Representations of the symmetric group in deformations of the free Lie algebra

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