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On the vanishing of a pure product in a (G,6) space Sing, Kuldip
Abstract
We begin by constructing a vector space over a field F , which we call a (G,σ) space of the set W = V₁xV₂... xVn , a cartesian product, where Vi is a finite-dimensional vector space over an arbitrary field F , G is a subgroup of the full symmetric group Sn and σ is a linear character of G . This space generalizes the spaces called the symmetry class of tensors defined by Marcus and Newman [1]. We can obtain the classical spaces, namely the Tensor space, the Grassman space and the symmetric space, by particularizing the group G and the linear character σ in our (G,σ) space. If (v₁,v₂,..., vn ) ∈ W , we shall denote the "decomposable" element in our space by v₁Δv₂…Δvn and call it the (G,σ) product or the Pure product if there is no confusion regarding G and σ, of the vectors v₁,v₂,..., vn . This corresponds to the tensor product, the skew symmetric product and the symmetric product in the classical spaces. The purpose of this thesis is to determine a necessary and sufficient condition for the vanishing of the (G,σ) product of the vectors v₁,v₂,..., vn in the general case. The results for the classical spaces are well-known and are deduced from our main theorem. We use the "universal mapping property" of the (G,σ) space to prove the necessity of our condition. These conditions are stated in terms of determinant-like functions of the matrices associated with the set of vectors v₁,v₂,...,vn.
Item Metadata
Title |
On the vanishing of a pure product in a (G,6) space
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1967
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Description |
We begin by constructing a vector space over a field F , which we call a (G,σ) space of the set W = V₁xV₂... xVn , a cartesian product, where Vi is a finite-dimensional vector space over an arbitrary field F , G is a subgroup of the full symmetric group Sn and σ is a linear character of G . This space generalizes the spaces called the symmetry class of tensors defined by Marcus and Newman [1]. We can obtain the classical spaces, namely the Tensor space, the Grassman space and the symmetric space, by particularizing the group G and the linear character σ in our (G,σ) space.
If (v₁,v₂,..., vn ) ∈ W , we shall denote the "decomposable" element in our space by v₁Δv₂…Δvn and call it the (G,σ) product or the Pure product if there is no confusion regarding G and σ, of the vectors v₁,v₂,..., vn . This corresponds to the tensor product, the skew symmetric product and the symmetric product in the classical spaces. The purpose of this thesis is to determine a necessary and sufficient condition for the vanishing of the (G,σ) product of the vectors v₁,v₂,..., vn in the general case. The results for the classical spaces are well-known and are deduced from our main theorem.
We use the "universal mapping property" of the (G,σ) space to prove the necessity of our condition. These conditions are stated in terms of determinant-like functions of the matrices associated with the set of vectors v₁,v₂,...,vn.
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Genre | |
Type | |
Language |
eng
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Date Available |
2011-09-13
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080561
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Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.