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Journal of Algebraic Combinatorics

, Volume 48, Issue 2, pp 179–225 | Cite as

The conjugacy action of \(S_n\) and modules induced from centralisers

  • Sheila SundaramEmail author
Article

Abstract

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.

Keywords

Conjugacy action Character tables Symmetric power Exterior power Plethysm Ramanujan sum 

Mathematics Subject Classification

20C05 20C15 20C30 05E18 06A07 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Pierrepont SchoolWestportUSA

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