Table Of ContentGeometric 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-
