On the Foundations of Combinatorial Theory: IX Combinatorial Methods in Invariant Theory

Doubilet, Peter; Rota, Gian‐Carlo; Stein, Joël

doi:10.1002/sapm1974533185

Cited by 327 publications

(172 citation statements)

References 12 publications

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“…The Grassmann-Cayley algebra G n is defined by adding a second product, the meet product, to the Grassmann algebra (or exterior algebra) of a vector space of dimension n, V , over a field K [6,1] where the join product (or the exterior product) is defined.…”

Section: The Underlying Vector Spacementioning

confidence: 99%

“…Usual presentation of the Grassmann-Cayley algebra [6,1,18] defines a bracket over the vector space V . Given n vectors a 1 , .…”

Section: The Meet Productmentioning

confidence: 99%

“…Bracket algebra and its relation with Grassmann-Cayley is a well-known topic [6,1]. Here, we are just going to explain how it has been introduced in our setting.…”

Section: Automating Proofsmentioning

confidence: 99%

“…This paper presents a formalization of Grassmann-Cayley algebra [6] that has been done in the Coq [2] proof assistant. The formalization is based on a data structure that represents elements of the algebra as complete binary trees.…”

mentioning

confidence: 99%

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A Formalization of Grassmann-Cayley Algebra in COQ and Its Application to Theorem Proving in Projective Geometry

Fuchs

Théry

2011

Automated Deduction in Geometry

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Abstract. This paper presents a formalization of Grassmann-Cayley algebra [6] that has been done in the Coq [2] proof assistant. The formalization is based on a data structure that represents elements of the algebra as complete binary trees. This allows to define the algebra products recursively. Using this formalization, published proofs of Pappus' and Desargues' theorem [7,1] are interactively derived. A method that automatically proves projective geometric theorems [11] is also translated successfully into the proposed formalization.

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Section: The Underlying Vector Spacementioning

confidence: 99%

“…Usual presentation of the Grassmann-Cayley algebra [6,1,18] defines a bracket over the vector space V . Given n vectors a 1 , .…”

Section: The Meet Productmentioning

confidence: 99%

“…Bracket algebra and its relation with Grassmann-Cayley is a well-known topic [6,1]. Here, we are just going to explain how it has been introduced in our setting.…”

Section: Automating Proofsmentioning

confidence: 99%

mentioning

confidence: 99%

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A Formalization of Grassmann-Cayley Algebra in COQ and Its Application to Theorem Proving in Projective Geometry

Fuchs

Théry

2011

Automated Deduction in Geometry

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“…Whiteley (1987Whiteley ( , 1989Whiteley ( , 1991Whiteley ( , 1992, Crapo and Ryan (1986), Crapo (1991), Crapo and Whiteley (1993) studied the realization problem with structure geometry (Crapo and Whiteley, 1982;White and Whiteley, 1983), invariant theory (Doubilet et al, 1974;White, 1975) and synthetic geometry (Bokowski and Sturmfels, 1989;Sturmfels, 1993;RichterGebert, 1996). Various necessary and sufficient conditions are established in terms of either Example 1.…”

Section: Introductionmentioning

confidence: 99%

A symbolic approach to polyhedral scene analysis by parametric calotte propagation

Zhao

Chen

2008

Robotica

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Abstract.Polyhedral scene analysis studies whether a 2D line drawing of a 3D polyhedron is realizable in the space, and if so, parameterizing the space of all possible realizations. For generic 2D data, symbolic computation with Grassmann-Cayley algebra is needed in the analysis. In this paper, we propose a method called parametric calotte propagation to solve the realization and parameterization problems for general polyhedral scenes at the same time. In algebraic manipulation, parametric propagation is more efficient than elimination. In application, it can lead to linear construction sequences for non-spherical polyhedra whose resolvable sequences do not exist.

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Quantum field theory meets Hopf algebra

Brouder

2009

Mathematische Nachrichten

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This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S-matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and lead to a correspondence between Feynman diagrams and semi-standard Young tableaux. Reciprocally, these concepts are used as models to derive Hopf algebraic constructions such as a connected coregular action or a group structure on the linear maps from S(V) to V. In most cases, noncommutative analogues are derived.Comment: 27 pages, 4 figures. Slightly edited version of the published pape

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On the Foundations of Combinatorial Theory: IX Combinatorial Methods in Invariant Theory

Cited by 327 publications

References 12 publications

A Formalization of Grassmann-Cayley Algebra in COQ and Its Application to Theorem Proving in Projective Geometry

A Formalization of Grassmann-Cayley Algebra in COQ and Its Application to Theorem Proving in Projective Geometry

A symbolic approach to polyhedral scene analysis by parametric calotte propagation

Quantum field theory meets Hopf algebra

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