,) = sgn(ai, A 2 ) [A1, B] A 2 (AI,A2)S(A;n-p,k+p-n)

19 1.2. TH XTRIOR ALGBRA OF A PANO SPAC g(al, a2,..., ak, bl, b2,..., b,) = sgn(bi, B 2 ) [A, B 2 ] B 1 (B 1,B 2 )S(B;k+p-n,n-k) where A = alva 2 V... ak and B = b 1 Vb 2 V... bp. Direct verification shows that f and g are (k +p)-multilinear functions in the vectors al, a2,..., ak, bi, b 2,.., bp. Hence f and g coincide if and only if they take the same values on any (k + p)-tuple of vectors taken from a given basis {el, e2,..., e,n } of V. Since f and g are alternating in the first k variables and in the last p variables, separately, it is sufficient to prove: f(eil,...eik..., ej ) = g(ei,... ek; ej,... ej ) in the case where il < i 2 <... < ik j1 < j2 <..." jp. Since f and g must agree on any basis, we may set d = k +p - n and simultaneously require that i 1 = jl,, id = jd. We then compute f(ei,... eik;ejl,...,ejp) = and (-1)d(k-d)[eid+l,... eik, ejl,.., ejp]eil V ei2 V... V eid g(eil,...,eik; ej1,...,ejp) = [ei,...,eik,ed+1,,...,ejp]ei, V ei, V.. V eid. Now since ej, e 2,..., ejd agree with ei 1, ei 2,..., eid the former vectors which are the first d vectors in ej,, ej,..., ej may each be shifted as far to the left as possible in the bracket in f to yield g, after d(k - d) sign changes. This proves the result. 0 We may now define the meet of two extensors A and B. Given extensors A = al V a2 V... V ak and B = bl V b 2 V... V bp with k,p > 1, we define the binary operation A by setting: 1. AAB=Oifk+p<n. 2. AA B = (A4,A2) sgn(ai, A 2 ) [A 1, B] A 2 = (B 1,B 2 ) sgn(b 1, B 2 ) [A, B 2 ] B 1 where the summations range over the splits S(al, a 2,... -, ak; n - p, k + p - n) and S(bl, b 2,..., bp; k + p - n, n - k) respectively. We remark that in the above equivalent formulas for the meet of two extensors A and B, the sign of the split extensor sgn(b 1, B 2 ) is computed as the sign of the ordered pair B 1, B 2 with respect to the order of vectors in B and not with respect to an underlying linear order on A or B. Geometrically, the meet of two extensors plays a similar role to the join.

20 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA Proposition 1.9 Let A and B be associated to the subspaces X and Y of V. If the union X U Y spans the whole space V and if X n Y 5 0 then A A B is the extensor associated to the subspace X n Y of V. PROOF. Suppose that X U Y spans V and let k be of step A and p of step B, with k + p > n. The claim holds trivially when k + p < n. Suppose therefore that k +p > n and take a basis c = {cl,2,..., cd} of the subspace X nfy where d = k +p - n. We then complete C to a basis {cl, c2,..., Cd, al,..., ak-d} associated to A and a basis {cl, C2,..., Cd, bi,..., bp-d of Y associated to B such that A = c1vc2v"..vcdvalv.. Vak-d B = cl Vc 2 V...VcdVblV...Vbp d. Compute the meet A A B by splitting either extensor. In particular AAB = :[cl, c21 * *, cd, a1, a2,..., ak-d, bl,..., bp-d]cl V c2 V... cd since all other terms vanish. Thus cl,..., cd is the extensor associated to the intersection subspace X n Y. 0 The following commutativity and associativity relations hold for the meet, their proofs can be found in [MB85]. Proposition 1.10 Let A and B be two extensors of steps k and p respectively. Then A A B = (-1)(n-k)(n-P)B A A. Proposition 1.11 (Linearity) Let A, A 1, A 2 and B be extensors with A = At + A 2. Then AAB = A AB+A 2 AB Proposition 1.12 (Associativity) Let A, B, C be extensors, then the associative law holds for the meet; (AAB) AC= AA(BAC) The definition of the meet of extensors can be extended to the sum of extensors as follows: Set T = A + C; then by Proposition 1.11 we define T A B = A A B + CA B. The following definition is fundamental to this thesis. Definition. A Peano space of step n equipped with the two operations of join V and meet A is called the Grassmann-Cayley algebra of step n and denoted GC(n).

21 1.3. BRACKT MTHODS IN PROJCTIV GOMTRY A few other notations will be useful as well. By a bracket polynomial on the alphabet A of letters, we mean a sum of terms, consisting of products of brackets [.], whose entries are selected from the letter set A. The content of a bracket is the set of elements in that bracket. By an extensor polynomial P(Ai, V, A) in the extensors {Ai} we mean a formal expression in Ai and the binary operations V and A. The step of the polynomial P is the step of the extensor obtained upon evaluating all join and meet operations in P. By Propositions 1.5 and 1.9 P is an extensor. 1.3 Bracket methods in Projective Geometry The methods of bracket algebra provide natural machinery for proving theorems of projective geometry. This connection, originally developed by Grassmann, was explored fully by Forder [For60], although the concept of identities involving only join and meet in a Grassmann-Cayley algebra was not made explicit there. Crapo [Cra91], Sturmfels [BS89, BS91], White [Whi75, Whi91] and others have worked extensively using these methods. A main problem in computational synthetic geometry is to find coordinates or non-realizability proofs for abstracting defined configurations. Of particular interest is Sturmfels method of final polynomials which is discussed in Bokowski and Sturmfels [BS87]. As an illustration of bracket method techniques we give a new proof of an n-dimensional generalization of Desargues' theorem [Jon54], and simplify several results of Fordor using the language of Peano space. Theorem 1.13 (Desargues) Let al,...,a, and bl,...,b, be tetrahedra in n - 1 dimensional projective space. If the lines aibi, 1 < i < n pass through a commmon point p, then the colines formed by the intersections of the pairs of hyperplanes al,...,ai,...,an and bl,...,bi,..., b,i all lie on a common hyperplane. PROOF. If p is the center of perspectivity, then the vertices bl,..., b,, lie on the lines pai,pa 2, -...,pa,. We first find an bracket expression for p. Let xi, 1 < i < n - 3 be variable points whose coordinates in a given basis are independent transcendentals (xi, 1, Xi,2, Zi,, X). Consider the expression alblxx2 Xn-3 A a 2 b 2 x 1 X 2... n-_ 3 = +([al, b, xl,.., x-3, a 2 ]b 2 l ", 3 - [al, bl,,..., xn 3, b 2 ]a 2 xl... Xn-3)

22 22 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA Since albl and a 2 b 2 have p as a common point of intersection we may also write alblzxl2.. Xn-3 A a 2 b 2 x 1 X 2.. Xn-3 = PX1X2... Xn-3 so that PX1X2.. Xn-3 = +([al,bi, xi,...,2 n-3, a 2 b 2 x 1 " Xn-3 - [a 1, b, x,..., Xn-3, b 2 ]a 2 xl. " Xn-3) From this one concludes that (p ± ([ai, b,21,..., -3, a2]b2 - [al,bl,,..., Xn-3, b 2 ]a 2 ) V X1X2 " Xn-3 = 0. Since the above linear combination of p, b 2 and a2 is a point in n - projective space and X1,..., Xn-3 is a basis for a flat of rank n - 3, 1 dimensional p = =([al, bl, xi,..., xn-3, a 2 ]b 2 - [al, bi, xl,..., n-3, b 2 ]a 2 ))+klix+- - "+kn-3xn-3 where the kl,..., kn- 3 are elements of the underlying field K. Since the brackets in this expression are non-zero, the sum of the first two terms represents a point on the line a 2 b 2 (it suffices to join b 2 - a2 with a 2 b 2 ) while the linear combination of the xi represents a point off the line a 2 b 2, since the indeterminates may be chosen in general position. Hence we conclude that ki = 0 for all i and that p = :([al, b, xi,..., Xn-3, a2]b2 - [al, bl, xi,...,,xn-3, b2]a2)), the entries 21,...,, -3 acting as arbitrary scale factors. In this way we may write k~bi = p + kiai for 1 < i < n, and hence n k:.-- k.. k'i bl...bi.b, = V (p + kiai). (1.2) xpanding the right side of 1.2 and transposing p to the left in each term one has k'... k'bk' b.. ;- i... b,, = j=1 isi pv( ki...k-...k^i..kn(-1)-1... ai --- an+ j=1 n kil.. L. l i kj... kn(-1)jal.. i...aj...an) + ki... ki.. kn al.. ai.. an j=i+l Computing the meet with the hyperplane al i... an, the last term in the above expression vanishs and the non-vanishing terms consist of a sum of terms of form (minus the scalars) pala2 & j i.. an A al... iai... an (1.3)

23 1.3. BRACKT MTHODS IN PROJCTIV GOMTRY 23 for j = 1,..., n, j # i. The result of computing the meet of 1.3 is a single non-zero term, and therefore the n - 2 dimensional coline k... k... kn bl... bi... bn A al " h i. a,, is given by a scalar multiple [p, al,...,pn] of the sum L - kl. j...ki... k (-1)J-lal... j... i....an+ j=1 n Ski..ki'.i'"kJ'.- kn(-1)j al'".. i'".aj " ' a n - (1.4) j=i+1 By Proposition 1.34 the linear combination of hyperplanes is again a hyperplane. We show that all colines of form (1.4) lie in the hyperplane H = k,.-. i k,,(-1)'-lal... i...an. (1.5) i=1 Since the sum of the steps of the hyperplane and coline span the underlying Peano space, it suffices to show, Proposition 1.9 that the meet of any coline 1.4 and the hyperplane 1.5 is zero. The coline (1.4) has the property that each subset of n - 3 vectors of {al,..., a,} occurs in exactly two terms or not at all. Therefore in computing the meet H A L by splitting L, in the sum of step n - 3 terms resulting, each non-zero extensor of step n - 3 on vector set {al,..., a, } occurs either twice or not at all. We show that each such pair of extensors occurs with opposite sign, that the coefficients are +1 is obvious. The terms of H A L contributing to the identical pair of extensors of step n - 3 may be written as (-1)-lal... aj.- -a, ldi... a, A (-1)-l al... a^j... di."al an, (1.6) (-1)j-lal... dj " " ai- al... an A (-1)/-lal 1i... aj "" ' l "' an (1.7) with I = j and where the vectors of L required by the bracket of H A L are al in the first term and aj in the second, and the extensors are otherwise identical. Since the signs of terms of H and L alternate, the difference in sign in 1.6 and 1.7 is equivalently given by the difference in sign of al "".aj.. "- ai... di. an A al.aj..."di"..al".. an, (1.8) al """al.."ai". a",-j -.-a,, A al..di.. "di".. aj.. "an. (1.9) where the positions of aj and a I are identical in the first extensor and identical in the second extensor of 1.8 and 1.9. An easy calculation using Proposition 1.8 shows that the sign of 1.8 and 1.9 alternate. O The following is a theorem in four-dimensional projective space.

24 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA Theorem 1.14 If A 1, A 2, A 3, A 4 are four lines in general position in a projective space of four dimensions, and C 1 is the line at the intersection of the solids A 2 U A 3, A 3 U A 4 and A 2 U A 4, and similarly for the lines C 2, C 3, C 4, then the solids A 1 U C 1, A 2 UC 2, A 3 UC 3, A 4 UC 4 intersect in a line, called the associate of Ai, A 2, A 3, A 4. PROOF. A projective space of 4 dimensions corresponds to a Peano space of step 5. Let A, B, C, D be lines. Let us denote the juxtaposition of lines as their join. By choosing representations A = ab, B = cd, C = ef, D = gh it follows by direct expansion using the definition of the meet that Similarly we have AB A CD = (AB A C) V D + (AB A D) V C AB A AC A AD = ((AB A C) V A) A AD = (ABACAAD) VA = A[A, B, e][f, A, D] - A[A, B, f][e, A, D] (1.10) where C = e V f. Since lines are of step 2, they commute with each other and we have AB = BA. denoting Aij = Ai If now A 1, A 2, A 3, A 4 V Aj and setting denote four lines in general position then B 1 = A 12 A A 13 A A 14 B 2 = A 23 A A 24 A A 2 1 B 3 = A 34 A A 32 A A 32 B 4 = A 4 1 A A42 A A43 By the definition of the meet, B 1 = [A12 A A 3 A A 4 1 ]AI B 2 = [A 23 A A 4 A Al' 2 ]A 2 C 1 = A 23 A A 3 4 A A 42, C 2 = A 34 A A 4 1 A A 13, C 3 = A 41 A A 1 2 A A 24, C,1 = A 12 A A 23 A A 31. B 3 = [A 3 4 A A 1 A A 3 2 ]A 3, B 4 = [A 4 1 A A 2 A A 34 ]A 1 I. The join of any six vectors in step 5 must vanish so we have the dual relation that the meet of the six covectors of type Aij is zero. By expanding the expression in two different ways using 1.15 we obtain (A 12 A A 23 A A 31 A A 14 A A 24 ) V A 34 (1.11) -[A,(12 ) A A,( 2 3 ) A Aa( 3 1) A A0(1 4 ) A Aa( 3 4 )]Aq( 24 ) = 0 or (1.12)

25 1.3. BRACKT MTHODS IN PROJCTIV GOMTRY where the sum a is taken over all permutations of the covectors. By expanding each of the expressions B 1 V C1, B 2 V C2, B 3 V C3, B 4 V C4 a direct calculation shows that 1.12 may be expressed as B 1 V C1 + B2 V C2 + B3 V C3 + B 4 V C4 = 0 Now resubstituting for Bi above we have [A 1 2 A A 3 A A 4 1 ]A1 V C1 + [A 2 3 A A 4 A A 12 ]A 2 V C2+ [A 3 4 A A 1 A A 2 3 ]A 3 V C3 + [A A 2 + A 3 4 ]A 4 V C4 = 0. (1.13) Since the brackets are scalars, the dependence of Ai V Ci, i = 1,..., 4 means that these subspaces intersect in a line. O We finally give a theorem holding in a projective space of any dimension. Theorem 1.15 Let R 1, R 2,..., IR,. be flats of dimension rl - 1, r 2-1,..., r- - 1 which span a projective space of dimension n - 1. Let p be any point which lies outside the span of any m - 1 of the Rj, 1 < j < m. Then p is incident with a unique flat of dimension m - 1 which intersects each of the Rj in one point. PROOF. Let the Rj for 1 < j < m be represented by extensors Rj = aa... a.) Since there is a point p outside the span of any m - 1 flats, the vectors a k),j 1,..., n form a basis. We may therefore expand p = X k)a k) where the k)are scalar coordinates. The required flat of step m may be described as M=(=( x i a i ) V... V ( xi)a )), (1.14) i=1 i=l1 where the scalars x () of M are the same as the coordinates of p. The flat M contains the point of Rj given by the jth factor of The flat also evidently contains the point p, and if M contained another point q of some Rj then we have q V M = 0. If the point q belongs to R 1 for example, we may write q = ya1) + + yra (1) Then computing the join q V M and expanding by linearity, leaves a relation between independent extensors of step m + 1 which is a contradiction. If any other flat of step m which cuts each Ri in just one point passes through p, let pl,..., Pm be the flat. Then x.(k) (k) flat. Then k)k) i V Pl A npmay be expanded to obtain another dependency amongst flats of step m + 1. The assumption that M is non-zero is valid as otherwise the factors of 1.14 are linearly dependent and hence p is on the join of m - 1 of the Rj, which is contrary to hypothesis. O

26 26 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA xample 1.16 In a Peano space of step 4, let 11,12 be lines spanning projective three space and let p be a point not on either line. Then there exists a unique line containing p and intersecting both lines. 1.4 Duality and Step Identities Several useful identities follow from the properties of the join and meet alone. In this section we give those which will are necessary in subsequent chapters. Let V be a Peano space of step n over the field K. We say that a linearly ordered basis {al, a2,..., an} of V is unimodular whenever [ala2,-...,an] = 1. Let al, a2,... an be a unimodular basis. The extensor = a V a2 V... an will be called the integral. The integral is well-defined and does not depend on the choice of unimodular basis. For details and properties of unimodular bases the reader is referred to [MB85]. The integral behaves like an identity in a GC algebra. Thus for every extensor B with step(b) > 0, we have BV=0, BA=B while for every scalar k, we have kv=k, kv=k. For every n-tuple (bl, b 2,..., bn) of vectors in V, we have the identity bl V b2 V... V bn = [bl, b2,..., bn]. The following propositions are example of what may be called step identities. These are identities in GC(n) which follow strictly fr'om the join and meet and the step of the extensors. Proposition 1.17 Let A, B be extensors such that step(a) + step(b) = n. Then AVB= (AAB)V.

27 1.4. DUALITY AND STP IDNTITIS Proposition 1.18 Let A, B, C be extensors such that step(a)+step(b)+step(c) = n; then AA (BVC) = [A,B,C] = (A V B) A C. The following proposition may be regarded as a generalization of Proposition The polynomial P is said to be properly step k if P is of step k with no proper subpolynomial of P evaluating to step 0. Proposition 1.19 Let {Ai} be an ordered set of extensors in a Grassmann-Cayley algebra of step n such that i step(ai) = n. Let P(Ai, V, A) be a properly step 0 extensor polynomial and let Q(V, A, Ai) be another properly step 0 extensor polynomial on the same ordered set {Ai } of extensors, where the operations of V and A have been interchanged at will, subject to the condition that step(q(v, A, Ai)) = 0. Then Q = P as extensors. PROOF. For any P(Ai, V, A) either P = RV S or P = RA S but as step(p) = 0 and step(p') : 0 for any proper subexpression P' of P, only P = R A S is possible. Further, we may assume step(r) = k and step(s) = n - k since step(p) = 0 iff R and S have complementary step. Assume that k < n - k. Let {Bi} C {Ai} be the subset of the extensors used to form extensor R. If -i step(bi) > k, then since i step(ai) = n the complementary set of extensors {Ci} = {Ai) \ {Bi} must satisfy >i step(c;) < n - k. Since step(a A B) 5 step(a), step(a A B) 5 step(b), and step(a V B) = step(a) + step(b), (assuming A V B non-zero) S({Ci}, V, A) must have step less than n - k = step(s). It follows that i step(bi) = k and Ci step(ci) = n - k. If R contains any operation A then step(r) < k, so R = ±Bi V B 2 V... V Bi, S = ±C 1 V C2 V... V Cj and, P = (B 1 V B 2 V... Bi) A (Ci V C 2 V... Cj) = [B1B2...- Bi, C1C2... Cj] = [B12,..., Bi, C1, C2,...,Cj]- Now let Q be any other extensor satisfying the hypothesis of the theorem. xtensor Q is a polynomial in V and A on the same ordered set of extensors {Ai} = {B1, B 2,... Bi, C 1, C 2,..., Cj }. Without loss of generality, we may write Q = TR' A S' with R' = B 1 V B 2 V... Bi, and i' < i. Then, Q = ±(BI V B 2 V... Bi,) A (Bi'+ 1 V... Bi V C V... V Cj) = [B(1B2... Bi', Bi'+I... BiCG1,, C2,, Cj] =- [B1, Bj2,..., Bi, Bi+, +I... Bi;, C1, C2,..., Cj] = -P.

28 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA Corollary 1.20 Let Ai, A 2,..., Am be extensors in GC(n) such that Cm=l step(ai) = n. Then for any i, j < n the following is an identity (A1 V A 2... V Ai) A (Ai+I V... V A,) = (Al V A 2... V Aj) A (Aj+i V... V Am) xample 1.21 If the sum of the steps of A, B, C, D,, F is n then ((A A B) V C) A (D V ( A F)) = (A A B) A (CV D V ( A F)) is a GC identity. Proposition 1.22 Let P be a non-zero extensor polynomial of step k > 0 in GC(n). Then P = V Zi V R where Zi and R are extensor polynomials with step(zi) = 0 for all i, and R is properly step k. PROOF. The polynomial P involves a set {Ai} of extensors in operations V and A. If subexpression Q g P has step 0, then if P is non-zero and Q : P, the next outermost operation in P involving Q must be a V so that Q V S C P for some S of step k > 0. If step(s) = 0 set Q +- QVS and repeat this step. Hence (QVS)AT C P occurs with step(s) > 0, step(t) > 0, unless the Proposition is true. In this case set [Q]SAT +- (QVS) AT. On the other hand, if Q is of step n, since P is non-zero, the next outermost parenthesization containing Q must be of the form Q A R = [Q]R. In either case factoring the scalar Q to the left, by induction, an extensor R of step k times a product of step 0 brackets remains. 0 Proposition 1.23 Let P(Ai, V, A) be a properly step 0 non-zero extensor polynomial in GC(n) in V, A and extensors A 1, A2,..., A,, such that i step(ai) = kn for some k > 0. Then P contains k occurences of the meet A, and 7n k occurences of the join V. PROOF. The operations V and A are binary operations over the alphabet of extensors {Ai, i = 1,..., n}, and P(Ai, V, A) is a parenthesized word. Hence the total number of occurences of V and A is mn - 1. We now show the number of A occurences must be k. If any join or meet of a subexpression of P vanishes then the extensor P = 0. Recursively evaluate step(p) by applying the rules step(r V S) = step(r) + step(s) and step(ra S) = step(r)+ step(s)-n. Let T be the binary tree corresponding to a parenthesized word, whose vertices represent recursively evaluated subexpressions, and a node Q has children R and S if Q = R V / A S. valuate the step of P recursively labeling each node of T by the step of the corresponding extensor. It is evident that the root of T has label step(p) = 0 = step(ai) - ( # of occurences of A) - n.

29 1.4. DUALITY AND STP IDNTITIS Since T step(ai) = kn it follows that there must be k occurences of meet in P. 0 Proposition 1.24 Let P(Ai, V, A) be a non-zero extensor polynomial in GC(n). Then step(p) = k if and only if i step(ai) - k (mod n). PROOF. By the formula derived in Proposition 1.23 we have step(ai) - k -n = step(p). Proposition 1.25 Any two extensor polynomials P(Ai, V, A) and Q(Bi, V, A) having the same sum of steps of extensors, consequently have the same number of join and the same number of meet operations. PROOF. Assume P has step k. By Proposition 1.23 step(p) = i step(ai) - (#meets).n which implies that the number of meets in P is (i step(ai)-step(p))/n. By Proposition 1.24 step(ai) = tn + k for some t > 0. Then the number of meets in P is in fact (t -n + k - k)/n = t. Similarly, the number of meets in Q is t and the number of joins in both P and Q must be equal as well. 0 Corollary 1.26 Let {Ai }, {Bj} be sets of extensors, each formed by joining vectors from a fixed alphabet A. Then any two non-zero polynomials P(Ai, V, A) and Q(Bj, V, A) having the same number of occurences of each vector from A have the same step and same number of meets and joins. PROOF. Provided A is non-zero, the step of an extensor Ai is the number of vectors joined in Ai. Since each vector occurs homogeneously i step(ai) = Cj step(bj) as both are non-zero. 0 The meet operation defines a second exterior algebra structure on the vector space G(V). The duality operator connecting the two is the Hodge Star Operator, [WVDH46, MB85] Given a linearly ordered basis {al,a2,.., a,n}, the associated cobasis of covectors of V is the set of covectors {a1,..., a,,} where ai= - [ai, al,.....,an]-lal V... VdiV...an. Let {al, a 2,..., a,, n} be a linearly ordered basis of V. The Hodge star operator relative to the basis {al, a2,..., a,n} is defined to be the (unique) linear operator *: G(V) -+ G(V)

30 CHAPTR 1. TH GRASSMANN-CAYLY ALGBRA such that, for every subset S of {1, 2,..., n}, 1 = *ail V... V ai = (-1)il+'"+ik-k(k+l)/ 2 [al,...,an]-lap, V ap-k where if S = {il,...,ik} with il < i2 < "' < ik, then S c = {Pl,...,Pn-k} and pl < " < Pn-k. This definition is equivalent to setting *1 = and *ail V. Vaik = ai, A.. A aik where {al,...,an} is the associated basis of covectors of {al,..., an}. We shall require the following two propositions whose proofs can be found in [MB85]. Proposition 1.27 A Hodge star operator is an algebra isomorphism between the exterior algebra of the join (G(V), V) and the exterior algebra of the meet (G(V), A). Moreover, it maps the set of extensors of Gk(V) onto the set of extensors of step Gn-k(V). When the basis is unimodular, the Hodge star operator implements the duality between join and meet. In this case we can state the duality principle: For any identity p(ai,..., Ap) = 0 between extensors Ai of steps ki, joins and meets holding in a Grassman-Cayley algebra, the identity 3(A 1,..., Ap) = 0 obtained by exchanging joins and meets and replacing Ai by an extensor Ai of step n - k holds. Proposition 1.28 Let the linearly ordered basis {al,..., an} be unimodular. Then the Hodge star operator * relative to {a1,...,a,} satisfies the following: 1. * maps extensors of step k to extensors of step n - k. 2. *(x V y) = (*x) A (*y) and *(x A y) = (*x) V (*y), for every x,y G(V). 3. *1 = and * = *(*x) = (l-)k(n-k)x, for every x Gk(V). 1.5 Alternative Laws Let X = (xi,j) be a generic 7n x d matrix over the complex numbers, and let C[xi,j] denote the corresponding polynomial ring in nd variables. The matrix X may be thought of as a configuration of n71 vectors in the vector space Cd. These vectors also represent a configuration of n points in d - 1 dimensional projective space pd-1. Consider the set A(n, d) = {[Ai,...,A d]l < A 1 < A 2 <. < Ad < n} of ordered d- tuples in [n], whose elements are the brackets. Define C[A(n, d)] to be the polynomial

31 1.5. ALTRNATIV LAWS ring generated by the (')-element set A(n, d). The algebra homomorphism 'n,d " C[A(n, d)] -+ C[xij] defined by, for bracket [A] = [Ai,..,,Ad], ( X, 1,1 XA1,2 "' X,\l,d [A] = det X2,,1 XA 2,2... X- 2,d XAd,1 X-d,2 " X)d,d maps each bracket [A] to the d x d subdeterminant of X whose rows are indexed by A. The image of the ring map On,d coincides with the subring Bn,d of C[xi,j] generated by the d x d minors of X, called the bracket ring. The ring map On,d is in general not injective and if In,d C C[A(n, d)] denotes the kernel of On,d, this ideal is called the ring of syzygies. It is not difficult to see that Bn,d - C[A(n, d)]/in,d. It is shown in Sturmfels [BS89], that an explicit Gr6bner basis exists for the ideal In,d and that the standard tableaux form a C-vector space basis for the bracket ring Bn,d. The group SL(Cd) of d x d-matriccs with determinant 1 act on the right on the ring C[xij] of polynomial functions on a general n x d-matrix X = (xi,j). The two fundamental theorems of invariant theory give an explicit description of the invariant ring C[xi,j] SL (Cd), although only the first is relevant here. very d x d minor of X is invariant under SL(Cd), and therefore Bn,d C C[xi,j]SL(Cd). The equality of these rings is given by the following. Theorem 1.29 (First Fundamental Theorem of Invariant Theory) The invariant ring C[xi,j] S L(Cd) is generated by the dxd-minors of the matrix X = (xi, xj). Alternative laws were first introduced by Rota in [MB85] and are useful for calculation in C[xi,j]sL(Cd). An alternative law is an identity which can be used in simplifying expressions containing joins and meets of extensors of different step. The two laws we use, given in Propositions 1.30 and 1.36 are most closely related to the Laplace expansion for determinants. We use the following notational convention throughout: juxtaposition of vectors al a2... as shall denote their join a Va 2 V-.. *Va, while juxtaposition of covectors X1X 2... Xk denotes their meet X 1 A X 2 A... Xk. Proposition 1.30 Let al, a2,..., ak be vectors and X 1, X 2,..., X, k > s. Set A = ala2... ak, then: A A(X AX 2 9 A... A X,) = Z (A 1,...,As+ 1 )S(A;1,...,1,k-s) [AI, Xi][A 2, X 2 ]... [As, Xs]A 8 +1 sgn(a1,a 2,..., A,+) X covectors, with