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), 315-333

MSC:
Primary 20C30; Secondary 05E10

DOI:
https://doi.org/10.1090/S0002-9947-1994-1153011-7

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 left-normed bracketing is defined recursively to be the bracketing sequence , where , , and . The linear subspace spanned by all multilinear left-normed bracketings of homogeneous degree , in the basis vectors of , is then an -module . Note that is the Lie representation of afforded by the th-degree 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1153011-7

Keywords:
Deformations,
free Lie algebra,
representation,
Poincaré-Birkhoff-Witt theorem

Article copyright:
© Copyright 1994
American Mathematical Society