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Some sign properties of symmetric functions

Published online by Cambridge University Press:  24 October 2008

D. B. Hunter  and
I. G. Macdonald
D. B. Hunter
Affiliation:
Department of Mathematics, University of Bradford
I. G. Macdonald
Affiliation:
School of Mathematical Sciences, Queen Mary College, London
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Abstract

This paper is concerned with the sign properties of the S-functions sλ for real arguments. We show first that sλ is indefinite if any part of the partition λ is odd. Thus it is only if all parts of λ are even that sλ can possibly be positive definite or semi-definite. In this case we show that sλ(x) is positive provided that at least l(λ) of the components of x are non-zero, where l(λ) is the number of parts of the partition λ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 2 , March 1989 , pp. 193 - 196
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Hunter, D. B.. The positive-definiteness of the complete symmetric functions of even order. Math. Proc. Camb. Philos. Soc. 82 (1977), 255–258.CrossRefGoogle Scholar
[2]Jacobson, N., Basic Algebra, vol. 1 (W. H. Freeman & Co., 1974).Google Scholar
[3]Macdonald, I. G., Symmetric Functions and Hall Polynomials (Clarendon Press, 1979).Google Scholar