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Existence of algebras of symmetryclasses of tensors with respect to translationinvariant pairs Hillel, Joel S.
Abstract
The notion of the 'classical' multilinear maps such as the symmetric and skewsymmetric maps, has the following generalization: given a vectorspace V and a pair (H[subscript n],X[subscript n]) where is a subgroup of the symmetric group S[subscript n] and X[subscript n] is a character of H[subscript n], we consider multilinear maps from V[superscript n] (nfold cartesian product of V ) into any other vector space, which are ‘symmetric with respect to (H[subscript n],X[subscript n])’, i.e., which have a certain symmetry in their values on permuted tuples of vectors, where the permutations are in H[subscript n]. Given a pair (H[subscript n],X[subscript n]) and a vectorspace V , we can construct a space V[superscript (n)] over V through which the maps 'symmetric with respect to (H[subscript n],X[subscript n])’ linearize. The space V[superscript (n)] is usually defined abstractly by means of a certain universal mapping property and gives the tensor, symmetric and Grassmarm spaces for the 'classical' maps. Given a sequence of pairs [formula omitted]and the corresponding spaces V[superscript (n)], we let [formula omitted] (where V[superscript b]) is the ground field). In the classical cases, A has a natural multiplicative operation which makes A an algebra, i.e., the Tensor, Symmetric and Grassmann algebras. This presentation has been motivated by the attempt to generalize the construction of an algebra A to a wider family of 'admissible' sequences of pairs [formula omitted]. This consideration has led us to investigate permutation groups on the numbers 1,2,3,… which are closed under a certain 'shift' of the permutations, i.e., if [formula omitted] is a permutation, we define [formula omitted] and we call a permutation group H 'translationinvariant' if for every [formula omitted] is also in H . We begin our presentation by characterising the 'translationinvariant' groups. We show that the study of these (infinite) groups can be reduced to the study of certain finite groups. Then, we proceed to discuss the lattice of the translationinvariant groups. Finally, we show that a translationinvariant group H , together, with an appropriate character X of H , represents an equivalence class of 'admissible' sequences of pairs [formula omitted]. For a particular choice of representatives of the equivalence class, we can construct an algebra of 'symmetry classes of tensors' which generalizes the Tensor, Symmetric and Grassmann algebras.
Item Metadata
Title  Existence of algebras of symmetryclasses of tensors with respect to translationinvariant pairs 
Creator  Hillel, Joel S. 
Publisher  University of British Columbia 
Date Issued  1968 
Description 
The notion of the 'classical' multilinear maps such as the symmetric and skewsymmetric maps, has the following generalization: given a vectorspace V and a pair (H[subscript n],X[subscript n]) where is a subgroup of the symmetric
group S[subscript n] and X[subscript n] is a character of H[subscript n], we consider
multilinear maps from V[superscript n] (nfold cartesian product of V ) into any other vector space, which are ‘symmetric with respect to (H[subscript n],X[subscript n])’, i.e., which have a certain symmetry in their values on permuted tuples of vectors, where the permutations are in H[subscript n].
Given a pair (H[subscript n],X[subscript n]) and a vectorspace V ,
we can construct a space V[superscript (n)] over V through which the maps 'symmetric with respect to (H[subscript n],X[subscript n])’ linearize. The space V[superscript (n)] is usually defined abstractly by means of a certain universal mapping property and gives the tensor, symmetric and Grassmarm spaces for the 'classical' maps.
Given a sequence of pairs [formula omitted]and the
corresponding spaces V[superscript (n)], we let [formula omitted] (where
V[superscript b]) is the ground field). In the classical cases, A has a natural multiplicative operation which makes A an algebra, i.e., the Tensor, Symmetric and Grassmann algebras.
This presentation has been motivated by the attempt to generalize the construction of an algebra A to a wider family of 'admissible' sequences of pairs [formula omitted].
This consideration has led us to investigate permutation groups on the numbers 1,2,3,… which are closed under a certain 'shift' of the permutations, i.e., if [formula omitted] is a permutation, we define
[formula omitted] and we call a permutation group H 'translationinvariant' if for every [formula omitted] is also in H .
We begin our presentation by characterising the 'translationinvariant' groups. We show that the study of these (infinite) groups can be reduced to the study of certain finite groups. Then, we proceed to discuss the lattice of the translationinvariant groups.
Finally, we show that a translationinvariant group H , together, with an appropriate character X of H , represents an equivalence class of 'admissible' sequences of pairs [formula omitted]. For a particular choice of representatives of the equivalence class, we can construct an algebra of 'symmetry classes of tensors' which generalizes the Tensor, Symmetric and Grassmann algebras.

Subject  Algebras, Linear 
Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20110615 
Provider  Vancouver : University of British Columbia Library 
Rights  For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. 
DOI  10.14288/1.0080518 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Campus  UBCV 
Scholarly Level  Graduate 
Aggregated Source Repository  DSpace 
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For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.