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On generating functions for the number of invariants of orthogonal tensors

Published online by Cambridge University Press:  26 February 2010

A. J. M. Spencer
A. J. M. Spencer
Affiliation:
Department of Theoretical Mechanics, University of Nottingham.
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Summary

Generating functions for the number of linearly independent invariants of a set of tensors under a given group of transformations are given by the theory of group representations. For the full and proper orthogonal groups these generating functions are in the form of definite integrals. The classical theory of algebraic invariants gives generating functions for the number of invariants of tensors under two-dimensional unimodular transformations, these generating functions being algebraic expressions. Because of a correspondence between the two-dimensional unimodular group and the three-dimensional proper orthogonal group, the corresponding generating functions are equivalent. The main result of this paper is an explicit demonstration of this equivalence. In addition, algebraic generating functions for the three-dimensional full orthogonal group are obtained and the use of the algebraic generating functions illustrated by applying them to a third order symmetric tensor.

Type
Research Article
Information
Mathematika , Volume 17 , Issue 2 , December 1970 , pp. 275 - 286
Copyright
Copyright © University College London 1970

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