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Prime power variations of higher Lien modules
Journal of Combinatorial Theory Series A ( IF 0.9 ) Pub Date : 2021-07-27 , DOI: 10.1016/j.jcta.2021.105512
Sheila Sundaram 1
Affiliation  

We define, for each subset S of the set P of primes, an Sn-module LienS with interesting properties. Lien is the well-known representation Lien of Sn afforded by the free Lie algebra, while LienP is the module Conjn of the conjugacy action of Sn on n-cycles. For arbitrary S the module LienS interpolates between the representations Lien and Conjn. We consider the symmetric and exterior powers of LienS. These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher LienS 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 fnT of homogeneous degree n. We show that the series λ,λiTpλ can be expressed as symmetrised powers of the functions fnT, 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 fnT, the conjugacy action and the Lie representation.



中文翻译:

更高留置权模块的主要功率变化

我们定义,每个子Ş 素数,一个 n-模块 一世电子n 具有有趣的特性。 一世电子n 是众所周知的表示 一世电子nn 由自由李代数提供,而 一世电子n 是模块 Cnjn 的共轭作用 nn 次循环上。对于任意S模块一世电子n 在表示之间进行插值 一世电子nCnjn. 我们考虑对称的和外部的权力一世电子n. 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 特征一世电子n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Fn齐次n。我们证明该系列λ,λ一世λ 可以表示为函数的对称幂 Fn,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Fn,共轭作用和谎言表示。

更新日期:2021-07-27
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