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Invariant, or coordinate-free methods provide a natural framework for many geometric questions. Invariant Methods in Discrete and Computational Geometry provides a basic introduction to several aspects of invariant theory, including the supersymmetric algebra, the Grassmann-Cayler algebra, and Chow forms. It also presents a number of current research papers on invariant theory and its applications to problems in geometry, such as automated theorem proving and computer vision.
Audience: Researchers studying mathematics, computers and robotics.



The Power of Positive Thinking

Supersymmetric algebra is introduced as a natural language to study tensors. A new symbolic method for the representation of skewsymmetric tensors is given. As an application, various properties of skewsymmetric tensors are derived.
Wendy Chan, Gian-Carlo Rota, Joel A. Stein

Introduction to Chow Forms

This article gives an elementary introduction to the theory of Chow forms. It is based on a sequence of four lectures given at the conference on “Invariant Methods in Discrete and Computational Geometry”, Curacao, Netherlands Antilles, June 1994. We thank the organizer, Neil White, for giving us the opportunity to participate and present this material. This work was supported in part by the National Science Foundation and the David and Lucile Packard Foundation.
John Dalbec, Bernd Sturmfels

Capelli’s Method of Variabili Ausiliarie, Superalgebras and Geometric Calculus

Capelli’s technique of variabili ausiliarie [8] — or “virtual variables”, as we prefer to call them — has been proved to be a useful tool in order to unify concepts and simplify computations in a variety of problems of invariant theory and its applications. In our opinion, Capelli’s technique finds its natural setting and acquires a much greater effectiveness in the context of supersymmetric algebras and Lie superalgebra actions; specifically, the technique of virtual variables acquires a special suppleness when the virtual variables are allowed to have a different signature than the signature of the variables one starts with.
Andrea Brini, Antonio G. B. Teolis

Letterplace Algebra and Symmetric Functions

A new class of determinantal identities in the letterplace algebra is derived. These new identities are the symbolic analogues of the various relations in symmetric functions. Meanwhile, new bases in the letterplace algebra are discovered, and they are further extended to the supersymnetric algebra.
Wendy Chan

A Tutorial on Grassmann-Cayley Algebra

The Grassmann-Cayley algebra is first and foremost a means of translating synthetic projective geometric statements into invariant algebraic statements in the bracket ring, which is the ring of projective invariants.
Neil L. White

Computational Symbolic Geometry

The aim of this work is to present a framework for symbolic computations in Geometry. More precisely, we are interested in problems coming from robotics and vision, therefore we focus on points, linear spaces, spheres, displacements and matrices. The approach chosen consists in dealing with intrinsic properties, in order that we (most of the time) manipulate invariant quantities (independent of the referential frame) and we (as much as possible) avoid using coordinates. The reason for this choice is that computations are done in a more simple, synthetic and natural way than if we used coordinates. For each class of object mentioned before, we give one or more possible formal representations and we describe the relations that exist between the quantities introduced to represent these objects. Here is the general scheme that we follow: if we want to work in a space A, we use a free algebra of polynomials (or a free module) F where the variables represent generators of A and we consider the relations K that exist between these objects:
$$ 0 \to K \to F\mathop{ \to }\limits^{\phi } A \to 0 $$
The map ∅ associates to a variable of F the object represented by this variable in A and K is the kernel of this map. We call such a map a representation of A. Notice that this representation is not unique. In order to be able to compute modulo K, we give a normal form algorithm which reduces every element of K to 0.
B. Mourrain, N. Stolfi

Invariant Theory and the Projective Plane

Motivated by an example of Doubilet, Rota, and Stein [5], we introduce a class of expressions in a Grassmann-Cayley algebra which represent projective invariants closely related to the Theorem of Desargues and its generalizations to higher-dimensional projective space. A classification of the rank 3 indecomposable planar identities is given.
Michael Hawrylycz

Automatic Proving of Geometric Theorems

In this talk, we describe a system for automatic generation of binomial proofs for geometry theorems. Beginning with proofs in projective geometry, we show how Felix Klein’s adjunction principle permits extension of these binomial proof methods to Euclidean, and eventually to other, geometries. The present report for the conference proceedings is simply a ‘verbatim’ record of the talk, and will serve as an outline of material to appear in a subsequent publication by the same authors1. References to earlier work will be found in §25.
Henry Crapo, Jürgen Richter-Gebert

The Resolving Bracket

In this talk, we define a projective complex and determine those complexes which are generically trivial, that is, which are homologically trivial in general position. A projective complex is basic when the space of first-order syzygies among the points of the configuration is the direct sum of the spaces of firstorder syzygies on the facets of the complex. We define the resolving bracket of a generic basis, a bracket polynomial which vanishes exactly when the generic basis fails, and permits the complex to be lifted to higher dimension. We determine the factorization of resolving brackets, and provide numerous examples. The present report for the conference proceedings is a summary of material to appear (with proofs) in a subsequent publication by the same authors. 1
Henry Crapo, Gian-Carlo Rota

Computation of the Invariants of a Point Set in P 3 from Its Projections in P 2

In this paper, some applications of classical projective invariants in computer vision are presented. The computation of (absolute projective) invariants of a set of points in projective three space from its projections in projective two space is concerned. After a brief review of some known results in computer vision for the computation from two projections, a new algorithm which allows the computation from three projections with fewer point correspondences is presented in this paper. Some important consequences of these results for computer vision are also presented.
L. Quan

Geometric Algebra and Möbius Sphere Geometry as a Basis for Euclidean Invariant Theory

Physicists have traditionally described their systems by means of explicit parametrizations of all their possible individual configurations. This makes a local description of the motion of the system relatively simple, but provides little insight into the global properties of its solution space. Geometers, on the other hand, tend to describe their systems implicitly in terms of their invariant geometric properties. This approach has the substantial advantage of enabling them to deal with entire sets of configurations simultaneously, and renders every theorem global in scope. In physical problems, an analogous approach would use invariants of the Lie group underlying the dynamical system in question, e.g. angular momentum in the case of rotationally invariant dynamics.
Timothy F. Havel

Invariants on G/U x G/U x G/U, G = SL(4,C)

With Applications to Tensor Products
Let G = SL(n,C). Let U be the standard maximal unipotent subgroup of G, i.e., the group of all upper triangular matrices with 1’s on the diagonal. The invariants of G acting by left translation on GIU x G/U x G/U are explicitly c alculated for n = 2,3 ,4. In each case, all relations among the invariantsriants are given along with a Stanley decomposition for the algebra of invariants. This theory is applied to the calculation of the decomposition of the tensor product of two finite-dimensional irreducible representations of G.
Frank D. Grosshans

On a Certain Complex Related to the Notion of Hyperdeterminant

The notion of determinant slowly emerged during a period of one hundred and twenty years, from the pioneering work of Leibniz to the celebrated 1812 memoir of Cauchy, which contained all the fundamental properties. The subsequent forty years were devoted to a further systematization of the theory, so that its importance came to be recognized by the mathematical community. A sign of that recognition was the appearance of the first complete treatise on determinants, written by Brioschi (Pavia 1854, French and German translations 1856).1
G. Boffi

On Cayley’s Projective Configurations an Algorithmic Study

In this paper Cayley’s configurations in projective r-limensional space are re-defined recursively using a combinatorial characterization of them and a recovery algorithm for a generating point set. Then, the loubletriple notation system characterization is justified also through an ad hoc algorithm.
Rodolfo San Agustin

On the Construction of Equifacetted 3-Spheres

The investigation of combinatorial 3-spheres is a vivid area of reseach. We discuss some problems within this field, and we provide classes of interesting samples (equifacetted 3-spheres) which are easier to handle und therefore they might be useful under various other aspects as well. We show that methods related to invariant theory such as the theory of oriented matroids can help in the investigation of 3-spheres.
Jürgen Bokowski

Depths and Betti Numbers of Homology Manifolds

In this paper, we characterize the set of all Betti sequences of compact triangulable homology manifolds. In addition, we characterize the Betti sequences of all Buchsbaum-Eulerian, Eulerian, and semi-Eulerian complexes, and the depths of their Stanley-Reisner rings.
Clara S. Chan, Douglas Jungreis, Richard Stong


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