The conjugacy action of $S_n$ and modules induced from centralisers
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2016
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| _version_ | 1866909774889615360 |
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| author | Sundaram, Sheila |
| author_facet | Sundaram, Sheila |
| contents | We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all parts odd. More generally, we study representations related to the conjugacy action of the symmetric group. These arise as sums of submodules induced from centraliser subgroups, and their Frobenius characteristics have elegant descriptions, often as a multiplicity-free sum of power-sum symmetric functions. We describe a general framework in which such representations, and consequently such linear combinations of power-sums, can be analysed. The conjugacy action for the symmetric group, and more generally for a large class of groups, is known to contain every irreducible. We find other representations of dimension $n!$ with this property, including a twisted analogue of the conjugacy action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1603_05898 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | The conjugacy action of $S_n$ and modules induced from centralisers Sundaram, Sheila Representation Theory Combinatorics Group Theory 20C05, 20C15, 20C30, 05E18, 06A07 We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all parts odd. More generally, we study representations related to the conjugacy action of the symmetric group. These arise as sums of submodules induced from centraliser subgroups, and their Frobenius characteristics have elegant descriptions, often as a multiplicity-free sum of power-sum symmetric functions. We describe a general framework in which such representations, and consequently such linear combinations of power-sums, can be analysed. The conjugacy action for the symmetric group, and more generally for a large class of groups, is known to contain every irreducible. We find other representations of dimension $n!$ with this property, including a twisted analogue of the conjugacy action. |
| title | The conjugacy action of $S_n$ and modules induced from centralisers |
| topic | Representation Theory Combinatorics Group Theory 20C05, 20C15, 20C30, 05E18, 06A07 |
| url | https://arxiv.org/abs/1603.05898 |