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 MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1995 ( 1995 Massachusetts Institute of Technology All rights reserved 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~§,(cid:127) MASSA(C)FH *UrSrErT-"T!S1 yIjNnSnT',I/TUTE MAY 23 1995 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 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. Emanuel 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. Contents 1 The Grassmann-Cayley Algebra 9 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The Exterior Algebra of a Peano Space ................ 12 1.3 Bracket methods in Projective Geometry . ............... 21 1.4 Duality and Step Identities . .................. .... 26 1.5 Alternative Laws ............................. 30 1.6 Geometric Identities ............... .......... .. 34 2 Arguesian Polynomials 43 2.1 The Alternative Expansion . .................. .... 44 2.2 The Theory of Arguesian Polynomials . ................ 48 2.3 Classification of Planar Identities . .................. . 57 2.4 Arguesian Lattice Identities . .................. .... 66 2.5 A Decomposition Theorem ....................... 74 3 Arguesian Identities 83 3.1 Arguesian Identities ................. ........ 83 3.2 Projective Geometry ........................... 93 3.3 The Transposition Lemma ............ ............ 98 4 Enlargement of Identities 105 CONTENTS 4.1 The Enlargement Theorem .................. .... . 105 4.2 Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 122 4.3 Geom etry ............ .. .. ............ ..... 126 5 The Linear Construction of Plane Curves 129 5.1 The Planar Conic ........... ... ........ ..... 130 5.2 The Planar Cubic ........... ............ ... .. 134 5.3 The Spacial Quadric and Planar Quartic . . . . . . . . . . . ..... 144 List of Figures 1.1 The Theorem of Pappus . . . . . . . . . . . . . . . 36 2.1 The Theorem of Desargues . . . . . . . . . . . . . 2.2 The graphs Bp, for i = 1,...,5 . . . . . . . . . . . 2.3 The Theorem of Bricard . . . . . . . . . . . . . . . 2.4 The Theorem of the third identity. . . . . . . . . . 2.5 The First Higher Arguesian Identity . . . . . . . . 4.1 Bp for P = (aV BC) A (bV AC) A (cV BCD) A (d V CD) and B2p. 109 4.2 The matrix representation of two polynomials P Q. ......... 125 Q 4.3 A non-zero term of an identity P - .... 126 5.1 Linear Construction of the Conic . . . . . . . 133 . . . . . . . . . . . . . 5.2 Linear Construction of the Cubic. .. . . . ... ............. 138 8 LIST OF FIGURES 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, 1911 1.1 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 CHAPTER 1. THE GRASSMANN-CAYLEY ALGEBRA 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 Extensions, Forder [For60], using pre- cursors to this method, develops thoroughly the geometry of the projective plane, with some attention to projective three space. The work of Forder contains implic- itly, 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 proposi- tions which interpret the join and meet geometrically. One problem in constructing Grassmann-Cayley algebra identities is that the usual expansion of the meet com- binatorially 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 configura- tions. This chapter concludes by defining precisely the notion of geometric identity in the Grassmann-Cayley algebra, and giving several examples of geometric iden- tities, 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 the- orem of Desargues in the projective plane and its many generalizations to higher-