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Trace polynomials of words in special linear groups

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  09 April 2009

J. B. Southcott
J. B. Southcott
Affiliation:
Department of Computing Science University of AdelaideG.P.O. Box 498 Adelaide 5001, Australia
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Abstract

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If w is a group word in n variables, x1,…,xn, then R. Horowitz has proved that under an arbitrary mapping of these variables into a two-dimensional special linear group, the trace of the image of w can be expressed as a polynomial with integer coefficients in traces of the images of 2n−1 products of the form xσ1xσ2…xσm 1 ≤ σ1 < σ2 <… <σm ≤ n. A refinement of this result is proved which shows that such trace polynomials fall into 2n classes corresponding to a division of n-variable words into 2n classes. There is also a discussion of conditions which two words must satisfy if their images have the same trace for any mapping of their variables into a two-dimensional special linear group over a ring of characteristic zero.

MSC classification

Secondary: 20G99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 4 , December 1979 , pp. 401 - 412
Copyright
Copyright © Australian Mathematical Society 1979

References

Cossey, J., Macdonald, S. O. and Street, A. P. (1970), ‘On the laws of certain finite groups’, J. Austral Math. Soc. 11, 441–489.CrossRefGoogle Scholar
Horowitz, R. D. (1972), ‘Characters of free groups represented in two-dimensional special linear group’, Comm. Pure Appl. Math. 25, 635–650.CrossRefGoogle Scholar
Southcott, J. B. (1974a), ‘A basis for the laws of a class of simple groups’, J. Austral. Math. Soc 17, 500–505.CrossRefGoogle Scholar
Southcott, J. B. (1974b), ‘Two-variable laws for a class of finite simple groups’, Bull. Austral. Math. Soc. 10, 85–89.CrossRefGoogle Scholar
Whittemore, A. (1973), ‘On special linear characters of free groups of rank n ≧ 4’, Proc. Amer. Math. Soc. 40, 383–388.Google Scholar