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XII.—Clebsch-Aronhold Symbols and the Theory of Symmetric Functions*

Published online by Cambridge University Press:  14 February 2012

H. W. Turnbull  and
A. H. Wallace
H. W. Turnbull
Affiliation:
The University, St Andrews.
A. H. Wallace
Affiliation:
The University, St Andrews.
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Synopsis

A square matrix A = (aij) is expressed symbolically in terms of Clebsch-Aronhold equivalent symbols aij = aiaj = βibj = …, and the symbolic expressions for symmetric functions of the latent roots of A are considered, the relation between these functions and projective invariants of the bilinear form uAx being noted. The Newton and Brioschi relations between the symmetric functions are obtained by reduction of symbolic determinants and permanents respectively, and the Wronskian relations are shown to be equivalent to certain identities between determinants and permanents due to Muir. Also the fundamental theorem of symmetric functions is obtained symbolically as a consequence of the first fundamental theorem of invariants. The paper concludes with a note on the symbolization of the h-bialternants, that is of the traces of irreducible invariant matrices of A.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 63 , Issue 2 , 1951 , pp. 155 - 173
Copyright
Copyright © Royal Society of Edinburgh 1951

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References

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