Lie Powers of Modules for Groups of Prime Order |
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Authors: | Bryant R M; Kovacs L G; Stohr Ralph |
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Institution: | Department of Mathematics UMIST, PO Box 88, Manchester M60 1QD bryant{at}umist.ac.uk, r.stohr{at}umist.ac.uk
Centre for Mathematics and its Applications, Australian National University Canberra ACT 0200, Australia kovacs{at}maths.anu.edu.au |
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Abstract: | Let L(V) be the free Lie algebra on a finite-dimensional vectorspace V over a field K, with homogeneous components Ln(V) forn 1. If G is a group and V is a KG-module, the action of Gextends naturally to L(V), and the Ln(V) become finite-dimensionalKG-modules, called the Lie powers of V. In the decompositionproblem, the aim is to identify the isomorphism types of indecomposableKG-modules, with their multiplicities, in unrefinable directdecompositions of the Lie powers. This paper is concerned withthe case where G has prime order p, and K has characteristicp. As is well known, there are p indecomposables, denoted hereby J1,...,Jp, where Jr has dimension r. A theory is developedwhich provides information about the overall module structureof LV) and gives a recursive method for finding the multiplicitiesof J1,...,Jp in the Lie powers Ln(V). For example, the theoryyields decompositions of L(V) as a direct sum of modules isomorphiceither to J1 or to an infinite sum of the form Jr J{p-1} J{p-1} ... with r 2. Closed formulae are obtained for the multiplicitiesof J1,..., Jp in Ln(Jp and Ln(J{p-1). For r < p-1, the indecomposableswhich occur with non-zero multiplicity in Ln(Jr) are identifiedfor all sufficiently large n. 2000 Mathematical Subject Classification:17B01, 20C20. |
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Keywords: | free Lie algebras cyclic groups modular representations |
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