Premium
On generating functions for the number of invariants of orthogonal tensors
Author(s) -
Spencer A. J. M.
Publication year - 1970
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300002941
Subject(s) - mathematics , unimodular matrix , orthogonal group , orthogonal functions , algebraic number , pure mathematics , generating set of a group , generating function , group (periodic table) , orthogonal basis , invariants of tensors , equivalence (formal languages) , algebra over a field , tensor (intrinsic definition) , combinatorics , mathematical analysis , geometry , chemistry , physics , organic chemistry , quantum mechanics
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.