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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
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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$.

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