Prime power variations of higher $Lie_n$ modules
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866918136888950784 |
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| author | Sundaram, Sheila |
| author_facet | Sundaram, Sheila |
| contents | We define, for each subset $S$ of the set $\mathcal{P}$ of primes, an $S_n$-module $Lie_n^S$ with interesting properties. $Lie_n^\emptyset$ is the well-known representation $Lie_n$ of $S_n$ afforded by the free Lie algebra, while $Lie_n^\mathcal{P}$ is the module $C\!onj_n$ of the conjugacy action of $S_n$ on $n$-cycles. For arbitrary $S$ the module $Lie_n^{S}$ interpolates between the representations $Lie_n$ and $C\!onj_n.$ We consider the symmetric and exterior powers of $Lie_n^S.$ These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher $Lie_n^S$ 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 $\sum_{λ, λ_i\in T} p_λ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_06389 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Prime power variations of higher $Lie_n$ modules Sundaram, Sheila Representation Theory Combinatorics 05E05 (primary) 20C30 (secondary) We define, for each subset $S$ of the set $\mathcal{P}$ of primes, an $S_n$-module $Lie_n^S$ with interesting properties. $Lie_n^\emptyset$ is the well-known representation $Lie_n$ of $S_n$ afforded by the free Lie algebra, while $Lie_n^\mathcal{P}$ is the module $C\!onj_n$ of the conjugacy action of $S_n$ on $n$-cycles. For arbitrary $S$ the module $Lie_n^{S}$ interpolates between the representations $Lie_n$ and $C\!onj_n.$ We consider the symmetric and exterior powers of $Lie_n^S.$ These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher $Lie_n^S$ 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 $\sum_{λ, λ_i\in T} p_λ$ 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. |
| title | Prime power variations of higher $Lie_n$ modules |
| topic | Representation Theory Combinatorics 05E05 (primary) 20C30 (secondary) |
| url | https://arxiv.org/abs/2107.06389 |