We investigate the representation of the symmetric group Sn derived from linearizing the action of Sn on the power set, P(Xn), of Xn := {1, . . . , n}, on the power
set of the power set, PP(Xn), of Xn, and, finally, on the set of all topologies on
Xn, Top(Xn). Moreover, we prove that the latter two cases give an algebra faithful
representation of the symmetric group Sn.
We decompose the representation of the symmetric group Sn on CY , into irreducibles, for some particular invariant subsets, Y , of P(Xn) and PP(Xn), in the
case n = 2, 3, 4. In the general case, we show that for some typical invariant subsets,
Y ⊂ PP(Xn) the representation on CY is explicitly a tensor product of representations that already have an explicit decomposition into irreducibles.
We reduce the action of Sn on Top(Xn) to an action on the set of reflexive,
transitive relation on Xn. We use this presentation to find orbits, O ⊂ Top(Xn),
such that CO is an algebra faithful representations, and orbits that are in bijection
with orbits of the action of Sn on the set of Young tabloids.
