Representations of the symmetric group in deformations of the free Lie algebra
Authors:
A. R. Calderbank, P. Hanlon and S. Sundaram
Journal:
Trans. Amer. Math. Soc. 341 (1994), 315333
MSC:
Primary 20C30; Secondary 05E10
MathSciNet review:
1153011
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Abstract: We consider, for a given complex parameter , the nonassociative product defined on the tensor algebra of dimensional complex vector space by . For symbols , the leftnormed bracketing is defined recursively to be the bracketing sequence , where , , and . The linear subspace spanned by all multilinear leftnormed bracketings of homogeneous degree , in the basis vectors of , is then an module . Note that is the Lie representation of afforded by the thdegree multilinear component of the free Lie algebra. Also, is the subspace of simple Jordan products in the free associative algebra as studied by Robbins [Ro]. Among our preliminary results is the observation that when is not a root of unity, the module is simply the regular representation. Thrall [T] showed that the regular representation of the symmetric group can be written as a direct sum of tensor products of symmetrised Lie modules . In this paper we determine the structure of the representations as a sum of a subset of these . The , indexed by the partitions of , are defined as follows: let be the multiplicity of the part in , let be the Lie representation of , and let denote the trivial character of the symmetric group . Let denote the character of the wreath product of acting on copies of . Then is isomorphic to the module Our theorem now states that when is a primitive th root of unity, the module is isomorphic to the direct sum of those , where runs over all partitions of such that no part of is a multiple of .
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 [T]
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199411530117
PII:
S 00029947(1994)11530117
Keywords:
Deformations,
free Lie algebra,
representation,
PoincaréBirkhoffWitt theorem
Article copyright:
© Copyright 1994
American Mathematical Society
