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Existence of algebras of symmetry-classes of tensors with respect to translation-in-variant pairs Hillel, Joel S.

Abstract

The notion of the 'classical' multilinear maps such as the symmetric and skew-symmetric maps, has the following generalization: given a vector-space 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] (n-fold 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 vector-space 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 'translation-invariant' if for every [formula omitted] is also in H . We begin our presentation by characterising the 'translation-invariant' 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 translation-invariant groups. Finally, we show that a translation-invariant 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.

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