奇数李表示的plethystic逆
The plethystic inverse of the odd Lie representations
AI总结：
本文证明了Richard Stanley关于奇数李表示之和的plethystic逆的猜想，并获得了正则表示的新的plethystic分解及交错和的逆。
摘要：
自由李代数多重线性分量所赋予的对称群$S_n$的表示$Lie_n$的Frobenius特征标已知满足许多有趣的plethystic恒等式。本文证明了Richard Stanley的一个猜想，该猜想确立了奇数李特征标之和$\sum_{n\geq 0} Lie_{2n+1}$的plethystic逆。我们获得了$S_n$的正则表示的一个看似新的plethystic分解，该分解以由钩子索引的不可约表示和李表示来表达。我们确定了奇数李特征标的交错和的plethystic逆。
英文摘要：
The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the multilinear component of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $\sum_{n\geq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations. We determine the plethystic inverse of the alternating sum of the odd Lie characteristics.
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