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Poincaré series for the occurrence of certain modular representations of GL(n,p) in the symmetric algebra

Published online by Cambridge University Press:  14 November 2011

David P. Carlisle  and
Grant Walker
David P. Carlisle
Affiliation:
Computer Science Department and University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
Grant Walker
Affiliation:
Mathematics Department, University of Manchester, Oxford Road, Manchester M13 9PL, U.K.
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Synopsis

The number of occurrences of the Steinberg representation St of GL(n, p) as a composition factor in the symmetric algebra Fp[x1, … xn] has been determined by several authors. We extend this result to the representations of GL(n, p) which are the closest neighbours of St in the SL(n, p) weight diagram. The method is to play off duality for GL(n, p)-modules against connectivity for M(n, p)-modules. The result is equivalent to determining the cohomology groups of the corresponding indecomposable stable summands of the localisation of an n-fold product of complex projective spaces at the prime p.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 27 - 41
Copyright
Copyright © Royal Society of Edinburgh 1989

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