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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

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