Lie powers and Witt vectors
R.M. Bryant
and Marianne Johnson
cdot M. Johnson School of Mathematics, University of Manchester, Manchester M13 9PL, UK
DOI: 10.1007/s10801-007-0117-9
Abstract
In the study of Lie powers of a module V in prime characteristic p, a basic role is played by certain modules B n introduced by Bryant and Schocker. The isomorphism types of the B n are not fully understood, but these modules fall into infinite families { B k, B pk, B p 2 k,... } \{B_{k},B_{pk},B_{p^{2}k},\dots \} , one family B( k) for each positive integer k not divisible by p, and there is a recursive formula for the modules within B( k). Here we use combinatorial methods and Witt vectors to show that each module in B( k) is isomorphic to a direct sum of tensor products of direct summands of the kth tensor power V \otimes k .
Pages: 169–187
Keywords: keywords free Lie algebra; Lie power; tensor power; Witt vector
Full Text: PDF
References
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2. Bryant, R.M.: Free Lie algebras and Adams operations. J. Lond. Math. Soc. (2) 68, 355-370 (2003)
3. Bryant, R.M., Schocker, M.: The decomposition of Lie powers. Proc. Lond. Math. Soc. (3) 93, 175- 196 (2006)
4. Bryant, R.M., Schocker, M.: Factorisation of Lie resolvents. J. Pure Appl. Algebra 208, 993-1002 (2007)
5. Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Wiley- Interscience, New York (1962)
6. Donkin, S., Erdmann, K.: Tilting modules, symmetric functions, and the module structure of the free Lie algebra. J. Algebra 203, 69-90 (1998)
7. Friedlander, E.M., Suslin, A.: Cohomology of finite group schemes over a field. Invent. Math. 127, 209-270 (1997)
8. Green, J.A.: Polynomial Representations of GLn. Lecture Notes in Mathematics, vol.
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9. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Clarendon, Oxford (1938)
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11. Kraśkiewicz, W., Weyman, J.: Algebra of coinvariants and the action of a Coxeter element. Bayreuth. Math. Schr. 63, 265-284 (2001)
12. Reutenauer, C.: Free Lie Algebras. Clarendon, Oxford (1993)
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14. Serre, J.-P.: Local Fields. Springer, New York (1979)
15. Witt, E.: Zyklische Körper und Algebren der Charakteristik p vom Grade pn. J. Reine Angew. Math.