Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 1153011
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider, for a given complex parameter $ \alpha $, the nonassociative product defined on the tensor algebra of $ n$-dimensional complex vector space by $ [x,y] = x \otimes y - \alpha y \otimes x$. For $ k$ symbols $ {x_1}, \ldots ,{x_k}$, the left-normed bracketing is defined recursively to be the bracketing sequence $ {b_k}$, where $ {b_1} = {x_1}$, $ {b_2} = [{x_1},{x_2}]$, and $ {b_k} = [{b_{k - 1}},{x_k}]$. The linear subspace spanned by all multilinear left-normed bracketings of homogeneous degree $ n$, in the basis vectors $ {v_1}, \ldots ,{v_n}$ of $ {\mathbb{C}^n}$, is then an $ {S_n}$-module $ {V_n}(\alpha )$. Note that $ {V_n}(1)$ is the Lie representation $ \operatorname{Lie}_n$ of $ {S_n}$ afforded by the $ n$th-degree multilinear component of the free Lie algebra. Also, $ {V_n}(- 1)$ 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 $ \alpha $ is not a root of unity, the module $ {V_n}(\alpha )$ is simply the regular representation.

Thrall [T] showed that the regular representation of the symmetric group $ {S_n}$ can be written as a direct sum of tensor products of symmetrised Lie modules $ {V_\lambda }$. In this paper we determine the structure of the representations $ {V_n}(\alpha )$ as a sum of a subset of these $ {V_\lambda }$. The $ {V_\lambda }$, indexed by the partitions $ \lambda $ of $ n$, are defined as follows: let $ {m_i}$ be the multiplicity of the part $ i$ in $ \lambda $, let $ \operatorname{Lie}_i$ be the Lie representation of $ {S_i}$, and let $ {\iota _k}$ denote the trivial character of the symmetric group $ {S_k}$. Let $ {\iota _{{m_i}}}[\operatorname{Lie}_i]$ denote the character of the wreath product $ {S_{{m_i}}}[{S_i}]$ of $ {S_{{m_i}}}$ acting on $ {m_i}$ copies of $ {S_i}$. Then $ {V_\lambda }$ is isomorphic to the $ {S_n}$-module

$\displaystyle ({\iota _{{m_1}}}[\operatorname{Lie}_1] \otimes \cdots \otimes {\... ...S_{m_1}}[{S_1}] \times \cdots \times {S_{{m_i}}}[{S_i}] \times \cdots }^{S_n}}.$

Our theorem now states that when $ \alpha $ is a primitive $ p$th root of unity, the $ {S_n}$-module $ {V_n}(\alpha )$ is isomorphic to the direct sum of those $ {V_\lambda }$, where $ \lambda $ runs over all partitions $ \lambda $ of $ n$ such that no part of $ \lambda $ is a multiple of $ p$.

References [Enhancements On Off] (What's this?)

  • [BS] H. Barcelo and S. Sundaram, On some submodules of the action of the symmetric group on the free Lie algebra, J. Algebra 154 (1993), 12-26. MR 1201910 (93j:20028)
  • [B] A. J. Brandt, The free Lie ring and Lie representations of the full linear group, Trans. Amer. Math. Soc. 56 (1944), 528-536. MR 0011305 (6:146d)
  • [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), 288-320. MR 850222 (87m:20042)
  • [G] A. M. Garsia, Combinatorics of the free Lie algebra and the symmetric group, Analysis: Research Papers Published in Honour of Jürgen Moser's 60th Birthday (P. H. Rabinowitz and E. Zehnder, eds.), Academic Press, San Diego, California, 1990, pp. 309-382. MR 1039352 (91a:17006)
  • [GR] A. M. Garsia and C. Reutenauer, A decomposition of Solomon's descent algebra, Adv. Math. 77 (1989), 189-262. MR 1020585 (91c:20007)
  • [J] N. Jacobson, Lie algebras, Dover, New York, 1979. MR 559927 (80k:17001)
  • [M] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Univ. Press, New York, 1979. MR 553598 (84g:05003)
  • [R] C. Reutenauer, Theorem of Poincaré-Birkhoff-Witt and symmetric group representations of degrees equal to Stirling numbers, Lecture Notes in Math., vol. 1234, Springer-Verlag, New York and Berlin, 1986. MR 927769 (89i:05029)
  • [Ro] D. P. Robbins, Jordan elements in the free associative algebra. I, J. Algebra 19 (1971), 354-378. MR 0281762 (43:7477)
  • [So] L. Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), 255-264. MR 0444756 (56:3104)
  • [Su] S. Sundaram, Decompositions of $ {S_n}$-submodules in the free Lie algebra, J. Algebra (1990) (to appear). MR 1206134 (93m:20020)
  • [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 (3:262d)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20C30, 05E10

Retrieve articles in all journals with MSC: 20C30, 05E10

Additional Information

Keywords: Deformations, free Lie algebra, representation, Poincaré-Birkhoff-Witt theorem
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society