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 .

**[BS]**Hélène Barcelo and Sheila Sundaram,*On some submodules of the action of the symmetric group on the free Lie algebra*, J. Algebra**154**(1993), no. 1, 12–26. MR**1201910**, https://doi.org/10.1006/jabr.1993.1002**[B]**Angeline Brandt,*The free Lie ring and Lie representations of the full linear group*, Trans. Amer. Math. Soc.**56**(1944), 528–536. MR**0011305**, https://doi.org/10.1090/S0002-9947-1944-0011305-0**[CHR]**A. R. Calderbank, P. Hanlon, and R. W. Robinson,*Partitions into even and odd block size and some unusual characters of the symmetric groups*, Proc. London Math. Soc. (3)**53**(1986), no. 2, 288–320. MR**850222**, https://doi.org/10.1112/plms/s3-53.2.288**[G]**Adriano M. Garsia,*Combinatorics of the free Lie algebra and the symmetric group*, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 309–382. MR**1039352****[GR]**A. M. Garsia and C. Reutenauer,*A decomposition of Solomon’s descent algebra*, Adv. Math.**77**(1989), no. 2, 189–262. MR**1020585**, https://doi.org/10.1016/0001-8708(89)90020-0**[J]**Nathan Jacobson,*Lie algebras*, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR**559927****[M]**I. G. Macdonald,*Symmetric functions and Hall polynomials*, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR**553598****[R]**Christophe Reutenauer,*Theorem of Poincaré-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers*, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) Lecture Notes in Math., vol. 1234, Springer, Berlin, 1986, pp. 267–284. MR**927769**, https://doi.org/10.1007/BFb0072520**[Ro]**David P. Robbins,*Jordan elements in a free associative algebra. I*, J. Algebra**19**(1971), 354–378. MR**0281762**, https://doi.org/10.1016/0021-8693(71)90095-0**[So]**Louis Solomon,*A Mackey formula in the group ring of a Coxeter group*, J. Algebra**41**(1976), no. 2, 255–264. MR**0444756**, https://doi.org/10.1016/0021-8693(76)90182-4**[Su]**Sheila Sundaram,*Decompositions of 𝑆_{𝑛}-submodules in the free Lie algebra*, J. Algebra**154**(1993), no. 2, 507–558. MR**1206134**, https://doi.org/10.1006/jabr.1993.1027**[T]**R. M. Thrall,*On symmetrized Kronecker powers and the structure of the free Lie ring*, Amer. J. Math.**64**(1942), 371–388. MR**0006149**, https://doi.org/10.2307/2371691

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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