We define, for each subset S of the set $\mathcal{P}$ of primes, an $S_n$-module $Li{e}_n^S$ with interesting properties. $Li{e}_n^{\varnothing }$ is the well-known representation $Li{e}_n$ of $S_n$ afforded by the free Lie algebra, while $Li{e}_n^{\mathcal{P}}$ is the module $Con{j}_n$ of the conjugacy action of $S_n$ on n-cycles. For arbitrary S the module $Li{e}_n^S$ interpolates between the representations $Li{e}_n$ and $Con{j}_n$. We consider the symmetric and exterior powers of $Li{e}_n^S$. These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher $Li{e}_n^S$ modules can be elegantly expressed as a multiplicity-free sum of power sums. In particular this establishes the Schur positivity of new classes of sums of power sums.

More generally, for each nonempty subset T of positive integers we define a sequence of symmetric functions $f_n^T$ of homogeneous degree n. We show that the series ${\sum }_{\lambda ,{\lambda }_i\in T}{p}_{\lambda }$ can be expressed as symmetrised powers of the functions $f_n^T$, analogous to the higher Lie modules first defined by Thrall. This in turn allows us to unify previous results on the Schur positivity of multiplicity-free sums of power sums, as well as investigate new ones. We also uncover some curious plethystic relationships between $f_n^T$, the conjugacy action and the Lie representation.

我们定义，每个子Ş集$\mathcal{磷}$ 素数，一个 ${秒}_n$-模块 $升\mathrm{一世}{\mathrm{电子}}_n^{秒}$ 具有有趣的特性。 $升\mathrm{一世}{\mathrm{电子}}_n^{\varnothing }$ 是众所周知的表示 $升\mathrm{一世}{\mathrm{电子}}_n$ 的 ${秒}_n$ 由自由李代数提供，而 $升\mathrm{一世}{\mathrm{电子}}_n^{\mathcal{磷}}$ 是模块 $C○n{j}_n$ 的共轭作用 ${秒}_n$在n 次循环上。对于任意S模块$升\mathrm{一世}{\mathrm{电子}}_n^{秒}$ 在表示之间进行插值 $升\mathrm{一世}{\mathrm{电子}}_n$ 和 $C○n{j}_n$. 我们考虑对称的和外部的权力$升\mathrm{一世}{\mathrm{电子}}_n^{秒}$. 这些是萨尔的高级谎言模块的类似物。我们证明了这些较高的 Frobenius 特征$升\mathrm{一世}{\mathrm{电子}}_n^{秒}$模块可以优雅地表示为幂和的无多重和。特别是这建立了新类别的幂和的舒尔正性。

更一般地，对于正整数的每个非空子集T，我们定义一个对称函数序列$F_n^{吨}$齐次n。我们证明该系列${\sum }_{\lambda ,{\lambda }_{\mathrm{一世}}\in 吨}{磷}_{\lambda }$ 可以表示为函数的对称幂 $F_n^{吨}$，类似于由 Thrall 首先定义的更高的 Lie 模块。这反过来又使我们能够统一先前关于无多重幂和之和的 Schur 正性的结果，并研究新的结果。我们还发现了一些奇怪的丰满关系$F_n^{吨}$，共轭作用和谎言表示。