THE CONJUGACY ACTION OF Sn AND MODULES INDUCED FROM CENTRALISERS

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 (from work of Frumkin and Heide-Saxl-ThiepZalesski, respectively) to contain every irreducible. We find other representations of dimension n! with this property, including a twisted analogue of the conjugacy action. We establish the positivity of the row sums indexed by irreducible characters of the symmetric group, when restricted to conjugacy classes of partitions with all parts odd. Another result asserts the positivity of all row sums (except for the one indexed by the sign character) when the columns exclude the partitions with distinct odd parts. Our work leads to a new proof that the conjugacy action of the alternating group also contains every irreducible. By considering two natural submodules of the conjugacy action, we obtain generalisations of the corresponding results for the symmetric group.